Operations on constructible functions and generalized valuations

OPERA TIONS ON CONSTR UCTIBLE FUNCTIONS AND GENERALIZED V ALUA TIONS ANDREAS BERNIG AND V ADIM LEBOVICI Abstract. Alesk er’s theory of generalized v aluations uniﬁes smooth measures and constructible functions on real analytic manifolds, extend- ing classical op erations on functions and measures. Alesk er show ed that these op erations agree with the sheaf-theoretic ones on constructible functions under restrictiv e assumptions, leaving key asp ects conjectural. In this article, we close this gap by pro ving that the t wo approaches indeed coincide on constructible functions under mild transv ersality as- sumptions. Our pro of is based on a comparison with the corresp onding op erations on characteristic cycles. As applications, we extend additive kinematic form ulas from conv ex bo dies to compact subanalytic sets in Euclidean spaces and derive new kinematic formulas on the 3-sphere. Contents 1. In tro duction 1 2. Preliminaries 7 3. W av e fron t set of subanalytic currents 18 4. Exterior pro duct 19 5. Pullbac k and pro duct 23 6. Pushforw ard and conv olution 28 7. Additiv e and multiplicativ e kinematic formulas 33 References 45 1. Introduction 1.1. Con text. A v aluation is a ﬁnitely additive measure deﬁned on suﬃ- cien tly regular subsets of some ﬁxed ambien t space. The theory of v aluations originates from Dehn’s solution to Hilb ert’s third problem on scissors con- gruence of p olytopes [ 31 ] and was ﬁrst formalized and systematically studied b y Hadwiger [ 41 ] for compact con v ex subsets of Euclidean spaces. In a series of pap ers, Alesk er [ 3 , 4 , 6 , 13 ] in tro duced a theory of v aluations on man- ifolds as a generalization of the classical theory of v aluations from con vex geometry . These smo oth v aluations encompass classical to ols from integral geometry , such as smo oth measures, the Euler characteristic, and intrinsic VL was supp orted in part b y FSMP and in part by EPSRC EP/R018472/1. AB w as supp orted by DFG grant BE 2484/10-1. 1 2 ANDREAS BERNIG AND V ADIM LEBOVICI v olumes. F ormally , they can b e described by a pair of diﬀeren tial forms, an n -form on the manifold and an ( n − 1)-form on its cosphere bundle [ 15 ]. An imp ortan t breakthrough b y Alesker and F u [ 13 ] is the discov ery of a pro duct structure on the space of smo oth v aluations which, combined with an integration functional introduced in [ 6 ], yields a p erfect bilinear pair- ing. This so-called Alesker-Poinc ar ´ e duality implies in particular that the space V ∞ ( X ) of smo oth v aluations on a smo oth manifold X is densely em- b edded in to the dual V −∞ ( X ) of the space of compactly supp orted smo oth v aluations, so that elements of this dual space can legitimately b e called gener alize d valuations [ 6 ]. It has b een sho wn in [ 16 ] that a generalized v al- uation can b e describ ed as a pair of curren ts, an n -current on the manifold and an ( n − 1)-current on its cosphere bundle. Although the most general family of sets on whic h a smo oth v aluation can b e ev aluated is not yet clear, Alesk er prov ed in [ 6 ] that it con tains the class of subanalytic sets when the manifold X is real analytic. Subanalytic sets ha v e a tame b eha viour: they can b e stratiﬁed (ev en satisfying Whitney’s conditions), they hav e lo cally ﬁnitely man y connected comp onen ts and so on, see [ 42 , 46 ]. Alesk er has constructed an embedding with sequen tially dense image [ 9 ]: [ − ] : CF ( X ) ⊗ C  → V −∞ ( X ) , where CF ( X ) is the group of c onstructible functions on X , that is, the group of functions ϕ : X → Z suc h that the sets ϕ − 1 ( m ) are subanalytic for all m ∈ Z and the family { ϕ − 1 ( m ) } m ∈ Z is lo cally ﬁnite. This em b edding is constructed using the notion of char acteristic cycle [ 34 , 46 ] of a constructible function, whic h is an n -current on the cotangen t bundle of the manifold that can easily b e turned into a pair of currents deﬁning a generalized v aluation. The groups of characteristic cycles and of constructible functions on a real analytic manifold are b oth isomorphic to the Grothendiec k group of constructible shea v es on this manifold, as sho wn b y Kashiw ara [ 45 , 46 ]. Moreo v er, Schapira has sho wn in [ 58 ] that this group isomorphism trans- lates fundamental op erations on sheav es (exterior pro duct, pullback, push- forw ard, Poincar ´ e-V erdier duality) into rich top ological op erations on con- structible functions. Notably , the pushforward of a constructible function b y a real analytic map integrates the constructible function on the ﬁb er of the map with resp ect to the Euler c haracteristic. These op erations found remark able applications to v arious areas of mathematics, from the general- ization of Akbulut-King’s num b er for real semi-algebraic sets in algebraic geometry [ 47 ], to the introduction of top ological integral transforms satisfy- ing p o werful injectivit y theorems [ 30 , 36 , 51 , 57 ] used for shap e description in applied geometry [ 29 ]. Alesk er deﬁnes P oincar ´ e-V erdier dualit y for smo oth and generalized v al- uations in [ 6 ] and the exterior pro duct, the pullback and the pushforw ard in a follo w-up paper [ 7 ]. These op erations generalize w ell-kno wn op erations from measure theory , such as the classical pushforw ard of smooth measures. OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 3 Moreo v er, these op erations can b e combined to induce ric h algebraic struc- tures on generalized v aluations. Namely , the pro duct of generalized v alu- ations [ 11 ] can b e deﬁned as the pullback of the exterior pro duct b y the diagonal embedding, and the conv olution [ 12 ] (sa y , on a Euclidean space) is deﬁned as the pushforward of the exterior pro duct b y the addition. F u and the ﬁrst author hav e sho wn in [ 19 ] that these operations are dual to the celebrated kinematic form ulas in in tegral geometry [ 28 ], and this result w as k ey to establishing explicit kinematic form ulas on complex spaces in [ 20 , 21 ]. As the structure of this in tro duction suggests, the op erations on gen- eralized v aluations were developed with the exp ectation that they w ould also generalize the op erations on constructible functions coming from sheaf theory . Alesk er prov es this fact for Poincar ´ e-V erdier dualit y , the exterior pro duct, and for the pullback and the pushforw ard under rather restrictive assumptions detailed b elo w [ 7 ]. 1.2. Con tributions. In this article, w e sho w that the op erations on gener- alized v aluations restrict on constructible functions to the operations coming from sheaf theory under mild transversalit y assumptions. W e state our re- sults b elo w, p ostp oning the precise deﬁnitions to relev an t sections. Since the generalized v aluation asso ciated with a constructible function is deﬁned using c haracteristic cycles, proving our result reduces to translating the op erations on characteristic cycles—viewed as Borel-Mo ore homology classes and pro vided in the v o cabulary of microlo cal sheaf theory in [ 46 ]— in to classical op erations from geometric measure theory on the currents rep- resen ting generalized v aluations. T o do so, we use a description of cycle op erations due to Schmid and Vilonen [ 59 ] in terms of classical op erations on Borel-Mo ore homology classes. The k ey is then that c haracteristic cycles can b e naturally considered as Borel-Mo ore homology classes or as curren ts, and that classical op erations on homology classes translate in to classical op erations on currents under mild transv ersality assumptions. Pul lb ack and pr o duct. Our ﬁrst result concerns the pullback f ∗ of generalized v aluations b y a real analytic map f , whic h is deﬁned for submersions and for immersions satisfying some transversalit y assumptions [ 7 ]. On constructible functions, the pullbac k is nothing but the precomp osition. W e pro ve our re- sult under a transv ersalit y assumption on the constructible function whic h implies that the pullback is well-deﬁned in the sense of generalized v alua- tions. Theorem 1. L et f : X → Y b e a r e al analytic map b etwe en r e al analytic manifolds and let ψ ∈ CF ( Y ) . If (i) f is a submersion, or (ii) f is an immersion which is tr ansverse to the str ata of a Whitney str atiﬁc ation of Y on which ψ is c onstant, then the pul lb ack f ∗ [ ψ ] ∈ V −∞ ( X ) is wel l-deﬁne d and f ∗ [ ψ ] = [ f ∗ ψ ] . 4 ANDREAS BERNIG AND V ADIM LEBOVICI This result is pro ven in [ 7 , Props. 3.3.4 and 3.5.12] in the setting of indi- cator functions of compact submanifolds with corners. While submanifolds with corners may not b e subanalytic, they satisfy regularity prop erties that arbitrary subanalytic subsets do not. Sp eciﬁcally , Alesk er’s proof of Theo- rem 1 in the case of submanifolds with corners crucially relies on the lo cal trivialit y of transverse intersections of submanifolds with corners. This prop- ert y is simply wrong for subanalytic sets, as shown b y the existence of ana- lytic families of non-diﬀeomorphic singularities; see [ 37 , Chap. I I, Ex. 2.1]. Our proof using Schmid and Vilonen’s description of the operations on c har- acteristic cycles circum ven ts this obstacle. Com bined with Alesker’s result on the restriction of the exterior pro d- uct [ 7 , Claim. 2.1.11], our result on the pullback implies a similar result for the restriction of the pro duct of generalized v aluations deﬁned in [ 11 ]. This result holds for constructible functions which are transv erse in the sense that there are tw o Whitney stratiﬁcations of the manifold whose strata are pairwise transverse and on which the constructible functions are resp ectiv ely constan t; see Section 5.3 . Corollary 2. If ϕ, ψ ∈ CF ( X ) ar e tr ansverse, then [ ϕ ] · [ ψ ] = [ ϕ · ψ ] . This result relates the pro duct on generalized v aluations asso ciated to subanalytic sets and the intersection of such sets. It is an analogue of the result known for submanifolds with corners [ 11 , Thm. 5] whic h pro of rested, as for the pullbac k, on the local triviality of transverse in tersections of such submanifolds. Pushforwar d and c onvolution. The pushforw ard f ∗ of generalized v aluations b y a real analytic map f is well-deﬁned when f is an immersion or when f is a submersion which is proper on the supp ort of ϕ and whose diﬀerential satisﬁes a transversalit y assumption with resp ect to the wa ve front set of the normal cycle of ϕ ; see [ 7 ]. T o pro v e our result on the pushforward, we ﬁrst need to assume transversalit y of the diﬀerential of f to the strata deﬁn- ing the normal cycle. How ev er, on the contrary to the pullbac k, this alone will not ensure that the pushforward is well-deﬁned in the sense of general- ized v aluations. As a consequence, we also require the stratiﬁcation of the normal cycle to b e angular , that is, to satisfy that the wa ve fron t set of the normal cycle is contained in the union of the conormal bundles to the strata; see Theorem 3.3 and Theorem 6.2 . W e show that angular stratiﬁcations of subanalytic currents alw a ys exist in Section 3 using desingularization. Theorem 3. L et f : X → Y b e a r e al analytic map b etwe en r e al analytic manifolds and ϕ ∈ CF ( X ) . If (i) f is an immersion, or (ii) f is a submersion which is pr op er on the supp ort of ϕ and whose diﬀer ential is tr ansverse to the str ata of an angular Whitney str ati- ﬁc ation deﬁning the normal cycle of ϕ , then f ∗ ϕ ∈ CF ( Y ) and f ∗ [ ϕ ] ∈ V −∞ ( Y ) ar e wel l-deﬁne d and f ∗ [ ϕ ] = [ f ∗ ϕ ] . OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 5 T o prov e our result, w e ﬁrst pro vide a correction to a form ula of [ 7 ] for the expression of the pushforw ard of generalized v aluations by immersions. In this case, the pushforward is an extension by zero and the pro of do es not presen t any serious diﬃculty . The interesting case is the case of submersions, where the pushforw ard of constructible functions is a non-trivial operation of in tegration along the ﬁbers with resp ect to the Euler characteristic. In [ 7 ], Alesk er prov es Theorem 3 under the restrictive assumption that the map f is a linear pro jection betw een Euclidean spaces and that the function ϕ is an indicator function of a compact conv ex subset with smo oth b oundary of strictly p ositiv e Gaussian curv ature. The pro of is based on an approximation pro cess which crucially relies on the restrictive assumptions on the subset. In contrast, our use of Schmid and Vilonen’s description of op erations on c haracteristic cycles allows us to bypass this obstacle once again. T reating the pushforward yields a similar result for the con v olution of generalized v aluations deﬁned in [ 12 ]. Recall that the theory of Lie groups (see for instance [ 62 , P art I I, Chap. IV, Sec. 5]) ensures that if G is a Lie group acting transitively on a smo oth manifold X , then G and X are naturally endow ed with unique real analytic structures and the action of G on X is real analytic. In such a situation, one can deﬁne the conv olution −∗ − (of constructible functions or of generalized v aluations) as the pushforw ard of the exterior pro duct by the action G × X → X . Note that the action is a submersion by transitivity . Combined with Alesk er’s result on the exterior pro duct, Theorem 3 implies: Corollary 4. L et G b e a Lie gr oup acting tr ansitively on a smo oth mani- fold X . F or ϕ ∈ CF( G ) and ψ ∈ CF( X ) , we have: [ ϕ ] ∗ [ ψ ] = [ ϕ ∗ ψ ] , whenever the action a : G × X → X and ϕ ⊠ ψ satisfy assumption (ii) of The or em 3 . This result relates the con volution of generalized v aluations with the con- v olution of constructible functions, which seems to b e the correct replace- men t for the notion of Minko wski sum. A dditive and multiplic ative kinematic formulas. Let V be an n -dimensional Euclidean v ector space and let G b e a closed subgroup of O( n ) that acts transitiv ely on the unit sphere. W e endow G with the Haar probabilit y measure and G := G ⋉ V with the pro duct of Haar probability measure and Leb esgue measure. The space of contin uous, translation- and G -inv ariant v aluations on con- v ex b o dies, to b e denoted by V al G , is ﬁnite-dimensional [ 2 ]. W e let µ 1 , . . . , µ N b e a basis of this space. Then, for an y con v ex bo dies K, L ⊂ R n there are additiv e kinematic formulas: Z G µ i ( K + g L ) dg = X k,l c i k,l µ k ( K ) µ l ( L ) , 6 ANDREAS BERNIG AND V ADIM LEBOVICI with constants c i k,l that can be determined (at least in some important c ases) b y either the template method or using algebraic in tegral geometry . It turns out that under our assumption on G , elemen ts of V al G are smo oth v alua- tions, and they can be ev aluated at compactly supp orted constructible func- tions. As we argued ab o v e, the correct replacement for the Mink owski sum of conv ex b o dies is the con volution pro duct of constructible functions. More- o v er, w e can use our result on the conv olution to pro v e additiv e kinematic form ulas for constructible functions: Theorem 5. If ϕ 1 , ϕ 2 ∈ CF( V ) ar e c omp actly supp orte d, then with the same c onstants c i k,l as ab ove we have: Z G µ i ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = X k,l c i k,l µ k ( ϕ 1 ) µ l ( ϕ 2 ) . The pro of of this theorem uses a map from the space of compactly sup- p orted generalized v aluations to translation-in v ariant generalized v aluations, that was considered earlier in the sp ecial cases of smo oth v aluations [ 12 ] as w ell as p olytopes [ 17 ]. One diﬃcult y is to pro ve that for almost all g ∈ G , the generalized v aluations asso ciated to ϕ 1 and g ∗ ϕ 2 are transversal, so that their conv olution pro duct exists. W e note that it may b e p ossible to prov e the theorem with a more clas- sical approac h, namely using a Hadwiger t yp e characterization of inv arian t v aluations. Such a Hadwiger t yp e theorem is indeed claimed in the con text of constructible functions for G = SO( n ) [ 14 , Lemma 12], but the notion of c onormal c ontinuity ma y b e hard to chec k for the kinematic integral. More- o v er, our approach has the adv an tage to w ork for other transitive groups, and also on the Lie groups S 3 and SO(3). These tw o groups are the only compact connected Lie groups of dimension ≥ 2 suc h that the space of bi-in v ariant smo oth v aluations is ﬁnite-dimensional [ 18 , Thm. B]. W e state the form ulas for S 3 , the case of SO(3) b eing similar. A natu- ral basis of the space of SO(4)-in v arian t smo oth v aluations consists of the Crofton v aluations ν i deﬁned for i ∈ { 0 , 1 , 2 , 3 } by ν i ( X ) = Z Geod 3 − i ( S 3 ) χ ( X ∩ E ) dE , where Geo d 3 − i ( S 3 ) is the manifold of totally geo desic submanifolds of S 3 of dimension (3 − i ), endow ed with the SO(4)-in v ariant probabilit y measure. Theorem 6. Ther e ar e c onstants d i k,l such that for ϕ 1 , ϕ 2 ∈ CF ( S 3 ) the fol lowing multiplic ative kinematic formulas hold: Z SO(4) ν i ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = X k,l d i k,l ν k ( ϕ 1 ) ν l ( ϕ 2 ) . OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 7 Setting m ( ν i ) = P d i k,l ν k ⊗ ν l , we obtain the fol lowing explicit formulas m ( ν 0 ) = ν 0 ⊗ ν 0 , m ( ν 1 ) = ν 0 ⊗ ν 1 + ν 1 ⊗ ν 0 − ( ν 1 ⊗ ν 2 + ν 2 ⊗ ν 1 ) + 2( ν 2 ⊗ ν 3 + ν 3 ⊗ ν 2 ) , m ( ν 2 ) = ν 0 ⊗ ν 2 + ν 2 ⊗ ν 0 + π 2 8 ν 1 ⊗ ν 1 − π 2 4 ( ν 1 ⊗ ν 3 + ν 3 ⊗ ν 1 ) − 2 ν 2 ⊗ ν 2 + π 2 2 ν 3 ⊗ ν 3 , m ( ν 3 ) = ν 0 ⊗ ν 3 + ν 3 ⊗ ν 0 + 1 2 ( ν 1 ⊗ ν 2 + ν 2 ⊗ ν 1 ) − ( ν 2 ⊗ ν 3 + ν 3 ⊗ ν 2 ) . 1.3. Plan of the pap er. Section 2 contains our notations and some pre- liminaries on Borel-Mo ore homology , curren ts, constructible functions, c har- acteristic cycles and generalized v aluations. In Section 3 , w e deﬁne angular stratiﬁcations of subanalytic currents and pro ve their existence. F or the sake of completeness, w e prov e in Section 4 that the exterior pro duct of general- ized v aluations restricts to the exterior pro duct of constructible functions [ 7 , Claim 2.1.11]. In Section 5 , w e recall the deﬁnition of the pullback of gen- eralized v aluations and pro v e Theorem 1 and Theorem 2 . In Section 6 , w e recall the deﬁnition of the pushforward of generalized v aluations and pro v e Theorem 3 and Theorem 4 . In Section 7 , w e prov e additive kine- matic form ulas for constructible functions on ﬂat spaces and multiplicativ e kinematic formulas for constructible functions on S 3 . Ac kno wledgemen ts. The ﬁrst named author wishes to thank the IHES for the kind and pro ductiv e atmosphere during his researc h stay in 2024, where large parts of this w ork ha ve b een work ed out. He also w ants to thank Dan Abramovic h, Semy on Alesk er and Thomas W annerer for useful discussions. The second named author wishes to thank Andre Belotto da Silv a, An toine Commaret, Jean-Marc Delort, Pierre Sc hapira and Shu Shen for useful discussions. 2. Preliminaries In this section, w e introduce our notations and recall known deﬁnitions and results used throughout the text. 2.1. Notations. Unless explicitly stated otherwise, manifolds are smo oth and without b oundary . W e use the following con v entions. • If V is a v ector space and V ∗ its dual space, then for all α ∈ V ∗ and x ∈ V w e denote the natural pairing by ⟨ α, x ⟩ = ⟨ x, α ⟩ = α ( x ). • X n for a smo oth manifold X of dimension n . • If X is a smo oth manifold and Y a submanifold of X , w e denote b y T ∗ Y X ⊂ T ∗ X the conormal bundle of Y . 8 ANDREAS BERNIG AND V ADIM LEBOVICI • If f : X → Y is a smo oth map b et w een smo oth manifolds, w e consider the follo wing classical diagram: T ∗ X X × Y T ∗ Y T ∗ Y d f τ . (2.1) • Given a real v ector bundle E ov er X , w e denote b y 0 X or simply 0 its zero section. Moreo v er, we denote by P + ( E ) the orien ted pro- jectivization of E , that is, the quotient ( E \ 0 X ) / R > 0 where R > 0 is the group of p ositive real n um b ers acting by m ultiplication on the ﬁb ers, i.e., by ( p, ξ ) 7→ ( p, λξ ) for an y λ ∈ R > 0 and ( p, ξ ) ∈ E \ 0 X . W e denote by ( p, [ ξ ]) the equiv alence class of ( p, ξ ). • When E is the cotangent bundle T ∗ X of a smooth manifold X , we simply denote by P X = P + ( T ∗ X ) the cosphere bundle of X and b y π X the canonical map P X → X . • The space of complex-v alued diﬀeren tial k -forms on X is denoted b y Ω k ( X ). • The indicator function of a subset S ⊂ X is denoted by 1 S . • The interior of S inside X is denoted b y Int( S ). • When dealing with real analytic manifolds, stratiﬁcations are alw ays assumed subanalytic. 2.2. Orien tations. T o keep the notation as simple as p ossible, we will as- sume that our manifolds are oriented. Without the choice of an orien tation, the normal cycle, the characteristic cycle, and the diﬀeren tial forms on X and P X ha v e to b e twisted b y a section of the orien tation bundle of X . With this modiﬁcation, all formulas and statemen ts remain true in the non- orien ted case. As in [ 7 , 11 ], w e use the follo wing conv entions on orientations. Boundary. If X is an oriented manifold with b oundary , then its b ound- ary ∂ X is endow ed with the Stok es orientation [ 39 , Sec. 3.2]. Quotient. Let G b e an orien ted Lie group acting smoothly , freely and prop- erly on an oriented manifold X . Assume further that the action of G on X is orien tation-preserving. Then, the quotien t manifold X/G is orientable. W e endo w X/G with the following orien tation. F or any p ∈ X , w e hav e a canonical isomorphism T p X ∼ = T p ( G · p ) ⊕ T π ( p ) ( X/G ). W e use this as a con v ention for X/G , namely or( T p ( G · p )) ∧ or( T π ( p ) ( X/G )) = or( T p X ). F or instance, the unit sphere can be orien ted as the b oundary of the unit ball or as the quotien t of the Euclidean space minus the origin by the action of the m ultiplicativ e group of positive real num b ers. Using the abov e conv entions, these tw o orientations coincide. Interse ction. Let X and Y be orien ted submanifolds of an oriented mani- fold Z . If the intersection X ∩ Y is transverse then it is induced an orientation as the preimage of X by the embedding Y  → Z follo wing the conv entions of [ 39 , Sec. 3.2]. This conv ention on orientations coincide with the one of OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 9 [ 25 , Ex. 11.12] which we recall here for the sake of clarity . Let z ∈ X ∩ Y . Cho ose a basis of T z ( X ∩ Y ). Complete this basis to a positive basis of T z Y . Then, add vectors of T z X so that the chosen basis of T z ( X ∩ Y ) follo wed by these vectors is a p ositiv e basis of T z X . By transversalit y , the so obtained collection of tangen t v ectors is a basis of T z Z . If it is p ositiv ely oriented, giv e X ∩ Y the orientation giv en by the basis chosen in the ﬁrst place. Otherwise, give it the opp osite orien tation. Fib er pr o duct. Let f : X → Z and g : Y → Z b e t wo smo oth maps b et w een orien ted manifolds where at most one of X and Y ma y hav e a b oundary and Z is b oundaryless. Assume further that f × g and ( f × g ) | ∂ ( X × Y ) is transv ersal to the diagonal ∆ Z ⊂ Z × Z . Then, the ﬁb er pr o duct deﬁned as X × Z Y = ( f × g ) − 1 (∆ Z ) is an orien table smo oth manifold endo w ed with the preimage orientation [ 39 , Sec. 3.2]. Informally , our con v entions are suc h that or( Z ) ∧ or( X × Z Y ) = or( X ) ∧ or( Y ). These conv entions agree for instance with [ 1 , Sec. 7.2] and constitutes an “associative” choice of orientations when considering successiv e ﬁb er pro ducts. In particular, when g : Y → Z is a ﬁber bundle with oriented base Z , oriented ﬁber F , and total space Y oriented b y the lo cal product orientations U × F with U ⊂ Z op en, then the orientation induced on the pullbac k bundle X × Z Y is also giv en by the lo cal pro duct orien tations of V × F with V ⊂ X op en. Normal bund le. If Y is a submanifold of an oriented manifold X , then the conormal bundle T ∗ Y X is orien ted as any open tubular neighborho o d of Y in X . This correctly deﬁnes an orientation of T ∗ Y X as the induced orienta- tion do es not dep end on the choice of auxiliary Riemannian metric used to construct tubular neigh b orho ods. If Y is orien ted, our choice of orien tation of T ∗ Y X and our con ven tions on the orien tation of ﬁb er bundles induce an orientation of the ﬁb ers of T ∗ Y X . These ﬁb ers are equipp ed with an action of R > 0 whic h preserv es the orien- tation, yielding an orien tation on the ﬁb ers of P + ( T ∗ Y X ) and hence of the total space P + ( T ∗ Y X ). Oriente d blowup. In the previous setting, the deﬁnition of the orien ted blo w- up e X of X along Y is recalled in [ 7 , Sec. 1.2]. As in loc. cit., we endow e X with the orien tation satisfying that the canonical embedding X \ Y  → e X is orien tation-preserving. 2.3. Curren ts. Let X be an n - dimensional manifold (possibly with bound- ary). The space Ω k c ( X ) of smooth and compactly supp orted k -forms on X has a natural top ology and its dual space is the space D k ( X ) of k -curren ts on X . If T ∈ D k ( X ) and ω ∈ Ω k c ( X ), we write ⟨ T , ω ⟩ or R T ω for the natural pairing. The latter notation is motiv ated b y the fact that an oriented k - dimensional closed submanifold Y ⊂ X deﬁnes a curren t J Y K b y ω 7→ R Y ω . The b oundary ∂ T of a current T is deﬁned b y R ∂ T ω = R T dω . Clearly ∂ J Y K = 10 ANDREAS BERNIG AND V ADIM LEBOVICI J ∂ Y K by Stokes’ theorem. A current T is called a cycle if ∂ T = 0. The sup- p ort of a current is deﬁned in the obvious w ay . The pullbac k and pushforw ard of curren ts are dual to the op erations of pushforw ard and pullback of diﬀerential forms resp ectiv ely , whenever the latter ones are deﬁned. In particular, w e can deﬁne the pushforward of a curren t by a map f : X → Y which is prop er on the supp ort of the curren t and the pullback b y any ﬁb er bundle map f : X n → Y m . More precisely , if ω ∈ Ω k c ( X ), the pushforwar d f ∗ ω ∈ Ω k + m − n c ( Y ), also called ﬁb er inte gr ation [ 24 ], is uniquely deﬁned by: Z Y α ∧ f ∗ ω = Z X f ∗ α ∧ ω , α ∈ Ω n − k ( Y ) . In that case, the pullback of a current T ∈ D k ( Y ) is given by ⟨ f ∗ T , ω ⟩ = ⟨ T , f ∗ ω ⟩ for an y ω ∈ Ω k + n − m c ( X ). More generally , the pullback of a curren t T b y a smo oth map is deﬁned under some conditions on a closed conical subset WF( T ) ⊂ T ∗ X \ 0 X as- so ciated to T called its wave fr ont set [ 44 , 55 , 56 ]. Here, we do not recall the deﬁnition of w av e front sets, but only a few k ey results used throughout the present article, referring to [ 44 , Chap. 8] and [ 17 , Sec. 2.2] for more de- tails. If Γ ⊂ T ∗ X \ 0 X is a closed conical subset, w e denote by D k, Γ ( X ) the space of k -curren t ov er X such that WF( T ) ⊂ Γ. Recall ( 2.1 ) and denote similarly d f | ∂ X : ( ∂ X ) × Y T ∗ Y → T ∗ ( ∂ X ). Prop osition 2.1 ([ 44 , Thm. 8.2.4]) . L et f : X n → Y m b e a smo oth map b etwe en oriente d smo oth manifolds, wher e X is p ossibly with b oundary and Y is without b oundary. L et Γ ⊂ T ∗ Y \ 0 X b e a close d c onic al subset such that: (i) for al l ( x, η ) ∈ X × Y T ∗ Y with τ ( x, η ) ∈ Γ , one has d f ∗ η  = 0 ; (ii) for al l ( x, η ) ∈ ∂ X × Y T ∗ Y with τ ( x, η ) ∈ Γ , one has ( d f | ∂ X ) ∗ η  = 0 . Then, denoting f ∗ Γ := d f  τ − 1 (Γ)  and Γ ′ = ( f ∗ Γ + T ∗ ∂ X X ) \ 0 X , ther e exists a unique se quential ly c ontinuous map f ∗ : D k, Γ ( Y ) → D k + n − m, Γ ′ ( X ) extending the pul lb ack of smo oth forms, c al le d the pullback of curr ents. The interse ction S ∩ T of tw o currents S and T on X is then deﬁned as the pullbac k of the exterior pro duct of t w o curren ts on X b y the diagonal em b edding δ : X  → X × X whenever it is well-deﬁned. W e sa y that the diagram of oriented smo oth manifolds: W Y X Z h h ′ g f (2.2) is a Cartesian squar e if W ∼ = X × Z Y as orien ted manifolds. In such a situation, we ha v e the following base change form ula: OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 11 Lemma 2.2 ([ 33 , Lem. 1.10]) . L et g : Y → Z b e a ﬁb er bund le and c onsider a Cartesian squar e as in ( 2.2 ) . If f is pr op er on the supp ort of T ∈ D k ( X ) , then g ∗ f ∗ T = h ′ ∗ h ∗ T . 2.4. Borel-Mo ore homology . If X n is a smo oth manifold, we denote b y H BM k ( X ) the k -th Borel-Mo ore homology C -vector space, whic h can b e deﬁned using the dualizing complex of the manifold [ 46 , Def. 3.1.16] or using the chain complex of lo cally ﬁnite singular chains with complex co eﬃcien ts [ 26 , Cor. V.12.21]. W e denote by C BM k ( X ) the vector space of lo cally ﬁnite singular k -c hains with complex co eﬃcien ts and by ∂ k : C BM k ( X ) → C BM k − 1 ( X ) the corresp onding b oundary op erator. W e thus ha v e H BM k ( X ) := k er ∂ k / im ∂ k +1 . If f : X → Y is a smo oth map which is prop er on the supp ort of C ∈ H BM k ( X ), then the pushforwar d f ∗ C ∈ H BM k ( Y ) is deﬁned in the obvious w ay as the image by f of the singular chains deﬁning C . The pullback of Borel-Mo ore homology classes is deﬁned via Poincar ´ e dualit y; see [ 25 , Sec. VI.11], [ 59 ]. Supp ose X oriented and let Z ⊂ X b e a closed subset. W e denote b y H k Z ( X ) = H k ( X , X \ Z ) the group of degree- k cohomology classes of X with supp ort in Z ; see for instance [ 35 , Sec. B.2]. Then, there is a P oincar´ e dualit y isomorphism [ 46 , Prop. 3.3.6]: PD : H k Z ( X ) → H BM n − k ( Z ) . If Z is only assumed lo cally closed, then H k Z ( X ) is deﬁned as H k Z ( X ) = H k ( U, U \ Z ) where U is an op en subset of X containing Z as a closed subset; see [ 46 , Def. 2.3.8]. Let Y m b e an orien ted smo oth manifold and Z ′ ⊂ Y be a closed subset. If f : X → Y is a smo oth map, w e denote by f ∗ : H k Z ′ ( Y ) → H k f − 1 ( Z ′ ) ( X ) the classical pullbac k of cohomology classes. Then, the pul lb ack of Borel-Mo ore homology classes on Z ′ is deﬁned as the comp osition: f ∗ : H BM k ( Z ′ ) H m − k Z ′ ( Y ) H m − k f − 1 ( Z ′ ) ( X ) H BM k + n − m ( f − 1 ( Z ′ )) . PD − 1 f ∗ PD The following lemma follo ws from the base c hange form ula for shea v es [ 46 , Prop. 3.1.9]. Lemma 2.3. If the fol lowing diagr am of oriente d smo oth manifolds: W Y X Z h h ′ g f , is a Cartesian squar e a nd if f is pr op er on the supp ort of C ∈ H BM k ( X ) , then g ∗ f ∗ C = h ′ ∗ h ∗ C . 12 ANDREAS BERNIG AND V ADIM LEBOVICI 2.5. Subanalytic currents and Borel-Mo ore homology . In the con text of real analytic manifolds, w e will consider subanalytic curren ts. W e refer to [ 42 ] for more information on this topic and to [ 46 , Chap. 8] for more details on subanalytic sets. Let X b e a real analytic manifold (p ossibly with b oundary). A k -current T on X is called sub analytic if there exists a lo cally ﬁnite stratiﬁcation { S α } α ∈ A of X together with a complex multiplicit y m α and a c hoice of orien ta- tion for each k -dimensional stratum S α suc h that T = P α ∈ A k m α J S α K , where A k indexes the strata of dimension k . In suc h a situation, the strat- iﬁcation { S α } α ∈ A is said to deﬁne T . W e denote by D sub k ( X ) the space of subanalytic k -currents on X and if S ⊂ X is a closed subanalytic subset, w e denote by D sub k ( S ) the subspace of currents supp orted on S . The triangulation theorem of Hardt ([ 43 ], [ 46 , Prop. 8.2.5]) provides an isomorphism b et w een the chain complex of subanalytic curren ts and the c hain complex of lo cally ﬁnite singular c hains with complex co eﬃcien ts; see [ 42 ] or [ 46 , Thm. 9.2.10]. In addition, this isomorphism is compatible with pushforwards and pullbacks under mild assumptions, as shown b y the follo wing tw o lemmas due to Hardt [ 42 ]; see also [ 50 , App. B]. In what follo ws, w e will denote by T the Borel-Mo ore homology class asso ciated to a subanalytic cycle T . Lemma 2.4. L et T ∈ D sub k ( X ) b e a cycle and let f : X → Y b e a r e al analytic map b etwe en oriente d r e al analytic manifolds which is pr op er on S = supp( T ) . Then, the pushforwar d f ∗ T is a sub ana lytic k -curr ent supp orte d on f ( S ) and f ∗ T = f ∗ T in H BM k ( f ( S )) . F or the pullback, it is prov en in [ 42 ] that the intersection of the subana- lytic curren t with the integration current on the embedded submanifold is w ell-deﬁned under mild transversalit y assumptions and corresponds to the classical intersection theory on Borel-Morel homology classes. This in ter- section theory of subanalytic curren ts is deﬁned using the theory of slicing, but it is easy to verify that the slicing is the same as the pullback b y the em b edding of the ﬁb er in the sense of Theorem 2.1 . Thus w e hav e: Lemma 2.5. L et f : X n  → Y m b e a close d emb e dding of oriente d r e al analytic manifolds and let T ∈ D sub k ( Y ) b e a cycle with supp ort S = supp( T ) . Assume that the map f is tr ansverse to the str ata of a Whitney str atiﬁc ation of Y deﬁning T . Then, the pul lb ack f ∗ T of the curr ent T is wel l-deﬁne d and one has f ∗ T = f ∗ T in H BM k + n − m ( f − 1 ( S )) . Note that the assumption of this last lemma is sligh tly stronger than what is stated in [ 42 ], but it is suﬃcient for our purp ose. 2.6. Constructible functions. As w e exp osed in the introduction, con- structible functions are deﬁned as integer-v alued functions on a real analytic manifold X whic h are piecewise constant on a lo cally ﬁnite stratiﬁcation of X . If ϕ : X → Z is constructible, the theory of stratiﬁcation of sub- analytic sets ([ 46 , Thm. 8.3.20, Exo. VI I I.12]) ensures that there exists a OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 13 Whitney stratiﬁcation of X b y lo cally closed relativ ely compact subanalytic submanifolds S i ⊂ X , i ∈ I such that ϕ is constant on each S i , i.e., there are m i ∈ Z such that: ϕ = X i ∈ I m i 1 S i . (2.3) It follows from basic prop erties of subanalytic sets that constructible func- tions form a commutativ e ring CF ( X ) for p oin t wise addition and multipli- cation (see [ 46 , Sec. 8.2]). Let us also denote by CF c ( X ) the subring of compactly supp orted constructible functions, for whic h the sum ( 2.3 ) can b e tak en ﬁnite and in v olving only compact subanalytic subsets. In this article, we fo cus on op erations on constructible functions, as devel- opp ed in [ 64 ] in the complex setting and b y [ 58 ] in the real analytic setting. First, consider another real analytic manifold Y and let ϕ ∈ CF ( X ) and ψ ∈ CF ( Y ). The exterior pr o duct ϕ ⊠ ψ ∈ CF ( X × Y ) is the constructible func- tion deﬁned b y ( ϕ ⊠ ψ )( x, y ) = ϕ ( x ) ψ ( y ) for any ( x, y ) ∈ X × Y . Moreov er, if f : X → Y is a real analytic map, the pul lb ack f ∗ ψ ∈ CF ( X ) is deﬁned simply by f ∗ ψ = ψ ◦ f . That these tw o op erations send constructible func- tions to constructible functions follows again from well-kno wn prop erties of subanalytic sets. Moreo ver, the p oint wise pro duct of t wo constructible func- tions on X clearly coincides with the pullbac k of their exterior product b y the diagonal em b edding δ : X  → X × X . A more in v olv ed op eration is the pushforw ard of a constructible function b y a real analytic map. First, if S ⊂ X is a lo cally closed relatively compact subanalytic subset of X , then w e denote b y χ ( S ) its Euler characteristic with compact supp ort; see [ 46 , Chap. 9]. In particular, when S is compact, then χ ( S ) is the usual Euler characteristic. Then, the inte gr al of a compactly supp orted function ϕ ∈ CF c ( X ) which can be written as ( 2.3 ) is the in teger: Z X ϕ d χ = X i ∈ I m i · χ ( S i ) . Finally , if f : X → Y is a real analytic map whic h is prop er on the supp ort of ϕ ∈ CF ( X ), the pushforwar d f ∗ ϕ ∈ CF ( Y ) is deﬁned for an y y ∈ Y b y: f ∗ ϕ ( y ) = Z X 1 f − 1 ( y ) ϕ d χ. That the function f ∗ ϕ is constructible is pro ven in [ 46 , Sec. 9.7] using sheaf theory , and can also b e seen as a consequence of the c ell decomp osition theorem in the o-minimal structure of globally subanalytic sets [ 63 , Chap. 4, (2.10)]. Using this op eration, one can naturally deﬁne the conv olution of constructible functions. F or that, supp ose that G is a Lie group acting on X so that the action a : G × X → X is real analytic. Then, the c onvolution of ϕ ∈ CF ( G ) and ψ ∈ CF ( X ) is deﬁned b y ϕ ∗ ψ = a ∗ ( ϕ ⊠ ψ ) whenev er the map a is prop er on the supp ort of ϕ ⊠ ψ . 14 ANDREAS BERNIG AND V ADIM LEBOVICI 2.7. Characteristic cycles. Let X n b e an oriented real analytic manifold. Denote b y C L ( X ) the group of L agr angian chains of X , that is, the sub- group of C BM n ( T ∗ X ) generated b y conical lo cally closed Lagrangian suban- alytic submanifolds. The group of L agr angian cycles is the subgroup L ( X ) of H BM n ( T ∗ X ) generated b y Lagrangian chains C ∈ C L ( X ) with ∂ n C = 0. Construction. T o any constructible function ϕ ∈ CF ( X ), one can asso ciate a Lagrangian cycle CC( ϕ ) ∈ L ( X ) called the char acteristic cycle of ϕ ; see [ 46 , Chap. 9] and [ 34 ]. The morphism CC : CF ( X ) → L ( X ) induces an iso- morphism b et ween constructible functions and Lagrangian cycles, with an explicit in v erse; see [ 45 , Thm. 8.3], [ 46 , Thm. 9.7.11]. W e brieﬂy recall its construction here. Denote by S a Whitney stratiﬁcation of X such that ϕ is constant on each stratum. W e denote: Λ = [ S ∈S T ∗ S X . Then, the subset Λ ◦ = [ S ∈S   T ∗ S X \ [ S ′ ⊋ S T ∗ S ′ X   is a smo oth subanalytic subset of T ∗ X which is op en and dense in Λ. W e denote b y { Λ α } α ∈ A the collection of connected comp onen ts of Λ ◦ . Note that for eac h connected comp onen t Λ α , there exists S α ∈ S such that Λ α ⊂ T ∗ S α X . The characteristic cycle of ϕ is deﬁned as: CC( ϕ ) = X α ∈ A m α [Λ α ] , (2.4) where [Λ α ] ∈ C BM n (Λ) and m α ∈ Z are certain integral m ultiplicities that can b e deﬁned lo cally using stratiﬁed Morse theory [ 38 ]; see [ 59 ]. Op er ations. The pushforw ard and pullback operations on characteristic cy- cles hav e been deﬁned in [ 46 ]. W e recall their description as exp osed in [ 59 ]. The maps from the classical diagram ( 2.1 ) allo w one to describ e the eﬀect of the pushforw ard of constructible functions on their characteristic cycles. Lemma 2.6 ([ 46 , Prop. 9.4.2], [ 59 , (2.17)]) . L et f : X n → Y m b e a r e al analytic map b etwe en oriente d r e al analytic manifolds and ϕ ∈ CF ( X ) such that f is pr op er on supp( ϕ ) . Then τ ∗ d f ∗ CC( ϕ ) ∈ H BM m ( T ∗ Y ) is wel l-deﬁne d and: CC( f ∗ ϕ ) = τ ∗ d f ∗ CC( ϕ ) . W e ha ve a similar statemen t for the pullbac k operation by real analytic maps satisfying some transversalit y assumptions [ 46 , 59 ]. See also [ 54 , Sec. 5] and [ 61 , Thm. 3.3] in the complex setting. W e recall the assumption of [ 59 ] under a simpler terminology: OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 15 Deﬁnition 2.7. A real analytic map f : X → Y is called tr ansverse to ψ ∈ CF ( Y ) (called normal ly non-singular in [ 59 ]) if there exists a Whitney strat- iﬁcation of Y suc h that ψ is constan t on eac h stratum and f is transv erse to each stratum. Under such transv ersalit y c onditions, Sc hmid and Vilonen state: Lemma 2.8 ([ 46 , Prop. 9.4.3], [ 59 , (2.20)]) . L et f : X n → Y m b e a r e al analytic map b etwe en oriente d r e al analytic manifolds and ψ ∈ CF ( Y ) such that f is tr ansverse to ψ . Then d f ∗ τ ∗ CC( ψ ) ∈ H BM n ( T ∗ X ) is wel l-deﬁne d and: CC( f ∗ ψ ) = d f ∗ τ ∗ CC( ψ ) . As explained in [ 59 ], the assumption under which [ 46 , Prop. 9.4.3] is pro v en is more general than the transversalit y assumption ab ov e, but this result will b e suﬃcient for our purp ose. 2.8. Generalized v aluations. Let X n b e a smo oth orien ted manifold, not necessarily real analytic. The manifold P X is a con tact manifold, with the con tact hyperplane at a p oin t ( x, [ ξ ]) giv en b y ker  dπ X | ( x, [ ξ ])  ∗ ξ . V ectors b elonging to the con tact plane are called horizon tal. A diﬀerential form is called v ertical if it v anishes whenever we plug in only horizontal vectors. Finally a Legendrian cycle is an ( n − 1)-cycle that v anishes on v ertical forms. Note that w e do not assume that T is rectiﬁable. W e denote by P ( X ) the set of compact submanifolds with corners, i.e. compact subsets that are lo cally diﬀeomorphic to a quadran t in R n . T o eac h P ∈ P ( X ) we can asso ciate its normal cycle N ( P ), whic h is an in tegral Legendrian cycle in P X ; see for instance [ 7 ]. A smo oth valuation on X is a map µ : P ( X ) → C suc h that there exist φ ∈ Ω n ( X ) and ω ∈ Ω n − 1 ( P X ) such that for all P ∈ P ( X ), we ha v e: µ ( P ) = Z P φ + Z N ( P ) ω . (2.5) The space of smo oth v aluations is denoted b y V ∞ ( X ). As a quotien t of the space Ω n ( X ) ⊕ Ω n − 1 ( P X ), this space is naturally endow ed with a F r´ ec het top ology . The supp ort of a smo oth v aluation is deﬁned in the obvious wa y and the space of compactly supp orted v aluations is denoted b y V ∞ c ( X ). It admits a lo cally con v ex top ology that is ﬁner than the induced top ology , see [ 6 , Section 5.1]. Any µ ∈ V ∞ c ( X ) can b e represen ted as in ( 2.5 ) by compactly supp orted forms ( φ, ω ) ∈ Ω n c ( X ) ⊕ Ω n − 1 c ( P X ), see [ 15 , Lemma 2.3]. Examples of smo oth v aluations are the Euler c haracteristic, the v olume (if X is a Riemannian manifold), or mixed volumes with smo oth reference b odies with p ositive curv ature on R n . An important breakthrough obtained b y Alesk er and F u [ 13 ], see also [ 11 ], is the introduction of a comm utative and asso ciativ e pro duct structure 16 ANDREAS BERNIG AND V ADIM LEBOVICI on V ∞ ( X ) such that the Euler characteristic is the unit element. More- o v er, it satisﬁes a version of Poincar ´ e duality , i.e. the map V ∞ ( X ) × V ∞ c ( X ) → C , ( µ 1 , µ 2 ) 7→ Z X µ 1 · µ 2 = ( µ 1 · µ 2 )( X ) is a p erfect pairing [ 6 , Thm. 6.1.1]; see also [ 15 ] for an alternative pro of. W e th us get an injection PD : V ∞ ( X )  →  V ∞ c ( X )  ∗ with dense image. This motiv ates the follo wing deﬁnition: Deﬁnition 2.9 ([ 6 , Deﬁnition 7.1.1]) . A gener alize d valuation is an element of the top ological dual: V −∞ ( X ) =  V ∞ c ( X )  ∗ , and this space is naturally endow ed with the weak top ology . The supp ort of a generalized v aluation is deﬁned in the ob vious wa y and the space of compactly supp orted generalized v aluations is denoted b y V −∞ c ( X ). It has a natural lo cally con vex top ology . By [ 6 , Prop. 7.3.10], there is a vector space isomorphism V −∞ c ( X ) ∼ = V ∞ ( X ) ∗ . A result of Br¨ oc ker and the ﬁrst named author [ 16 ] implies the following description of generalized v aluations. Prop osition 2.10. Ther e is a close d emb e dding V −∞ ( X )  → D n ( X ) ⊕ D n − 1 ( P X ) with image the sp ac e of curr ents ( C , T ) such that: (i) T is a L e gendrian cycle, (ii) π X ∗ T = ∂ C . If ( C , T ) is the pair corresp onding to ζ ∈ V −∞ ( X ) and µ ∈ V ∞ c ( X ) is giv en by the forms ( φ, ω ), then ⟨ ζ , µ ⟩ = C ( φ ) + T ( ω ) . (2.6) The wave fr ont set of ζ is deﬁned by WF( ζ ) = (WF( C ) , WF( T )). Given closed conical sets Λ ⊂ T ∗ X \ 0 and Γ ⊂ T ∗ P X \ 0, w e denote by V −∞ Λ , Γ ( X ) the set of generalized v aluations ζ with WF( ζ ) ⊂ (Λ , Γ). This space is endo w ed with the H¨ ormander top ology , see [ 27 ]. The space V ∞ ( X ) also injects in this space of curren ts. If µ ∈ V ∞ ( X ) is giv en by the forms ( φ, ω ) as in ( 2.5 ), then by [ 15 ] we ha v e: C = π X ∗ ω ∈ C ∞ ( X ) ⊂ D n ( X ) , T = s ∗ ( D ω + π ∗ X φ ) ∈ Ω n ( P X ) ⊂ D n − 1 ( P X ) . Here D : Ω n − 1 ( P X ) → Ω n ( P X ) is the Rumin op erator (see [ 53 ]), and s : P X → P X is the in volution ( x, [ ξ ]) 7→ ( x, [ − ξ ]). The follo wing lemma will b e useful in the pro of of Theorem 3 in Section 6 . Lemma 2.11. Assume that X is c onne cte d and let ψ ∈ V −∞ ( X ) b e a gener- alize d valuation with T = 0 . Then ψ is a multiple of the Euler char acteristic. OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 17 Pr o of. Since ∂ C = π X ∗ T = 0, the constancy theorem [ 32 , (4.1.7)] implies that C = λ J X K . As noted ab ov e, a compactly supported smo oth v aluation µ can b e represen ted by a pair ( φ, ω ) of compactly supp orted forms. Therefore, ⟨ ψ , µ ⟩ = C ( φ ) + T ( ω ) = λ Z X φ = λ Z X µ = λ Z X χ · µ = ⟨ PD( λχ ) , µ ⟩ . □ 2.9. Constructible functions as generalized v aluations. In [ 6 ], Alesker deﬁnes an embedding CF ( X )  → V −∞ ( X ) of the group of constructible functions into the space of generalized v aluations whic h extends to an em- b edding [ − ] : CF ( X ) ⊗ C  → V −∞ ( X ) with sequentially dense image [ 6 , Prop. 8.2.2]. The construction of this em b edding uses the characteristic cy- cle construction. F ollo wing Theorem 2.10 , the generalized v aluation [ ϕ ] ∈ V −∞ ( X ) asso ciated to ϕ ∈ CF ( X ) is more naturally describ ed using the normal cycle N ( ϕ ), whic h is a subanalytic Legendrian ( n − 1)-cycle on the cosphere bundle P X ; see [ 34 ]. W e recall the relationship b etw een normal and c haracteristic cycles de- tailed in [ 34 , Sec. 4.7]. F ollo wing ( 2.3 ), write ϕ = P i ∈ I m i 1 S i for in te- gers m i ∈ Z and lo cally closed subanalytic subsets S i of X . Since X is orien ted, the constructible function ϕ can th us naturally b e seen as a cur- ren t C φ in tegrating compactly supp orted diﬀerential n -forms on the in terior of the subanalytic subsets S i that are n -dimensional, and this current is clearly indep enden t of the decomp osition of ϕ c hosen suc h that ( 2.3 ) holds. Denoting q X : T ∗ X \ 0 X → P X the quotient map and j X : T ∗ X \ 0 X  → T ∗ X the inclusion, we can deﬁne the c oniﬁc ation of a subanalytic cur- ren t T ∈ D k ( P X ) as follo ws. If S ∈ D sub k ( T ∗ X \ 0 X ) is a subanalytic current deﬁned by conical strata, then these strata are also subanalytic in T ∗ X b y [ 46 , Prop. 8.3.8 (i)], so that S naturally deﬁnes a subanalytic current on T ∗ X , denoted b y j X ∗ S . W e deﬁne the c oniﬁc ation of a subanalytic cur- ren t T ∈ D sub k ( P X ) as the current cone( T ) = j X ∗ q ∗ X T ∈ D sub k +1 ( T ∗ X ). With these notations, the c haracteristic and normal cycles are related b y the formula: CC( ϕ ) = C φ + s ∗ cone( N ( ϕ )) , (2.7) where s : T ∗ X → T ∗ X denotes the multiplication by − 1 in the ﬁb ers of T ∗ X . Deﬁnition 2.12. The generalized v aluation [ ϕ ] asso ciated to ϕ ∈ CF ( X ) corresp onds to the pair of curren ts ( C , T ) = ( C φ , N ( ϕ )) ∈ D n ( X ) ⊕D n − 1 ( P X ). Note that the abov e deﬁnition of the em b edding [ − ] : CF ( X )  → V −∞ ( X ) corresp onds to the sign conv entions of [ 8 ]. Example 2.13. Let ϕ = 1 S for some subanalytic subset S ⊂ X . The generalized v aluation [ ϕ ] ∈ V −∞ ( X ) is represen ted b y the pair of cur- ren ts ( J Int( S ) K , N ( S )) where Int( S ) denotes the in terior of S inside X (whic h is empty when dim( S ) < n ). If in addition S is a compact submanifold with corners, then for each µ ∈ V ∞ c ( X ) we hav e ⟨ [ ϕ ] , µ ⟩ = µ ( S ). This motiv ates 18 ANDREAS BERNIG AND V ADIM LEBOVICI to write µ ( ϕ ) instead of ⟨ [ ϕ ] , µ ⟩ for ϕ ∈ CF ( X ). T o sp ell this out: compactly supp orted smo oth v aluations can b e ev aluated on constructible functions. 3. W a ve front set of subanal ytic currents When considering pullbac ks of subanalytic curren ts, w e will use tw o no- tions of transversalit y . One is form ulated in terms of geometric transver- salit y of real analytic maps with the strata deﬁning the curren t and the other form ulated in terms of w av e front sets. In this section, w e sho w using desingularization that one can alw ays stratify the supp ort of a subanalytic curren t so that its w a v e fron t set is con tained in the union of conormal bun- dles to the strata. F or such a stratiﬁcation, the ﬁrst notion of transversalit y implies the second. Deﬁnition 3.1. Let T be a subanalytic current on X . A lo cally ﬁnite stratiﬁcation { S α } α ∈ A of X deﬁning T is called angular if w e hav e: WF( T ) ⊂ [ α ∈ A T ∗ S α X \ 0 S α . Note that if a giv en stratiﬁcation is angular, then so is any reﬁnemen t of this stratiﬁcation. In particular, one can alw ays reﬁne an angular stratiﬁca- tion to mak e it also satisfy Whitney’s conditions. Note also that angularity is preserved under diﬀeomorphisms. Lemma 3.2. If T 1 is a sub analytic curr ent on X 1 with an angular str at- iﬁc ation { S 1 α } α ∈ A , and T 2 is a sub analytic curr ent on X 2 with an angular str atiﬁc ation { S 2 β } β ∈ B , then the pr o duct str atiﬁc ation { S 1 α × S 2 β } α ∈ A,β ∈ B is an angular str atiﬁc ation of the sub analytic curr ent T 1 ⊠ T 2 on X 1 × X 2 . Pr o of. This is obvious from the inclusion WF( T 1 ⊠ T 2 ) ⊂ (WF( T 1 ) × WF( T 2 )) ∪ (WF( T 1 ) × 0 X 2 ) ∪ (0 X 1 × WF( T 2 )) , see [ 44 , Thm. 8.2.9]. □ Deﬁnition 3.3. W e sa y that a real analytic map f : X → Y is angularly tr ansverse to a subanalytic curren t T ∈ D sub k ( Y ) if there exists an angular Whitney stratiﬁcation of Y deﬁning T suc h that f is transv erse to each stratum. If f is angularly transv erse to the subanalytic curren t T , then f and Γ = WF( T ) satisfy the assumptions of Theorem 2.1 . W e no w prov e that angular stratiﬁcations alwa ys exist. Prop osition 3.4. A ny sub analytic curr ent admits an angular str atiﬁc ation. W e do not know if assuming the transversalit y of f with the strata of some stratiﬁcation en tails the existence of an angular stratiﬁcation with the same prop ert y . Our pro of of Theorem 3.4 is a com bination of a functoriality prop ert y (Theorem 3.5 ) and of global smo othings of subanalytic sets prov en b y Bier- stone and P arusi ´ nski (Theorem 3.6 ). OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 19 Lemma 3.5. If a sub analytic curr ent admits an angular str atiﬁc ation, then so do es its pushforwar d by a pr op er r e al analytic map. Pr o of. Let f : X → Y b e a prop er analytic map b etw een real analytic man- ifolds and T be a subanalytic curren t on X . Denote b y S = ( S α ) α ∈ A an angular stratiﬁcation of X deﬁning T . Up to reﬁning S if necessary , one can consider a stratiﬁcation R = ( R β ) β ∈ B of Y such that eac h f − 1 ( R β ) is a union of strata of S and eac h restriction f | S α : S α → R β is an analytic submersion; see e.g. [ 38 , P art I, Sec. 1.7]. By [ 40 , Chap. VI, Prop. 3.9] w e ha v e that: WF( f ∗ T ) ⊂ f ∗ WF( T ) := τ  d f − 1 (WF ( T ) ∪ 0 X )  \ 0 Y . (3.1) It then follo ws directly from the facts that S is angular and that f | S α : S α → R β is a submersion that the ﬁb er of the righ t-hand side ab o ve any y ∈ R β is contained in ( T ∗ R β Y ) y . □ W e only state a sp ecial case of the result of Bierstone and P arusi ´ nski whic h is suﬃcient for our purp ose. Theorem 3.6 ([ 22 , Thm. 1.2]) . L et X b e a r e al analytic manifold of dimen- sion n and let S b e a close d sub analytic subset of X of dimension k . Then, ther e exist a r e al analytic manifold e X of dimension n , a pr op er r e al analytic map ϕ : e X → X such that ϕ induc es an isomorphism fr om a disjoint union of r elative interiors of k -dimensional submanifolds with c orners of e X onto an op en sub analytic subset U ⊂ S such that dim( S \ U ) < k . Pr o of of The or em 3.4 . Let T b e a subanalytic k -current on X . W e can assume without loss of generalit y that T is the integration o v er a closed subanalytic subset S ⊂ X of dimension k . Note that w e will not care ab out orien tations in this pro of as the result pro ven is in v ariant by a change of sign. W e denote by e U the disjoin t union of relativ e interiors of k -dimensional subanalytic submanifolds with corners of e X given b y Theorem 3.6 and use the same notations as in this statemen t. The curren t e T of in tegration o v er e U b eing a sum of curren ts integrating o ver submanifolds with corners, it clearly admits an angular stratiﬁcation. Moreov er, we claim that T = ϕ ∗ e T . Indeed, for an y compactly supp orted smo oth k -diﬀerential form ω ∈ Ω k c ( M ), w e ha v e: D ϕ ∗ e T , ω E = D e T , ϕ ∗ ω E = Z e U ϕ ∗ ω = Z U ω = Z S ω = ⟨ T , ω ⟩ . Hence the existence of an angular stratiﬁcation for T by Theorem 3.5 . □ 4. Exterior pr oduct In this section, w e pro v e that the exterior pro duct of generalized v alua- tions restricts to the exterior pro duct of constructible functions, whic h w as claimed in [ 7 , Claim 2.1.11]. 20 ANDREAS BERNIG AND V ADIM LEBOVICI W e follow the notations of [ 7 , 11 ]. Let X 1 and X 2 b e t w o real analytic manifolds and denote X = X 1 × X 2 . The pro jections are denoted b y e p i : X 1 × X 2 → X i for i = 1 , 2. Let M 1 = { ( x 1 , x 2 , [ ξ 1 : 0]) } ⊂ P X and M 2 = { ( x 1 , x 2 , [0 : ξ 2 ]) } ⊂ P X . Consider the oriented blo wup F : b P X → P X along M := M 1 ∪ M 2 . The space b P X is the closure inside P X × X ( P X 1 × P X 2 ) of the set { ( x 1 , x 2 , [ ξ 1 : ξ 2 ] , [ ξ 1 ] , [ ξ 2 ]) } . Then b P X is a manifold of dimension 2(dim X 1 + dim X 2 ) − 1 with b oundary , where the b oundary is given by N := N 1 ∪ N 2 with: N 1 = { ( x 1 , x 2 , [ ξ 1 : 0] , [ ξ 1 ] , [ η 2 ] } , N 2 = { ( x 1 , x 2 , [0 : ξ 2 ] , [ η 1 ] , [ ξ 2 ] } . The map F : b P X ↠ P X is given b y: F ( x 1 , x 2 , [ ξ 1 : ξ 2 ] , [ η 1 ] , [ η 2 ]) := ( x 1 , x 2 , [ ξ 1 : ξ 2 ]) . Moreo v er, w e hav e a map Φ : b P X ↠ P X 1 × P X 2 giv en by: Φ( x 1 , x 2 , [ ξ 1 : ξ 2 ] , [ η 1 ] , [ η 2 ]) := ( x 1 , [ η 1 ] , x 2 , [ η 2 ]) . Finally , w e hav e the ob vious maps: P X 1 × X 2 P X X 1 × P X 2 P X 1 X P X 2 i 1 p 1 π X i 2 p 2 The op eration ⊠ : V −∞ ( X 1 ) × V −∞ ( X 2 ) → V −∞ ( X ) is deﬁned as follo ws. Deﬁnition 4.1 ([ 7 , Sec. 2]) . F or any pair of generalized v aluations ( ζ 1 , ζ 2 ) ∈ V −∞ ( X 1 ) × V −∞ ( X 2 ) resp ectiv ely describ ed by pairs of currents ( C 1 , T 1 ) and ( C 2 , T 2 ), the exterior pr o duct ζ 1 ⊠ ζ 2 ∈ V −∞ ( X ) is deﬁned by the pair of currents: C = C 1 ⊠ C 2 , T = F ∗ Φ ∗ ( T 1 ⊠ T 2 ) + ( e p 1 ◦ π X ) ∗ C 1 ∩ i 2 ∗ p ∗ 2 T 2 + i 1 ∗ p ∗ 1 T 1 ∩ ( e p 2 ◦ π X ) ∗ C 2 . W e can now pro v e the main result of this section: Prop osition 4.2 ([ 7 , Claim 2.1.11]) . L et ϕ 1 ∈ CF ( X 1 ) and ϕ 2 ∈ CF ( X 2 ) b e c onstructible functions on r e al analytic manifolds. We have: [ ϕ 1 ] ⊠ [ ϕ 2 ] = [ ϕ 1 ⊠ ϕ 2 ] . Pr o of. F ollowing Theorem 2.12 , denote ( C i , T i ) = ( C φ i , N ( ϕ i )) for i = 1 , 2. W e m ust prov e: C φ 1 ⊠ φ 2 = C 1 ⊠ C 2 , (4.1) N ( ϕ 1 ⊠ ϕ 2 ) = F ∗ Φ ∗ ( T 1 ⊠ T 2 ) + ( e p 1 ◦ π X ) ∗ C 1 ∩ i 2 ∗ p ∗ 2 T 2 + i 1 ∗ p ∗ 1 T 1 ∩ ( e p 2 ◦ π X ) ∗ C 2 . (4.2) OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 21 The ﬁrst equalit y ( 4.1 ) follows readily from the deﬁnition of C φ 1 ⊠ φ 2 . F or ( 4.2 ), rec all that c haracteristic cycles b ehav e well with resp ect to ex- terior pro duct [ 34 , Thm. 4.5], [ 46 , (9.4.1)]: CC( ϕ 1 ⊠ ϕ 2 ) = CC( ϕ 1 ) ⊠ CC( ϕ 2 ) . By the formula relating c haracteristic and normal cycles ( 2.7 ), and denoting again by s : T ∗ X → T ∗ X the multiplication b y − 1 in the ﬁb ers, we ha v e: s ∗ CC( ϕ 1 ⊠ ϕ 2 ) = C 1 ⊠ C 2 + C 1 ⊠ cone( T 2 )+cone( T 1 ) ⊠ C 2 +cone( T 1 ) ⊠ cone( T 2 ) , and the sum of the last three terms is cone( N ( ϕ 1 ⊠ ϕ 2 )). Thus ( 4.2 ) will follo w from: cone( T 1 ) ⊠ cone( T 2 ) = cone  F ∗ Φ ∗ ( T 1 ⊠ T 2 )  , (4.3) C 1 ⊠ cone( T 2 ) = cone  ( e p 1 ◦ π X ) ∗ C 1 ∩ i 2 ∗ p ∗ 2 T 2  , (4.4) cone( T 1 ) ⊠ C 2 = cone  i 1 ∗ p ∗ 1 T 1 ∩ ( e p 2 ◦ π X ) ∗ C 2  . (4.5) T o pro ve ( 4.3 ), denote b y b π : [ T ∗ X → b P X the pullbac k of the ﬁb er bun- dle q X : T ∗ X \ 0 X → P X b y the smooth map F : b P X → P X . More explicitly , this pullback bundle can b e describ ed as follows. The total space [ T ∗ X is the manifold with b oundary of dimension 2 dim( X 1 ) + 2 dim( X 2 ) suc h that: In t( [ T ∗ X ) =  ( x 1 , x 2 , ξ 1 , ξ 2 ) ∈ T ∗ X : ξ 1  = 0 and ξ 2  = 0  , ∂ [ T ∗ X =  T ∗ X 1 \ 0 X 1  × P X 2 ∪ P X 1 ×  T ∗ X 2 \ 0 X 2  , and the map b π : [ T ∗ X → b P X is given b y: b π ( x 1 , x 2 , ξ 1 , ξ 2 ) = ( x 1 , x 2 , [ ξ 1 : ξ 2 ] , [ ξ 1 ] , [ ξ 2 ]) on Int( [ T ∗ X ) , b π ( x 1 , x 2 , ξ 1 , [ η 2 ]) = ( x 1 , x 2 , [ ξ 1 : 0] , [ ξ 1 ] , [ η 2 ]) on ( T ∗ X 1 \ 0 X 1 ) × P X 2 , b π ( x 1 , x 2 , [ η 1 ] , ξ 2 ) = ( x 1 , x 2 , [0 : ξ 2 ] , [ η 1 ] , [ ξ 2 ]) on P X 1 × ( T ∗ X 2 \ 0 X 2 ) . Moreo v er, the ﬁber bundle [ T ∗ X is naturally equipped with a map e F : [ T ∗ X → T ∗ X \ 0 X giv en by: e F ( x 1 , x 2 , ξ 1 , ξ 2 ) = ( x 1 , x 2 , ξ 1 , ξ 2 ) on Int( [ T ∗ X ) , e F ( x 1 , x 2 , ξ 1 , [ η 2 ]) = ( x 1 , x 2 , ξ 1 , 0) on ( T ∗ X 1 \ 0 X 1 ) × P X 2 , e F ( x 1 , x 2 , [ η 1 ] , ξ 2 ) = ( x 1 , x 2 , 0 , ξ 2 ) on P X 1 × ( T ∗ X 2 \ 0 X 2 ) . 22 ANDREAS BERNIG AND V ADIM LEBOVICI These maps and bundles assemble into the following commutativ e diagram whose b ottom square is Cartesian: In t( [ T ∗ X ) [ T ∗ X T ∗ X \ 0 X T ∗ X P X 1 × P X 2 c P X P X ι j j X ◦ ι p Φ ◦ b π b π e F q X j X Φ F (4.6) where j , j X and ι denote canonical inclusions and p := Φ ◦ b π ◦ j . Note also that w e hav e canonical identiﬁcations In t( [ T ∗ X ) = ( T ∗ X 1 \ 0 X 1 ) × ( T ∗ X 2 \ 0 X 2 ), p = q X 1 × q X 2 : ( T ∗ X 1 \ 0 X 1 ) × ( T ∗ X 2 \ 0 X 2 ) → P X 1 × P X 2 , j X ◦ ι = j X 1 × j X 2 : ( T ∗ X 1 \ 0 X 1 ) × ( T ∗ X 2 \ 0 X 2 ) → T ∗ X , whic h entail: cone( T 1 ) ⊠ cone( T 2 ) = j X ∗ ι ∗ p ∗ ( T 1 ⊠ T 2 ) = j X ∗ e F ∗ j ∗ p ∗ ( T 1 ⊠ T 2 ) . Moreo v er, one has the equality of maps j ∗ p ∗ = (Φ ◦ b π ) ∗ from the space of curren ts on P X 1 × P X 2 to the space of currents on [ T ∗ X . Indeed, for an y compactly supp orted smo oth diﬀeren tial form ω on [ T ∗ X we ha v e that (Φ ◦ b π ) ∗ ω = p ∗ j ∗ ω . This last equalit y follo ws readily from the fact that the ﬁb ers of Φ ◦ b π are the closure of the ﬁb ers of p in [ T ∗ X so that the in tegration along the ﬁb ers of Φ ◦ b π of the smo oth form ω and the integration along the ﬁb ers of p of the restriction j ∗ ω of ω to the interior of [ T ∗ X coincide. Since the b ottom square of ( 4.6 ) is Cartesian, one has that q ∗ X F ∗ Φ ∗ ( T 1 ⊠ T 2 ) = e F ∗ (Φ ◦ b π ) ∗ ( T 1 ⊠ T 2 ) , so that w e hav e prov en: cone( T 1 ) ⊠ cone( T 2 ) = j X ∗ e F ∗ (Φ ◦ b π ) ∗ ( T 1 ⊠ T 2 ) = j X ∗ q ∗ X F ∗ Φ ∗ ( T 1 ⊠ T 2 ) = cone( F ∗ Φ ∗ ( T 1 ⊠ T 2 )) . T o prov e ( 4.4 ) (and ( 4.5 ) by symmetry), consider the comm utative dia- gram: T ∗ X 1 × T ∗ X 2 X 1 × T ∗ X 2 T ∗ X 2 T ∗ X 1 × ( T ∗ X 2 \ 0 X 2 ) X 1 × ( T ∗ X 2 \ 0 X 2 ) T ∗ X 2 \ 0 X 2 X 1 P X X 1 × P X 2 P X 2 p ′′ 2 i ′′ 2 p 1 ◦ π X ◦ q X q X j X id X 1 × q X 2 p ′ 2 i ′ 2 id X 1 × j X 2 q X 2 j X 2 p 1 ◦ π X i 2 p 2 OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 23 where j X 2 , i ′ 2 and i ′′ 2 are the canonical injections, and p ′ 2 , p ′′ 2 and q X 2 the canonical submersions. The b ottom left and the top right comm utative squares of the ab ov e diagram are Cartesian squares. Therefore, we ha v e: j X ∗ q ∗ X i 2 ∗ p ∗ 2 T 2 = j X ∗ i ′ 2 ∗ (id X 1 × q X 2 ) ∗ p ∗ 2 T 2 = i ′′ 2 ∗ ( p ′′ 2 ) ∗ cone( T 2 ) . Moreo v er, it follows from the deﬁnitions of the op erations that i ′′ 2 ∗ ( p ′′ 2 ) ∗ cone( T 2 ) = J 0 X 1 K ⊠ cone( T 2 ) , and, denoting e π X 1 : T ∗ X 1 → X 1 the canonical submersion, we ha ve: j X ∗ q ∗ X ( e p 1 ◦ π X ) ∗ C 1 = ( e π X 1 ) ∗ C 1 ⊠ J T ∗ X 2 K . Therefore, we ha v e: j X ∗ q ∗ X  ( e p 1 ◦ π X ) ∗ C 1 ∩ i 2 ∗ p ∗ 2 T 2  = j X ∗ q ∗ X  ( e p 1 ◦ π X ) ∗ C 1  ∩ j X ∗ q ∗ X  i 2 ∗ p ∗ 2 T 2  = ( e π X 1 ) ∗ C 1 ⊠ J T ∗ X 2 K ∩ J 0 X 1 K ⊠ cone( T 2 ) = C 1 ⊠ cone( T 2 ) . □ 5. Pullback and product In this section, w e recall the deﬁnition of the pullbac k op eration on gener- alized v aluations as deﬁned in [ 7 ] and pro v e that it restricts on constructible functions to the usual pullback given by precomp osition under mild transver- salit y assumptions. As a corollary , we obtain a similar result for the pro duct of constructible functions. 5.1. Deﬁnition. W e follow closely the exp osition and the notations of [ 7 ]. Let X and Y be t w o oriented real analytic manifolds, let f : X → Y b e a real analytic map and let ζ ∈ V −∞ ( Y ) b e represen ted by the pair of curren ts ( C , T ). Submersions. In [ 7 ], the pullback of generalized v aluations b y submersions is deﬁned as the adjoint op eration to the pushforward of compactly supported smo oth v aluations. The explicit expression of the pullbac k as op erations on the pair of currents represen ting the generalized v aluation is prov en in [ 7 , Prop. 3.3.3]. T o b e more consistent with the deﬁnition of the other op er- ations on generalized v aluations, we will take this explicit expression as a deﬁnition of the pullback. Consider then the diagram of canonical maps: P X d f ← − X × Y P Y τ − → P Y . (5.1) Deﬁnition 5.1 ([ 7 , Prop. 3.3.3]) . Supp ose that f is a submersion. The pul lb ack of ζ b y f is the generalized v aluation f ∗ ζ ∈ V −∞ ( X ) represen ted b y the pair of currents: ( C ′ , T ′ ) =  f ∗ C, d f ∗ τ ∗ T  . 24 ANDREAS BERNIG AND V ADIM LEBOVICI Immersions. The pullback of generalized v aluations by immersions is de- ﬁned under the following transv ersality assumptions: Deﬁnition 5.2 ([ 7 , Def. 3.5.2]) . Denote WF( ζ ) = (Λ , Γ). (i) A closed embedding f : X → Y is tr ansverse to ζ if: Λ ∩ T ∗ X Y = ∅ , Γ ∩ T ∗ X × Y P Y P Y = ∅ , Γ ∩ T ∗ P + ( T ∗ X Y ) P Y = ∅ . (ii) An immersion f : X → Y is tr ansverse to ζ if for every x ∈ X , there exists an op en neighborho o d U of x in X and an op en neigh b or- ho od V of f ( x ) in Y suc h that f | U : U → V is a closed em b edding whic h is transverse to ζ | V ∈ V −∞ ( V ). Consider the oriented blowup ^ X × Y P Y of X × Y P Y along the submani- fold P + ( T ∗ X Y ). The total space of this blo wup is a manifold with b oundary of dimension dim( X ) + dim( Y ) − 1 and such that: In t  ^ X × Y P Y  = n ( x, [ ξ ] , [ ξ ′ ]) ∈ ( X × Y P Y ) × X P X : d f ∗ ξ  = 0 ; [ d f ∗ ξ ] = [ ξ ′ ] o , ∂  ^ X × Y P Y  = P + ( T ∗ X Y ) × X P X . (5.2) W e denote by α : ^ X × Y P Y → X × Y P Y  → P Y the comp osition of the blo wup map and of the canonical inclusion and by β : ^ X × Y P Y → P X the map induced b y d f . These maps yield a diagram: P X β ← − ^ X × Y P Y α − → P Y . (5.3) Deﬁnition 5.3 ([ 7 , Claim 3.5.4]) . Supp ose that f is an immersion whic h is transv erse to ζ . The pul lb ack of ζ by f is the generalized v aluation f ∗ ζ ∈ V −∞ ( X ) represented b y the pair of currents: ( C ′ , T ′ ) = ( f ∗ C, β ∗ α ∗ T ) . 5.2. Restriction to constructible functions. In the case of a generalized v aluation associated to a constructible function, the curren ts ( C, T ) are both subanalytic. In this case, the op erations in Theorem 5.3 are th us w ell-deﬁned under the assumptions that f and α are transv erse to the strata of a Whitney stratiﬁcation deﬁning C ψ and N ( ψ ) resp ectively , and it is not necessary to require that the conditions of Theorem 5.2 are met. Moreo ver, in view of the construction of normal and characteristic cycles (Section 2.7 ), it is clear that these transversalit y assumptions are satisﬁed when f is transverse to ψ in the sense of Theorem 2.7 . Theorem 5.4 (see Theorem 1 ) . L et f : X → Y b e a r e al analytic map b e- twe en r e al analytic manifolds and let ψ ∈ CF ( Y ) . If any one of the fol lowing two c onditions is satisﬁe d: OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 25 (i) f is a submersion, or (ii) f is an immersion which is tr ansverse to ψ , then the pul lb ack f ∗ [ ψ ] ∈ V −∞ ( X ) is wel l-deﬁne d and f ∗ [ ψ ] = [ f ∗ ψ ] . Pr o of. (i) Suppose ﬁrst that f is a submersion. In that case, the pullbac k is alw a ys w ell-deﬁned. By [ 7 , Prop. 3.3.3], we m ust prov e: C f ∗ ψ = f ∗ C ψ , (5.4) N ( f ∗ ψ ) = d f ∗ τ ∗ N ( ψ ) . (5.5) One can reduce to lo cal computations on an op en neigh b orhoo d U of a p oin t x ∈ X . Using the (real analytic) normal form of submersions, one can reduce to pro ving the result for X = R m + n , Y = R n and f : R m + n → R n the pro jection on the last co ordinates. In that case, the pullback has the simple form f ∗ ψ = 1 R m ⊠ ψ . In these local co ordinates, equality ( 5.4 ) b ecomes ob vious. F or ( 5.5 ), note that N ( 1 R m ) = 0, so that the result on the exterior pro duct (Theorem 4.2 ) ensures that: N ( f ∗ ψ ) = ( e p 1 ◦ π X ) ∗ J R m K ∩ i 2 ∗ p ∗ 2 N ( ψ ) = J P R m + n K ∩ i 2 ∗ p ∗ 2 N ( ψ ) = i 2 ∗ p ∗ 2 N ( ψ ) . The ﬁber pro duct X × Y P Y is isomorphic to R m × P R n and under this iden tiﬁcation the diagram P X d f ← − X × Y P Y τ − → P Y in v olved in the deﬁnition of the pullback coincides with the diagram P R m + n i 2 ← − R m × P R n p 2 − → P R n in v olved in the form ula for the exterior pro duct, hence the result. (ii) Supp ose now that f is an immersion which is transverse to ψ . W e m ust prov e that f ∗ C ψ = C f ∗ ψ and N ( f ∗ ψ ) = β ∗ α ∗ N ( ψ ). Since everything is lo cal, we can assume without loss of generalit y that f is a closed embedding whic h is transverse to ψ . T o do so, we use Schmid and Vilonen’s description of the pullbac k op er- ation on characteristic cycles which ensures that CC( f ∗ ψ ) = d f ∗ τ ∗ CC( ψ ) as Borel-Mo ore homology classes; see Theorem 2.8 . Again, the construc- tion of c haracteristic cycles and our assumption on the transversalit y of f to ψ imply that τ is transv erse to the strata of a subanalytic Whitney strat- ifcation of T ∗ Y deﬁning CC( ψ ). Therefore, Theorems 2.4 and 2.5 ensure that CC( f ∗ ψ ) = d f ∗ τ ∗ CC( ψ ) as subanalytic currents. F rom this last equal- it y and the relationship b et ween c haracteristic and normal cycles w e deduce that C f ∗ ψ = d f ∗ τ ∗ C ψ and cone( N ( f ∗ ψ )) = d f ∗ τ ∗ cone( N ( ψ )). The result will then follo w from: d f ∗ τ ∗ C ψ = f ∗ C ψ , d f ∗ τ ∗ cone( N ( ψ )) = cone ( β ∗ α ∗ N ( ψ )) . 26 ANDREAS BERNIG AND V ADIM LEBOVICI The ﬁrst equality is obvious. W e will pro ve the second in tw o steps. F or that, let us denote by e q : ( X × Y T ∗ Y ) \ T ∗ X Y → ^ X × Y P Y the map ( x, ξ ) 7→ ( x, [ ξ ] , [ d f ( ξ )]) and by e j : ( X × Y T ∗ Y ) \ T ∗ X Y  → X × Y T ∗ Y the canonical inclusion. Denote also the canonical m aps q X : T ∗ X \ 0 → → P X and j X : T ∗ X \ 0  → T ∗ X , and similarly for q Y and j Y . These maps ﬁt into the following comm utativ e diagram: P Y ^ X × Y P Y T ∗ Y \ 0 ( X × Y T ∗ Y ) \ 0 ( X × Y T ∗ Y ) \ T ∗ X Y T ∗ Y X × Y T ∗ Y α j Y q Y e j ′ τ ′ e j e q τ ′′ τ (5.6) where τ ′ , τ ′′ , e j ′ are the canonical inclusions. The ﬁrst step will b e to pro ve that, denoting T = N ( ψ ), we hav e τ ∗ cone( T ) = e j ∗ e q ∗ α ∗ T . The comm uta- tivit y of ( 5.6 ) implies that: e j ∗ e q ∗ α ∗ T = e j ′ ∗ τ ′′ ∗ ( τ ′ ◦ τ ′′ ) ∗ q ∗ Y T . By our transversalit y assumption, w e hav e supp( τ ′∗ q ∗ Y T ) ⊂ ( X × Y T ∗ Y ) \ T ∗ X Y , so that: τ ′′ ∗ ( τ ′ ◦ τ ′′ ) ∗ q ∗ Y T = τ ′∗ q ∗ Y T . Therefore, using that the b ottom left square of ( 5.6 ) is Cartesian, we get: e j ∗ e q ∗ α ∗ T = e j ′ ∗ τ ′∗ q ∗ Y T = τ ∗ j Y ∗ q ∗ Y T = τ ∗ cone( T ) . The second step will to prov e that d f ∗ e j ∗ e q ∗ α ∗ T = cone( β ∗ α ∗ T ). T o do this, consider the Cartesian square: ^ ^ X × Y P Y ^ X × Y P Y T ∗ X \ 0 P X π ′ π ′′ β q X where the total space ^ ^ X × Y P Y is a smo oth manifold with b oundary of dimension dim( X ) + dim( Y ) giv en by: In t  ^ ^ X × Y P Y  = n ( x, [ ξ ] , ξ ′ ) ∈ ( X × Y P Y ) × ( T ∗ X \ 0) : d f ( ξ )  = 0 ; [ d f ( ξ )] = [ ξ ′ ] o , ∂  ^ ^ X × Y P Y  = P + ( T ∗ X Y ) × X ( T ∗ X \ 0) , the map π ′′ is the canonical pro jection and the map π ′ is ( x, [ ξ ] , ξ ′ ) 7→ ( x, [ ξ ] , [ ξ ′ ]). Note that the map ι : ( X × Y T ∗ Y ) \ T ∗ X Y  → ^ ^ X × Y P Y giv en OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 27 b y ( x, ξ ) 7→ ( x, [ ξ ] , d f ( ξ )) is an op en em b edding onto the interior of ^ ^ X × Y P Y . Therefore, we ha v e a commutativ e diagram: ^ X × P Y P X ( X × Y T ∗ Y ) \ T ∗ X Y ^ ^ X × Y P Y T ∗ X \ 0 X × Y T ∗ Y T ∗ X β e q e j ι π ′′ π ′ j X q X d f (5.7) Denoting T ′ = α ∗ T , we must prov e that d f ∗ e j ∗ e q ∗ T ′ = j X ∗ q ∗ X β ∗ T ′ . Using the commutativit y of ( 5.7 ), we ha ve: d f ∗ e j ∗ e q ∗ T ′ = j X ∗ π ′′ ∗ ι ∗ ι ∗ π ′∗ T ′ . Again, our transversalit y assumptions entails supp( π ′∗ T ′ ) ⊂ Int( ^ ^ X × Y P Y ), so that ι ∗ ι ∗ π ′∗ T ′ = π ′∗ T ′ . Using that the top right square of ( 5.7 ) is Carte- sian, we ﬁnally get: d f ∗ e j ∗ e q ∗ T ′ = j X ∗ π ′′ ∗ π ′∗ T ′ = j X ∗ q ∗ X β ∗ T ′ , whic h concludes the pro of. □ 5.3. Pro duct. In this section, we consider the pro duct of generalized v alu- ations deﬁned in [ 11 ]. Deﬁnition 5.5. Let ζ 1 , ζ 2 ∈ V −∞ ( X ) b e tw o generalized v aluations suc h that the diagonal embedding δ : X  → X × X is transverse to ζ 1 ⊠ ζ 2 ∈ V −∞ ( X × X ) in the sense of Theorem 5.2 . Then, the pr o duct of ζ 1 and ζ 2 is the generalized v aluation ζ 1 · ζ 2 ∈ V −∞ ( X ) deﬁned b y: ζ 1 · ζ 2 = δ ∗ ( ζ 1 ⊠ ζ 2 ) . The deﬁnition of the pro duct of generalized v aluations is deﬁned in [ 11 ] in terms of op erations on the curren ts ( C, T ). That the t wo deﬁnitions coincide follo ws from the facts that (i) the transv ersality assumption of Theorem 5.2 in the case of the diagonal em b edding coincides with the transv ersalit y as- sumption for the product in [ 11 , Thm. 8.3], (ii) the pro duct of generalized v aluations deﬁned in [ 11 , Thm. 8.3] restricts to the product of smo oth v al- uations, (iii) the pro duct of smo oth v aluations is deﬁned as a pullback of the exterior pro duct by the diagonal em b edding, (iv) smooth v aluations are dense in the space of generalized v aluations and (v) op erations on general- ized v aluations are sequentially contin uous ([ 7 , Sec. 2],[ 7 , Prop. 3.5.5]). Pr o of of The or em 2 . Alesker’s result on the restriction of the exterior prod- uct of generalized v aluations to constructible functions (Theorem 4.2 ) and our result on the restriction of the pullback (Theorem 1 ) imply the result for the restriction of the pro duct to constructible functions. □ 28 ANDREAS BERNIG AND V ADIM LEBOVICI 6. Pushfor w ard and convolution In this section, we recall the deﬁnition of the pushforward of generalized v aluations introduced in [ 7 ] with a correction to the formula for immersions. Then, we pro v e that the pushforw ard of generalized v aluations restricts on constructible functions to the pushforward coming from sheaf theory under mild transversalit y assumptions. 6.1. Deﬁnition. W e follow closely the exp osition and the notations of [ 7 ]. The pushforward of generalized v aluations is deﬁned as the adjoint of the pullbac k of smo oth v aluations using Alesker-P oincar´ e duality . More pre- cisely , consider a real analytic map f : X n → Y m b et ween orien ted real an- alytic manifolds and ζ ∈ V −∞ ( X ) represen ted by the pair of curren ts ( C, T ). The pushforward f ∗ ζ ∈ V −∞ ( Y ) is deﬁned via the formula: Z Y f ∗ ζ · ψ = Z X ζ · f ∗ ψ , for all ψ ∈ V ∞ c ( Y ) , whenev er it make sense. Immersions. Supp ose that f is an immersion and that n < m , for otherwise the description is trivial. In that case, the pullbac k of a smo oth v alua- tion ψ ∈ V ∞ ( Y ) b y f is again smo oth, and it given for an y P ∈ P ( X ) b y f ∗ ψ ( P ) = ψ ( f ( P )). Moreo v er, recall the maps: P X β ← − ^ X × Y P Y α − → P Y from ( 5.3 ) and consider the natural maps: X β ′ ← − P + ( T ∗ X Y ) α ′ − → P Y . If ψ ∈ V ∞ c ( Y ) is given by the pair of diﬀeren tial forms ( φ, ω ) ∈ Ω n ( Y ) ⊕ Ω n − 1 ( P Y ) as in ( 2.5 ), then [ 7 , Prop. 3.1.2] ensures that the pullback f ∗ ψ is describ ed by the pair ( β ′ ∗ α ′∗ ω , β ∗ α ∗ ω ). It follo ws then from the deﬁnition of the pushforward that: ⟨ f ∗ ζ , ψ ⟩ = ⟨ ζ , f ∗ ψ ⟩ =  C, β ′ ∗ α ′∗ ω  + ⟨ T , β ∗ α ∗ ω ⟩ =  α ′ ∗ β ′∗ C + α ∗ β ∗ T , ω  . Therefore, we can correct a formula from [ 7 , Prop. 3.4.2]: Prop osition 6.1. If f : X n → Y m is an immersion with n < m , then the gener alize d valuation f ∗ ζ ∈ V −∞ ( Y ) is r epr esente d by the p air of curr ents: ( C ′ , T ′ ) =  0 , α ∗ β ∗ T + α ′ ∗ β ′∗ C  . (6.1) Submersions. When f is a submersion which is proper on the supp ort of ζ , t w o rem arks are in order. First, since the pullbac k of a smo oth v aluation b y a submersion ma y not b e smo oth, the pro of that such a deﬁnition makes sense requires to chec k that the pro duct of generalized v aluations ζ · f ∗ ψ is well-deﬁned. Denoting WF( ζ ) = (Λ , Γ), this will b e the case when Γ ∩ T ∗ X × Y P Y P X = ∅ ; see [ 7 , Sec. 3.6]. Second, the pushforward of generalized v aluations is deﬁned in lo c. cit. for prop er submersions or for compactly OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 29 supp orted generalized v aluations. Y et, the deﬁnition clearly mak e sense when the map f is prop er on the supp ort of ζ . Moreo v er, denoting by ( C ′ , T ′ ) the pair of currents representing f ∗ ζ , it is pro v en in [ 12 , Prop. 4.6] that: T ′ = τ ∗ d f ∗ T . (6.2) More precisely , lo c. cit. pro ves this formula when f is prop er, but the assumption that f is prop er on the supp ort of ζ is suﬃcien t once again. Finally , w e point out that the description of C ′ is left as an op en question in lo c cit.. 6.2. Restriction to constructible functions. In con trast to the pullbac k situation, the deﬁnition of pushforw ard of generalized v aluations by sub- mersions require a condition on w av e fron t sets ev en when the generalized v aluation come from a constructible function. This is due to the fact that the curren ts associated to the generalized v aluation f ∗ ψ for ψ ∈ V ∞ c ( Y ) are not subanalytic. As a consequence, we require a transv ersalit y assumption with resp ect to an angular stratiﬁcation of the normal cycle. Theorem 6.2 (see Theorem 3 ) . L et ϕ ∈ CF ( X ) and let f : X → Y b e a r e al analytic map b etwe en r e al analytic manifolds. If any one of the fol lowing two c onditions is satisﬁe d: (i) f is an immersion, or (ii) f is a submersion which is pr op er on the supp ort of ϕ and such that d f : X × Y P Y  → P X is angularly tr ansverse to N ( ϕ ) , then f ∗ ϕ ∈ CF ( Y ) and f ∗ [ ϕ ] ∈ V −∞ ( Y ) ar e wel l-deﬁne d and f ∗ [ ϕ ] = [ f ∗ ϕ ] . Pr o of. Recall that ( C ′ , T ′ ) are asso ciated to f ∗ [ ϕ ] and denote ( C, T ) = ( C φ , N ( ϕ )) and ( C ′′ , T ′′ ) = ( C f ∗ φ , N ( f ∗ ϕ )). (i) Assume ﬁrst that f is an immersion. By additivit y and prop erness of f on the support of ϕ , w e can reduce to the case w ere f is a closed embedding with n < m , the case n = m b eing trivial. W e clearly hav e C ′′ = 0 = C ′ , so w e are left to prov e that: T ′′ = α ∗ β ∗ T + α ′ ∗ β ′∗ C. (6.3) Reducing to lo cal computations and using the (real analytic) normal form of immersions, one can reduce to the case where X = R n , Y = R n + p and f : R n → R n + p is the inclusion along the ﬁrst co ordinates x 7→ ( x, 0). In that case, w e hav e f ∗ ϕ = ϕ ⊠ 1 { 0 } p . W e are thus left to prov e that ( 6.3 ) follo ws from Theorem 4.2 . F ollo wing Section 4 , consider the diagrams of canonical maps P R n + p i 2 ← − R n × P R p p 2 − → P R p and P R n × P R p b P R n + p P R n + p Φ F . 30 ANDREAS BERNIG AND V ADIM LEBOVICI Let us also consider the manifold P 0 = P + ( T ∗ { 0 } p R p ) and the em b edding i : P 0  → P R p , the normal cycle T 0 = N ( 1 { 0 } p ) = i ∗ J P 0 K and denote b y e p : P R n + p → R n the composition of the canonical maps P R n + p → R n + p → R n . Then, Theorem 4.2 ensures that: T ′′ = F ∗ Φ ∗ ( T ⊠ T 0 ) + e p ∗ C ∩ i 2 ∗ p ∗ 2 T 0 . The result will then follo w from the equalities: F ∗ Φ ∗ ( T ⊠ T 0 ) = α ∗ β ∗ T , (6.4) e p ∗ C ∩ i 2 ∗ p ∗ 2 T 0 = α ′ ∗ β ′∗ C. (6.5) T o pro ve ( 6.4 ), consider the following comm utativ e diagram: P R n P R n × P 0 P R n × P R p In t  b P 0  b P 0 b P R n + p ^ R n × R n + p P R n + p P R n + p j q e α j ′′ ι e Φ e β Φ 0 j ′ F 0 Φ F β α (6.6) where j = id P R n × i , the smo oth manifold with boundary b P 0 and the maps Φ 0 and j ′ are deﬁned as the pullbac k of the ﬁb er bundle Φ : b P R n + p ↠ P R n × P R p b y j and all other maps are canonical maps. More precisely , follo wing the notations of Section 4 , the manifold b P 0 is the closure of the set { ( x 1 , 0 , [ ξ 1 : ξ 2 ] , [ ξ 1 ] , [ ξ 2 ]) } inside P R n + p × R n + p ( P R n × P R p ). Similarly , in the co ordinates of ( 5.2 ) the map ι is the ob vious map: ι ( x 1 , 0 , [ ξ 1 : ξ 2 ] , [ ξ 1 ] , [ ξ 2 ]) = ( x 1 , [ ξ 1 : ξ 2 ] , [ ξ 1 ]) , and the other maps of ( 6.6 ) are uniquely deﬁned b y the commutativit y of the diagram. Using that the top righ t square of ( 6.6 ) is Cartesian, we can compute: F ∗ Φ ∗ ( T ⊠ T 0 ) = F ∗ Φ ∗ j ∗ q ∗ T = F ∗ j ′ ∗ Φ ∗ 0 q ∗ T . (6.7) Arguing as in the pro of of Theorem 4.2 , w e hav e the equality of op erations on currents Φ ∗ 0 = j ′′ ∗ e Φ ∗ and β ∗ = ι ∗ e β ∗ . Indeed, for any compactly supp orted smo oth diﬀerential form ω on b P 0 the equality Φ 0 ∗ ω = e Φ ∗ j ′′∗ ω follows from the fact the ﬁb ers of Φ 0 are the closure of the ﬁbers of e Φ in b P 0 so that the in tegration along the ﬁb ers of Φ 0 of the smo oth form ω and the in tegration along the ﬁb ers of e Φ of the restriction j ′′∗ ω of ω to the in terior In t( b P 0 ) OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 31 coincide. A similar pro of holds for β ∗ = ι ∗ e β ∗ . Thus ( 6.7 ) implies: F ∗ Φ ∗ ( T ⊠ T 0 ) = F ∗ j ′ ∗ j ′′ ∗ e Φ ∗ q ∗ T = e α ∗ e β ∗ T = α ∗ ι ∗ e β ∗ T = α ∗ β ∗ T . T o pro ve ( 6.5 ), consider the following diagram: R n × P 0 P 0 R n P R n + p R n × P R p P R p q 2 i ′ α ′ β ′ i e p p 2 i 2 where the square is Cartesian up to a change of orientation b y ( − 1) n ( p − 1) (corresp onding to exchanging the tw o factors). Moreov er, one has that q ∗ 2 J P 0 K = ( − 1) n ( p − 1) J R n × P 0 K , so that: i 2 ∗ p ∗ 2 T 0 = ( − 1) n ( p − 1) i 2 ∗ i ′ ∗ q ∗ 2 J P 0 K = α ′ ∗ ( J R n × P 0 K ) . Therefore, we ha v e: e p ∗ C ∩ i 2 ∗ p ∗ 2 T 0 = e p ∗ C ∩ α ′ ∗ ( J R n × P 0 K ) = α ′ ∗  α ′∗ e p ∗ C ∩ J R n × P 0 K  = α ′ ∗ β ′∗ C. (ii) Assume that f is a submersion such that d f is angularly transverse to N ( ϕ ). This transv ersalit y assumption ensures that f ∗ [ ϕ ] is w ell-deﬁned b y [ 12 , Prop. 4.5]. Since f is prop er on supp( ϕ ), the pushforward f ∗ ϕ is also w ell-deﬁned. Again, using additivit y of the pushforward and the prop erness of f on the supp ort of ϕ , w e can reduce to the case where ϕ is supp orted in an op en subset where the submersion f can b e put in real analytic normal form. It is th us suﬃcient to prov e the result for X = R n , Y = R m and f : R n → R m is the pro jection on the ﬁrst co ordinates. No w, let us pro ve that T ′′ = τ ∗ d f ∗ T , i.e., that N ( f ∗ ϕ ) = τ ∗ d f ∗ N ( ϕ ). T o do so, we use the formula for the pushforw ard op eration on character- istic cycles (Theorem 2.6 ) which ensures that CC( f ∗ ϕ ) = τ ∗ d f ∗ CC( ϕ ) as Borel-Mo ore homology classes. Using the canonical metrics on X = R n and Y = R m (whic h amoun ts to ha v e c hosen lo cal real analytic metrics on X and Y ), we can iden tify the pro jectiv e bundles P X and P Y with unit cosphere bundles S ∗ X ⊂ T ∗ X and S ∗ Y ⊂ T ∗ Y , yielding a commutativ e 32 ANDREAS BERNIG AND V ADIM LEBOVICI diagram: P X X × Y P Y P Y T ∗ X X × Y T ∗ Y T ∗ Y ι X d f τ ι ι Y d f τ (6.8) Note that the square on the righ t of ( 6.8 ) is Cartesian. Therefore, we ha v e the equalities of Borel-Mo ore homology classes: ι ∗ Y CC( f ∗ ϕ ) = ι ∗ Y τ ∗ d f ∗ CC( ϕ ) = τ ∗ ι ∗ d f ∗ CC( ϕ ) = τ ∗ d f ∗ ι ∗ X CC( ϕ ) . Note that due to conicalit y , the map ι Y is transv erse to the curren t CC( f ∗ ϕ ) and ι X to CC( ϕ ). Moreov er, our transversalit y assumption ensures that the map d f is transverse to the current ι ∗ X CC( ϕ ) = N ( ϕ ), so that Theorems 2.4 and 2.5 ensure that w e ha ve ι ∗ Y CC( f ∗ ϕ ) = τ ∗ d f ∗ ι ∗ X CC( ϕ ) as subanalytic curren ts, that is, N ( f ∗ ϕ ) = τ ∗ d f ∗ N ( ϕ ). Therefore, the generalized v aluation f ∗ [ ϕ ] − [ f ∗ ϕ ] ∈ V −∞ ( Y ) is repre- sen ted by the pair of currents ( C ′ − C ′′ , 0). By additivit y , we can assume without loss of generalit y that X is connected, so that Theorem 2.11 en- sures f ∗ [ ϕ ] − [ f ∗ ϕ ] = λχ for some λ ∈ C . W e are th us left to prov e that λ = 0. It follo ws from the deﬁnitions that the support of f ∗ [ ϕ ] − [ f ∗ ϕ ] is included in f (supp( ϕ )); see e.g. [ 11 , Eq. (28)]. Therefore, if f (supp( ϕ ))  = Y , then we must ha ve λ = 0. More generally , if dim( Y ) > 0, we can use [ 6 , Prop. 6.2.1] to write ϕ = ϕ 1 + ϕ 2 with f (supp( ϕ j ))  = Y for j = 1 , 2. By additivit y of pushforward op erations, w e can reduce to the previous case and prov e again that λ = 0. If dim( Y ) = 0, we can assume Y = { pt } without loss of generality so that f ∗ ϕ is constan t equal to µ = R X ϕ d χ , and hence [ f ∗ ϕ ] = µ · χ ∈ V −∞ ( Y ). Similarly , w e hav e f ∗ [ ϕ ] = µ · χ by [ 6 , Prop. 8.3.1], so f ∗ [ ϕ ] = [ f ∗ ϕ ] once again. □ 6.3. Con v olution. W e recall the deﬁnition of the conv olution of generalized v aluations introduced in [ 12 ]. Let G b e a Lie group acting transitiv ely on a smo oth manifold X . As explained in the introduction, the manifolds G and X are then equipp ed with canonical real analytic structures and the action a : G × X → X is real analytic. The c onvolution of the generalized v aluations µ ∈ V −∞ ( G ) and ζ ∈ V −∞ ( X ) is deﬁned as µ ∗ ζ := a ∗ ( µ ⊠ ζ ) ∈ V −∞ ( X ), provided that the pushforw ard on the right-hand side is well- deﬁned. W e can now prov e our result on the restriction of the conv olution of generalized v aluations to constructible functions. Pr o of of The or em 4 . The conv olution of the constructible functions ϕ ∈ CF ( G ) and ψ ∈ CF ( X ) is deﬁned as so on as the action a : G × X → X is proper on the supp ort of ϕ ⊠ ψ , which is assumed in (ii) of Theorem 3 . OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 33 Therefore, under this assumption, the con v olution [ ϕ ] ∗ [ ψ ] is well-deﬁned in the sense of generalized v aluations and so is ϕ ∗ ψ in the sense of con- structible function s. The result then directly follows from Alesk er’s result on the exterior product (Theorem 4.2 ) and our result on the pushforw ard (The- orem 3 ). □ 7. Additive and mul tiplica tive kinema tic f ormulas In this section w e pro ve additiv e and multiplicativ e kinematic form ulas for constructible functions. As w e ha ve noted earlier, the con v olution product of constructible functions is a goo d replacemen t for the Minko wski sum of con v ex b odies. W e will pro ve suc h formulas in tw o diﬀeren t cases. The ﬁrst case is on a Euclidean vector space acting on itself by addition. W e consider v aluations that are translation-inv arian t and in v arian t with re- sp ect to a subgr oup of the orthogonal group that acts transitively on the unit sphere. In this case, additive kinematic form ulas for conv ex b o dies are w ell- kno wn. T o prov e their extension to the setting of constructible functions w e ﬁrst prov e a statement of indep enden t interest (Theorem 7.9 ) that relates compactly supp orted generalized v aluations, translation-inv arian t general- ized v aluations and their resp ectiv e conv olution products. It generalizes results of the ﬁrst named author with Alesker in the smo oth case [ 12 ] and with F aifman in the p olytopal case [ 17 ]. The second case is that of constructible functions on the Lie group S 3 . The con volution pro duct of tw o geo desically conv ex subsets of S 3 is in gen- eral not geo desically con vex. Hence, there are no multiplicativ e kinematic form ulas for geo desically con vex sets and w e are forced to state our form ulas with constructible functions. In both cases the con volution of compactly supp orted constructible func- tions is well-deﬁned. Ho wev er, the conv olution pro duct of the asso ciated generalized v aluations is only deﬁned under some conditions. W e will sho w that these conditions are satisﬁed for constructible functions in generic p o- sition. 7.1. Compactly supp orted versus translation-in v ariant v aluations. In order to distinguish b etw een R n and its dual, it will b e con venien t to write V := R n . Deﬁnition 7.1. Tw o compactly supp orted generalized v aluations ζ 1 , ζ 2 ∈ V −∞ c ( V ) are tr ansverse if there are closed conical sets Λ i ⊂ T ∗ V \ 0 and Γ i ⊂ T ∗ P V \ 0 with WF( ζ i ) ⊂ (Λ i , Γ i ) such that the follo wing condition holds: There are no ( x i , [ ξ ]) ∈ P V , i = 1 , 2 and η ∈ T ∗ [ ξ ] P + ( V ∗ ) ⊂ T ∗ ( x i , [ ξ ]) P V suc h that ( x 1 , [ ξ ] , η ) ∈ Γ 1 and ( x 2 , [ ξ ] , − η ) ∈ Γ 2 . Prop osition 7.2. If ζ 1 and ζ 2 ar e tr ansverse, then the c onvolution ζ 1 ∗ ζ 2 is wel l-deﬁne d. Mor e pr e cisely, denoting (Λ i , Γ i ) as in the pr evious deﬁnition, the c onvolution on c omp actly supp orte d smo oth valuations on V extends as 34 ANDREAS BERNIG AND V ADIM LEBOVICI a jointly se quential ly c ontinuous map V −∞ c , Λ 1 , Γ 1 ( V ) × V −∞ c , Λ 2 , Γ 2 ( V ) → V −∞ c ( V ) . Pr o of. With (Λ , Γ) as in [ 12 , Prop. 4.1], the exterior product extends to a join tly sequentially con tinuous map V −∞ c , Λ 1 , Γ 1 ( V ) × V −∞ c , Λ 2 , Γ 2 ( V ) → V −∞ c , Λ , Γ ( V × V ) . Moreo v er, the pushforward extends to a sequentially con tin uous map V −∞ c , Λ , Γ ( V × V ) → V −∞ c ( V ) , pro vided that Γ is disjoin t from T ∗ ( V × V ) × V P V P V × V [ 12 , Prop. 4.5]. T o c hec k this last condition, notice that the submanifold ( V × V ) × V P V consists of all p oin ts ( x 1 , x 2 , [ ξ : ξ ]) ∈ P V × V . Then, it is easily chec ked that Γ ( x 1 ,x 2 , [ ξ : ξ ]) = { ( ρ 1 , ρ 2 ) : ρ i ∈ Γ i | ( x i , [ ξ ]) } , where we use the injection T ∗ ( x 1 , [ ξ ]) P V ⊕ T ∗ ( x 2 , [ ξ ]) P V  → T ∗ ( x 1 ,x 2 , [ ξ : ξ ]) P V × V that is dual to the pro jection d (Φ ◦ F − 1 ) : T ( x 1 ,x 2 , [ ξ : ξ ]) P V × V ↠ T ( x 1 , [ ξ ]) P V ⊕ T ( x 2 , [ ξ ]) P V from [ 12 , Section 4.1]. Let us write ρ i = ρ ′ i + ρ ′′ i according to the decomp osition T ∗ ( x i , [ ξ ]) P V = T ∗ x i V ⊕ T ∗ [ ξ ] P + ( V ∗ ) . Then ( ρ 1 , ρ 2 ) v anishes on T ( x 1 ,x 2 , [ ξ : ξ ]) ( V × V ) × V P V if and only if ρ ′ 1 = ρ ′ 2 = 0 and ρ ′′ 1 = − ρ ′′ 2 . This will nev er b e the case when ( ρ 1 , ρ 2 ) ∈ Γ ( x 1 ,x 2 , [ ξ : ξ ]) b y our transversalit y assumptions, hence the result. □ W e w ant to compare the conv olution of compactly supp orted v aluations with the conv olution of translation-inv arian t v aluations and hav e to recall some deﬁnitions ﬁrst. W e refer to [ 10 , 60 ] for more information. The set of conv ex b o dies in V will b e denoted by K n . A v aluation on V is a map µ : K n → R that satisﬁes µ ( K ∪ L ) + µ ( K ∩ L ) = µ ( K ) + µ ( L ) whenev er K , L, K ∪ L ∈ K n . The vector space of translation-inv ariant and con tin uous v aluations is denoted b y V al. It admits a decomp osition V al = L n k =0 V al k , where V al k denotes the subspace of k -homogeneous elements. The space V al is a Banach space with a natural action of the group GL( n ). The smo oth vectors of this ac tion form a dense subspace V al ∞ that admits a F r ´ echet space top ology . A smo oth translation-inv arian t v aluation µ ∈ V al ∞ can b e written as µ ( K ) = Z K φ + Z N ( K ) ω , OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 35 where N ( K ) is the normal cycle of the conv ex b o dy K (see for instance [ 52 ]) and φ ∈ Ω n ( V ) tr and ω ∈ Ω n − 1 ( P V ) tr are translation-in v ariant forms. Com- paring with ( 2.5 ) it is easy to show that V al ∞ ( V ) = ( V ∞ ( V )) tr , i.e. that the tw o notions of “smo oth and translation-in v ariant v aluations” coincide. The con v olution of smooth translation-inv arian t v aluations is a commu- tativ e and asso ciative pro duct on V al ∞ ⊗ Dens( V ∗ ) that is characterized by the prop ert y [v ol( • + A ) ⊗ vol ∗ ] ∗ [vol( • + B ) ⊗ vol ∗ ] = [v ol( • + A + B ) ⊗ v ol ∗ ] for all con vex bo dies A, B with smo oth b oundary of p ositive curv ature. Alesk er, using the pro duct of smooth translation-in v arian t v aluations, constructed in [ 6 , Thm. 6.1.1] an injectiv e map PD : V al ∞ ( V )  → (V al ∞ ( V )) ∗ ⊗ Dens( V ) =: V al −∞ ( V ) with dense image. Elemen ts on the right-hand side are called translation- in v ariant generalized v aluations. Similar to Theorem 2.10 they can b e uniquely represented b y a pair of currents ( C, T ), where C is a translation- in v ariant n -curren t on V (i.e. a multiple of the integration curren t) and T is a translation-in v ariant Legendrian ( n − 1)-cycle on P V . The wa ve front set of a v aluation is deﬁned as the w av e front set of T , whic h can b e represen ted as a closed conical subset of T ∗ P + ( V ∗ ) \ 0 by translation-in v ariance. The subset of V al −∞ ( V ) consisting of v aluations with w av e fron t set contained in some closed conical set Γ ⊂ T ∗ P + ( V ∗ ) \ 0 is denoted by V al −∞ Γ ( V ). It is endo w ed with the H¨ ormander topology . Deﬁnition 7.3. Tw o elements µ 1 , µ 2 ∈ V al −∞ ( V ) ⊗ Dens( V ∗ ) are tr ansverse if there are closed conical sets Γ i ⊂ T ∗ P + ( V ∗ ) \ 0 with WF( µ i ) ⊂ Γ i suc h that Γ 1 ∩ s (Γ 2 ) = ∅ where s : T ∗ P + ( V ∗ ) → T ∗ P + ( V ∗ ) is the multiplication b y − 1 in the ﬁb ers. Prop osition 7.4 ([ 17 , Prop. 4.7]) . If µ 1 , µ 2 ∈ V al −∞ ( V ) ⊗ Dens( V ∗ ) ar e tr ansverse, then the c onvolution pr o duct µ 1 ∗ µ 2 is deﬁne d. Mor e pr e cisely, if Γ 1 , Γ 2 satisfy the c ondition of the pr evious deﬁnition, then the c onvolution extends as a jointly se quential ly c ontinuous map  V al −∞ Γ 1 ( V ) ⊗ Dens( V ∗ )  ×  V al −∞ Γ 2 ( V ) ⊗ Dens( V ∗ )  → V al −∞ ( V ) ⊗ Dens( V ∗ ) . T o motiv ate the next theorem, let us recall t w o results related to conv olu- tion of v aluations. W e denote by vol an y choice of Lebesgue measure on V , and v ol ∗ the dual Lebesgue measure on V ∗ , so that v ol ⊗ vol ∗ is indep enden t of the c hoices. Theorem 7.5 ([ 12 , Thm. 2]) . Deﬁne a map e F : V ∞ c ( V ) → V ∞ ( V ) tr ⊗ Dens( V ∗ ) , ζ 7→ Z V ζ ( • − x ) d vol( x ) ⊗ v ol ∗ 36 ANDREAS BERNIG AND V ADIM LEBOVICI and let F : V ∞ c ( V ) → V al ∞ ( V ) ⊗ Dens( V ∗ ) b e the c omp osition of e F with the natur al isomorphism V ∞ ( V ) tr ⊗ Dens( V ∗ ) ∼ = V al ∞ ( V ) ⊗ Dens( V ∗ ) . Then F is a surje ctive homomorphism with r esp e ct to the c onvolution pr o ducts on b oth sides. The p olytop e algebra Π( V ) is the algebra generated by all symbols b P , where P is a p olytop e, sub ject to the relations b P + b Q = \ P ∪ Q + \ P ∩ Q whenev er P , Q, P ∪ Q are p olytop es, and \ P + x = b P for all x ∈ V . The pro duct is deﬁned on generators by b P · b Q := \ P + Q , see [ 48 ]. Note that the indicator function of a p olytop e is a constructible function, hence it deﬁnes a generalized v aluation [ 1 P ]. This is why we deviate from the standard notation which uses [ P ] instead of b P . Theorem 7.6 ([ 17 , Thm. 2.5]) . The map on p olytop es M ( P ) := [ µ 7→ µ ( P )] ∈ V al ∞ ( V ) ∗ ∼ = V al −∞ ( V ) ⊗ Dens( V ∗ ) extends to an inje ctive map M : Π( V ) → V al −∞ ( V ) ⊗ Dens( V ∗ ) . Mor e over, if x, y ∈ Π( V ) ar e in gener al p osition, then M ( x ) and M ( y ) ar e tr ansverse and M ( x · y ) = M ( x ) ∗ M ( y ) . In the following we will simply write µ ( ζ ) for the natural pairing b e- t w een µ ∈ V al ∞ ( V ) ⊂ V ∞ ( V ) and ζ ∈ V −∞ c ( V ). W e deﬁne the map M : V −∞ c ( V ) → V al −∞ ( V ) ⊗ Dens( V ∗ ) ∼ = V al ∞ ( V ) ∗ , ζ 7→ [ µ 7→ µ ( ζ ) , µ ∈ V al ∞ ( V )] . Note that for p olytop es, we ha ve M ([ 1 P ]) = M ( P ), where the righ t hand side refers to the map from Theorem 7.6 , which justiﬁes the use of the same letter. Next, we deﬁne a map f M : V −∞ c ( V ) → V −∞ ( V ) tr ⊗ Dens( V ∗ ) , ζ 7→ Z V ζ ( • − x ) d vol( x ) ⊗ v ol ∗ . Here ζ ( • − x ) ∈ V −∞ c ( V ) is deﬁned as ( t x ) ∗ ζ for the translation t x : V → V . OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 37 Lemma 7.7. The tr ansp ose F ∗ of the map F fr om The or em 7.5 is an iso- morphism. The fol lowing diagr am c ommutes V −∞ c ( V ) V al −∞ ( V ) ⊗ Dens( V ∗ ) V −∞ ( V ) tr ⊗ Dens( V ∗ ) V ∞ c ( V ) V al ∞ ( V ) ⊗ Dens( V ∗ ) V ∞ ( V ) tr ⊗ Dens( V ∗ ) M f M F ∗ ∼ = F e F ∼ = . Pr o of. The statemen t ab out F ∗ , as w ell as the comm utativit y of the righ t hand square is prov ed in [ 17 , Prop. 2.5]. Let us show that f M = F ∗ ◦ M . F or p olytop es this w as sho wn in [ 17 , Lemma 3.2] and the general case follows the same lines. Let ζ ∈ V −∞ c ( V ) and β ∈ V ∞ c ( V ). Then ⟨ F ∗ ◦ M ( ζ ) , β ⟩ = ⟨ M ( ζ ) , F ( β ) ⟩ = F ( β )( ζ ) = e F ( β )( ζ ) = Z V ⟨ β ( • − x ) , ζ ⟩ dvol( x ) ⊗ v ol ∗ = Z V ⟨ β , ζ ( • + x ) ⟩ dvol( x ) ⊗ v ol ∗ = Z V ⟨ β , ζ ( • − x ) ⟩ dvol( x ) ⊗ v ol ∗ = ⟨ f M ( ζ ) , β ⟩ . It is clear from the deﬁnitions that the restriction of f M to V ∞ c ( V ) equals e F , whic h implies the commutativit y of the left hand square. □ Lemma 7.8. L et (Λ 1 , Γ 1 ) with Λ 1 ⊂ T ∗ V \ 0 , Γ 1 ⊂ T ∗ P V \ 0 . We set b Γ :=  ([ ξ ] , η ′′ ) ∈ T ∗ P + ( V ∗ ) : ∃ e x ∈ V such that (0 , η ′′ ) ∈ Γ 1 | ( e x, [ ξ ])  . Then, for ζ ∈ V −∞ c , Λ 1 , Γ 1 ( V ) one has WF( M ( ζ )) ⊂ b Γ and the map M : V −∞ c , Λ 1 , Γ 1 ( V ) → V al −∞ b Γ ( V ) ⊗ Dens( V ∗ ) is se quential ly c ontinuous. Pr o of. T o simplify the notation, w e ﬁx a Leb esgue measure on V and use it to iden tify Dens( V ∗ ) ∼ = C . Let ( C 1 , T 1 ) be the pair of curren ts corresponding 38 ANDREAS BERNIG AND V ADIM LEBOVICI to ζ . In particular, we ha ve WF( T 1 ) ⊂ Γ 1 . Let τ ′ : P V × V → P V b e the map ( x, [ ξ ] , y ) 7→ ( x + y , [ ξ ]) and set T := ( τ ′ ) ∗ ( T 1 ⊠ vol) . Fix ( x, [ ξ ] , y ) ∈ P V × V and v ∈ T ( x, [ ξ ]) P V . Then ( v , 0) ∈ T ( x, [ ξ ] ,y ) ( P V × V ) and d τ ′ ( v , 0) = dt y ( v ), where t y : P V → P V is translation b y y . Let ω b e a compactly supp orted ( n − 1)-form on P V . W e can consider t ∗ y ω as an ( n − 1)-form on P V × V . More precisely , its v alue at the p oint ( x, [ ξ ] , y ) is given by ( t ∗ y ω ) | ( x, [ ξ ]) ∈ ∧ n − 1 T ∗ ( x, [ ξ ]) P V ⊂ ∧ n − 1 T ∗ ( x, [ ξ ] ,y ) ( P V × V ). It follo ws from the ab ov e that ( τ ′ ) ∗ ω | ( x, [ ξ ] ,y ) = t ∗ y ω | ( x, [ ξ ]) mo dulo the ideal generated by the forms dy j for j = 1 , . . . , n . Since T 1 ⊠ v ol v anishes on this ideal, we obtain ⟨ T , ω ⟩ = ⟨ T 1 ⊠ vol , ( τ ′ ) ∗ ω ⟩ =  T 1 , Z V t ∗ y ω dy  =  Z V ( t y ) ∗ T 1 dy , ω  . Therefore, the current T is the Legendrian cycle corresp onding to the gen- eralized v aluation f M ( ζ ) = R V t y ∗ ζ dy . With vol b eing a smo oth current, [ 27 , Prop. 2.16] implies that WF( T 1 ⊠ vol) | ( x, [ ξ ] ,y ) = { ( κ, 0) : κ ∈ WF( T 1 ) | ( x, [ ξ ]) } ⊂ { ( κ, 0) : κ ∈ Γ 1 | ( x, [ ξ ]) } . W e can write κ = ( κ ′ , κ ′′ ) with κ ′ ∈ T ∗ x V and κ ′′ ∈ T ∗ [ ξ ] P + ( V ∗ ). It follo ws from [ 27 , Prop. 2.15] that WF( T ) | ( x, [ ξ ]) ⊂ n ( η ′ , η ′′ ) ∈ T ∗ ( x, [ ξ ]) P V : ∃ e x ∈ V , ( κ ′ , κ ′′ ) ∈ Γ 1 | ( e x, [ ξ ]) , ( dτ ′ | ( e x, [ ξ ] ,x − e x ) ) ∗ ( η ′ , η ′′ ) = ( κ ′ , κ ′′ , 0) o . Since ( dτ ′ | ( e x, [ ξ ] ,x − e x ) ) ∗ ( η ′ , η ′′ ) = ( η ′ , η ′′ , η ′ ), the last set is precisely the image of b Γ under the embedding T ∗ P + ( V ∗ ) \ 0  → T ∗ P V \ 0 and hence the w a ve fron t set of M ( ζ ) is contained in b Γ. Since exterior pro duct and pushforw ard of currents are sequen tially con tin uous operations with resp ect to the corresp onding H¨ ormander topologies [ 27 , Props. 2.15 and 2.16], the sequen tial contin uit y of M follows. □ The next theorem generalizes Theorems 7.5 and 7.6 . Theorem 7.9. If ζ 1 , ζ 2 ∈ V −∞ c ( V ) ar e tr ansverse, then M ( ζ 1 ) and M ( ζ 2 ) ar e tr ansverse as wel l and M ( ζ 1 ∗ ζ 2 ) = M ( ζ 1 ) ∗ M ( ζ 2 ) . Pr o of. Let Λ i ⊂ T ∗ V \ 0 and Γ i ⊂ T ∗ P V \ 0 be as in Deﬁnition 7.1 . Let b Γ i ⊂ T ∗ P + ( V ∗ ) \ 0 b e the corresp onding sets deﬁned in Theorem 7.8 . Clearly b Γ 1 and b Γ 2 satisfy the condition from Deﬁnition 7.3 . This shows that M ( ζ 1 ) and M ( ζ 2 ) are transv erse. Using [ 11 , Lemma 8.2] we can appro ximate ζ j in V −∞ c , Λ j , Γ j ( V ) b y a sequence of smo oth v aluations ζ i j ∈ V ∞ c ( V ) for j = 1 , 2. OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 39 By Lemma 7.8 , M ( ζ i j ) conv erges to M ( ζ j ) in V al −∞ b Γ j ( V ) ⊗ Dens( V ∗ ) and hence Theorem 7.4 implies that M ( ζ i 1 ) ∗ M ( ζ i 2 ) conv erges to M ( ζ 1 ) ∗ M ( ζ 2 ) in V al −∞ ( V ) ⊗ Dens( V ∗ ). On the other hand, ζ i 1 ∗ ζ i 2 con v erges to ζ 1 ∗ ζ 2 in V −∞ c ( V ) by Theorem 7.2 . Theorem 7.8 (with Γ 1 = T ∗ P V \ 0) then implies that M ( ζ i 1 ∗ ζ i 2 ) con verges to M ( ζ 1 ∗ ζ 2 ) in V al −∞ ( V ) ⊗ Dens( V ∗ ). Since M ( ζ i 1 ) ∗ M ( ζ i 2 ) = F ( ζ i 1 ) ∗ F ( ζ i 2 ) = F ( ζ i 1 ∗ ζ i 2 ) = M ( ζ i 1 ∗ ζ i 2 ) b y Theorem 7.5 , the pro of is complete. □ 7.2. Constructible functions in general p osition. The aim of this sub- section is to prov e the following result. Prop osition 7.10. L et G ⊂ O( n ) b e a Lie sub gr oup that acts tr ansitively on the unit spher e and ϕ 1 , ϕ 2 ∈ CF( R n ) b e c omp actly supp orte d. Then for almost al l ( g 1 , g 2 ) ∈ G × G , the c onstructible functions ( g 1 ) ∗ ϕ 1 and ( g 2 ) ∗ ϕ 2 satisfy the assumptions of Cor ol lary 4 . In p articular, the gener alize d valuations [( g 1 ) ∗ ϕ 1 ] and [( g 2 ) ∗ ϕ 2 ] ar e tr ansverse in the sense of The or em 7.1 . Pr o of. W e wan t to show that for almost all ( g 1 , g 2 ) the normal cycle of N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) is transversal to the diﬀerential of the addition map, i.e. to the set X × Y P Y , where X = R n × R n and Y = R n . Note that X × Y P Y ⊂ P X \ ( M 1 ∪ M 2 ) consists of all tuples ( x 1 , x 2 , [ η : η ]) ∈ P X with x 1 , x 2 ∈ R n and η ∈ T ∗ x 1 + x 2 R n \ { 0 } . First we construct an angular stratiﬁcation of the normal cycle. T ake angular stratiﬁcations of N ( ϕ 1 ) and N ( ϕ 2 ). Then a stratiﬁcation of the normal cycle N ( ϕ 1 ⊠ ϕ 2 ) is giv en by all b S of the form b S := { ( x 1 , x 2 , [ ξ 1 : λξ 2 ]) : ( x i , [ ξ i ]) ∈ S i for i = 1 , 2 , λ ∈ R + } , (7.1) where S i is a stratum of N ( ϕ i ). Similarly , a stratiﬁcation of N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) is given b y all strata of the form { ( x 1 , x 2 , [ ξ 1 : λξ 2 ]) , ( x i , [ ξ i ]) ∈ ( g i , g i ) S i for i = 1 , 2 , λ ∈ R + } , (7.2) Using the Euclidean metric, w e can iden tify P Y with R n × S n − 1 , by send- ing ( x, v ) ∈ R n × S n − 1 to ( x, [ ξ ]) where ξ = ⟨ v , •⟩ . W e can iden tify P X \ ( M 1 ∪ M 2 ) with R n × R n × S n − 1 × S n − 1 ×  0 , π 2  b y mapping ( x 1 , x 2 , v 1 , v 2 , ϕ ) to ( x 1 , x 2 , [cos( ϕ ) ξ 1 : sin( ϕ ) ξ 2 ]) ∈ P X \ ( M 1 ∪ M 2 ), where ξ i = ⟨ v i , •⟩ . Up to this identiﬁcation, our stratiﬁcation is the product of the angular stratiﬁcation of ϕ 1 , the one of ϕ 2 and the obvious stratiﬁcation of the op en in terv al  0 , π 2  , hence it is angular by Lemma 3.2 . Strictly sp eaking, since the in terv al is op en, we only obtain that the stratiﬁcation is angular outside any op en neigh b orho o d of M 1 ∪ M 2 , whic h is clearly suﬃcien t for our purpose. Fix S 1 , S 2 and b S as ab ov e. W e claim that the map Ξ : b S × G × G → P X ( x 1 , x 2 , [ ξ 1 : λξ 2 ] , g 1 , g 2 ) 7→ ( g 1 x 1 , g 2 x 2 , [ g 1 ξ 1 : λg 2 ξ 2 ]) 40 ANDREAS BERNIG AND V ADIM LEBOVICI is transversal to X × Y P Y . F or this, we use our iden tiﬁcations and consider a p oin t p = ( x 1 , x 2 , v 1 , v 2 , ϕ, g 1 , g 2 ) ∈ Ξ − 1 ( X × Y P Y ) and let q = Ξ( p ) = ( g 1 x 1 , g 2 x 2 , v , v , ϕ ) with v = g 1 v 1 = g 2 v 2 . Then T x 1 R n and T x 2 R n are con tained in T q ( X × Y P Y ). By deﬁnition of b S , the sub- space T φ  0 , π 2  b elongs to the image of d Ξ | p . By our assumption on G , the map g ∈ G 7→ g v ∈ S n − 1 is a submersion. Therefore the image of d Ξ | p con- tains a v ector of the form ( ∗ , 0 , u, 0 , 0) and a vector of the form (0 , ∗ , 0 , u, 0) for each u ∈ T v S n − 1 . This prov es the claim. By Thom’s transv ersality theorem [ 39 ], it follo ws that for almost all ( g 1 , g 2 ) the image of Ξ( • , g 1 , g 2 ) is transversal to X × Y P Y . This image is precisely the s et ( 7.2 ). Doing this for all strata, w e obtain that for almost all ( g 1 , g 2 ), all strata of N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) are transv ersal to X × Y P Y . Giv en such a pair ( g 1 , g 2 ), we set ψ i := ( g i ) ∗ ϕ i ∈ CF c ( R n ). Then ψ 1 and ψ 2 satisfy the assumption of Corollary 4 . Let us show that ψ 1 and ψ 2 are also transv ersal in the sense of Deﬁnition 7.1 . Let Γ i b e the wa ve fron t set of N ( ψ i ). Supp ose that there are ( x 1 , [ ξ ] , η ) ∈ Γ 1 and ( x 2 , [ ξ ] , − η ) ∈ Γ 2 with η ∈ T ∗ [ ξ ] P + ( V ∗ ) ⊂ T ∗ ( x i , [ ξ ]) P V . Let S i b e the stratum of N ( ψ i ) that contains ( x i , [ ξ ]). By angularity of the stratiﬁcations, η v anishes on T ( x i , [ ξ ]) S i . The stratum of N ( ψ 1 ⊠ ψ 2 ) that contains ( x 1 , x 2 , [ ξ : ξ ]) is b S from ( 7.1 ). Since ( η , − η ) v anishes on T ( x 1 ,x 2 , [ ξ : ξ ]) X × Y P Y and on T ( x 1 ,x 2 , [ ξ : ξ ]) b S , w e get a contradiction to the transv ersality of X × Y P Y and b S . □ Remark 7.11. It is clear that the k ernel of the map M con tains ϕ − t ∗ x ϕ for all x ∈ V . In view of the results on the p olytop e algebra men tioned ab o ve, it is tempting to conjecture that the k ernel of M is generated b y suc h elemen ts. W e could then consider the algebra of constructible functions mo dulo translations and inject it into V al −∞ ( V ) ⊗ Dens( V ∗ ). Ho w ever, the conjecture is not true, as the following easy example shows. T ake n = 2 and let ϕ 1 = 2 · 1 S 1 b e twice the indicator function of the unit circle and ϕ 2 = 1 S 1 2 the indicator function of the circle with radius 2. If µ is a smo oth translation in v arian t v aluation on V , then we can decomp ose it into homogeneous components µ = µ 0 + µ 1 + µ 2 , with µ 0 a m ultiple of the Euler c haracteristic and µ 2 a multiple of the Leb esgue measure. Clearly Euler c haracteristic and Leb esgue measure b oth v anish on ϕ 1 and ϕ 2 . Since µ 1 is 1-homogeneous, w e ha v e µ 1 ( ϕ 2 ) = 2 µ 1 ( S 1 ) = µ 1 ( ϕ 1 ). Hence ϕ 1 − ϕ 2 is in the kernel of M . On the other hand, ϕ 1 − ϕ 2 can not b e written as a ﬁnite sum of functions of the form ϕ − t ∗ x ϕ , since the curv ature of each arc of S 1 is 1, while the curv ature of each arc of S 1 2 is 1 2 . 7.3. Additiv e kinematic form ulas. In this section, w e assume that G is a closed subgroup of O( n ) that acts transitively on the unit sphere. Using the Leb esgue measure of R n w e can iden tify Dens( V ) ∼ = C . The connected OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 41 groups acting eﬀectiv ely and transitiv ely on the unit sphere w ere classiﬁed b y Montgomery-Samelson and Borel [ 23 , 49 ]: SO( n ) , U( n/ 2) , SU( n/ 2) , Sp( n/ 4) , Sp( n/ 4) · U(1) , Sp( n/ 4) · Sp(1) , G 2 , Spin(7) , Spin(9) . It was observed by Alesk er [ 5 ] that the vector space V al G of contin uous, translation-in v ariant and G -inv arian t v aluations on V is ﬁnite-dimensional and that all such v aluations are smo oth. Giv en a basis µ 1 , . . . , µ N of V al G , there are additive kinematic formulas: Z G µ i ( K + g L ) dg = X k,l c i k,l µ k ( K ) µ l ( L ) , for all K, L ∈ K n . (7.3) It is a diﬃcult problem to determine such a basis and to compute the constan ts c i k,l explicitly . In the case G = O( n ) this can b e achiev ed using Hadwiger’s theorem and the template metho d (i.e. computing the constants b y plugging in enough diﬀerent examples). F or the other kno wn cases from the list ab ov e, in particular for G = U( n/ 2), the following link b et ween the additiv e kinematic form ulas and the con volution pro duct was the key ingredien t. Let us deﬁne a map a : V al G → V al G ⊗ V al G b y a ( µ i ) = P k,l c i k,l µ k ⊗ µ l . It is easily c heck ed that this map do es not dep end on the c hoice of a basis, as we ha v e a ( µ )( K, L ) = R G µ ( K + g L ) dg . It w as shown in [ 19 ] that the restriction of PD to G -in v ariant v aluations, denoted by PD G , is an isomorphism and that the following diagram com- m utes: V al G V al G ⊗ V al G V al G ∗ V al G ∗ ⊗ V al G ∗ a PD G PD G ⊗ PD G c ∗ (7.4) where c ∗ denotes the adjoint of the conv olution pro duct c : V al G ⊗ V al G → V al G . The aim of this section is to prov e a v ersion of these formulas for com- pactly supp orted constructible functions. Note that we can ev aluate a v alu- ation µ ∈ V al G on a compactly supp orted constructible function b y Example 2.13 . Theorem 7.12 (see Theorem 5 ) . L et G ⊂ O( n ) b e a close d sub gr oup that acts tr ansitively on the spher e. The formula ( 7.3 ) holds for ϕ 1 , ϕ 2 ∈ CF c ( R n ) with the same c onstants: Z G µ i ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = X k,l c i k,l µ k ( ϕ 1 ) µ l ( ϕ 2 ) . 42 ANDREAS BERNIG AND V ADIM LEBOVICI Pr o of. Let ϕ 1 , ϕ 2 ∈ CF c ( R n ) and µ i ∈ V al G . Then by G -inv ariance of µ i , Z G µ i ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = Z G × G µ i (( g 1 ) ∗ ϕ 1 ∗ ( g 2 ) ∗ ϕ 2 ) dg 1 dg 2 = Z G × G ⟨ µ i , M (( g 1 ) ∗ ϕ 1 ∗ ( g 2 ) ∗ ϕ 2 ) ⟩ dg 1 dg 2 = Z G × G ⟨ µ i , M (( g 1 ) ∗ ϕ 1 ) ∗ M (( g 2 ) ∗ ϕ 2 ) ⟩ dg 1 dg 2 (Cor. 4 , Thm. 7.9 , Prop. 7.10 ) =  µ i , Z G M (( g 1 ) ∗ ϕ 1 ) dg 1 ∗ Z G M (( g 2 ) ∗ ϕ 2 ) dg 2  =  c ∗ ◦ PD G ( µ i ) , Z G M (( g 1 ) ∗ ϕ 1 ) dg 1 ⊗ Z G M (( g 2 ) ∗ ϕ 2 ) dg 2  ( 7.4 ) =  (PD G ⊗ PD G ) ◦ a ( µ i ) , Z G M (( g 1 ) ∗ ϕ 1 ) dg 1 ⊗ Z G M (( g 2 ) ∗ ϕ 2 ) dg 2  = a ( µ i )( ϕ 1 ⊗ ϕ 2 ) (b y ( G × G )-in v ariance of a ( µ i )) = X k,l c i k,l µ k ( ϕ 1 ) µ l ( ϕ 2 ) . □ 7.4. Multiplicativ e kinematic formula on S 3 . In this section, we con- sider the case of the action of G = SO(4) on the 3-sphere. Prop osition 7.13. L et ϕ 1 , ϕ 2 ∈ CF( S 3 ) . Then for almost al l ( g 1 , g 2 ) ∈ G × G , the gener alize d valuations ( g 1 ) ∗ ϕ 1 and ( g 2 ) ∗ ϕ 2 satisfy the assumptions of The or em 4 . Pr o of. The general strategy is as in Prop osition 7.2 , with some minor ad- justmen ts. Iden tifying S 3 with unit quaternions, the 3-sphere H := S 3 is a Lie group. W e write L h , R h : H → H for left and righ t multiplication by h . Let us denote X = H × H and Y = H and consider the multiplication map X → Y . W e wan t to ﬁnd an angular stratiﬁcation of N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) which is transversal to the submanifold X × Y P Y . Since left and right multiplications b y elements of H are diﬀeomorphisms, w e hav e X × Y P Y ⊂ P X \ ( M 1 ∪ M 2 ), so that it is enough to study the strata of N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) inside the op en set P X \ ( M 1 ∪ M 2 ). T o do so, we ﬁrst pro ceed to some iden tiﬁcations. Using left translations, w e can iden tify P H and H × P + ( h ∗ ). Explicitly , an element ( h, [ ξ ]) ∈ P H is mapp ed to ( h, [ dL ∗ h ξ ]) ∈ H × P + ( h ∗ ). Moreov er, using the metric, w e may iden tify P + ( h ∗ ) and the unit sphere S 2 ⊂ h , by mapping a vector v ∈ S 2 OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 43 to [ ⟨• , v ⟩ ]. As usual, the adjoint action of H on h is denoted by Ad, it pre- serv es the scalar pro duct and hence acts on S 2 . Similarly , the bundle P H × H can be identiﬁed with H × H × P + ( h ∗ ⊕ h ∗ ), by mapping ( h 1 , h 2 , [ ξ 1 : ξ 2 ]) to ( h 1 , h 2 , [ dL ∗ h 1 ξ 1 : dL ∗ h 2 ξ 2 ]). Next P + ( h ∗ ⊕ h ∗ ) can b e identiﬁed with S 5 . Under our identiﬁcations, w e ha v e P X \ ( M 1 ∪ M 2 ) = H × H × S 5 ++ where: S 5 ++ :=  x ∈ S 5 : ( x 1 , x 2 , x 3 )  = 0 , ( x 4 , x 5 , x 6 )  = 0  . This set can b e describ ed using the following diﬀeomorphism, called spher- ical join: S 2 × S 2 ×  0 , π 2  → S 5 ++ , ( v 1 , v 2 , ψ ) 7→ (cos ψ v 1 , sin ψ v 2 ) . Then, a stratiﬁcation deﬁning N ( ϕ 1 ⊠ ϕ 2 ) on the op en set H × H × S 5 ++ is given b y the strata: b S = n ( h 1 , h 2 , cos ψ v 1 , sin ψ v 2 ) : ( h i , v i ) ∈ S i for i = 1 , 2 , ψ ∈  0 , π 2 o , where S i runs ov er all strata of angular stratiﬁcations of H × S 2 that re- sp ectiv ely deﬁne N ( ϕ i ) for i = 1 , 2. T o get a stratiﬁcation deﬁning N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) on the op en set H × H × S 5 ++ , consider the group homomorphism H × H → G , sending ( e, f ) to the map x 7→ exf . Its kernel is { (1 , 1) , ( − 1 , − 1) } and H × H is a double co ver of G . An elemen t ( e, f ) ∈ H 2 acts on H × S 2 b y L e R f ⊗ Ad f , and the iden tiﬁcation P H ∼ = H × P + ( h ∗ ) is ( H × H )-equiv ariant for this action. Then, considering a lift ( e i , f i ) ∈ H 2 of g i ∈ G , the strata n  e 1 h 1 f 1 , e 2 h 2 f 2 , cos ψ Ad f 1 v 1 , sin ψ Ad f 2 v 2  : ( h i , v i ) ∈ S i for i = 1 , 2 , ψ ∈  0 , π 2  o (7.5) deﬁne N (( g 1 ) ∗ ϕ 1 ⊠ ( g 2 ) ∗ ϕ 2 ) on the op en set H × H × S 5 ++ . Finally , w e claim that the map Ξ : b S × H 4 → H × H × S 5 ++ sending ( h 1 , h 2 , (cos ψ v 1 , sin ψ v 2 ) , e 1 , f 1 , e 2 , f 2 ) to ( e 1 h 1 f 1 , e 2 h 2 f 2 , (cos ψ Ad f 1 v 1 , sin ψ Ad f 2 v 2 )) is transversal to X × Y P Y . Indeed, this map is a submersion, since the left action of H on itself and the coadjoin t action of H on P + ( h ∗ ) = S 2 are b oth transitiv e. Therefore, Thom’s transversalit y theorem [ 39 ] ensures that for almost all ( e 1 , f 1 , e 2 , f 2 ) ∈ H 4 , the subanalytic subset Ξ( b S , e 1 , f 1 , e 2 , f 2 ), whic h is equal to the set in ( 7.5 ), is transv ersal to X × Y P Y . A similar argumen t as in the pro of of Theorem 7.10 shows that the strat- iﬁcation given b y ( 7.5 ) is an angular stratiﬁcation. □ Since G acts transitively on the unit sphere bundle of S 3 , the space V ( S 3 ) G of G -inv arian t smo oth v aluations on S 3 is ﬁnite-dimensional. It is w ell- kno wn that it is spanned by the Crofton v aluations ν 0 , ν 1 , ν 2 , ν 3 . 44 ANDREAS BERNIG AND V ADIM LEBOVICI Let c : V ( S 3 ) G ⊗ V ( S 3 ) G → V ( S 3 ) G b e the conv olution pro duct on S 3 . More precisely , for ζ 1 , ζ 2 ∈ V ( S 3 ) G the pushforward of the exterior pro d- uct ζ 1 ⊠ ζ 2 ∈ V ∞ ( S 3 × S 3 ) under the m ultiplication map S 3 × S 3 → S 3 is again a smooth v aluation b y [ 12 , Thm. 1] that will b e denoted by c ( ζ 1 , ζ 2 ) = ζ 1 ∗ ζ 2 . Let us denote by PD G the comp osition of the natural maps V ( S 3 ) G  → V ∞ ( S 3 ) PD  → V −∞ ( S 3 ) ↠ ( V ( S 3 ) G ) ∗ . This map is an isomorphism of vector spaces, moreov er (PD G ) ∗ = PD G , see [ 21 , Deﬁnition 2.14 and Corollary 2.18]. Prop osition 7.14. Ther e is a map m : V ( S 3 ) G → V ( S 3 ) G ⊗ V ( S 3 ) G such that for ϕ 1 , ϕ 2 ∈ CF( S 3 ) m ( µ )( ϕ 1 , ϕ 2 ) = Z G µ ( ϕ 1 ∗ g ∗ ϕ 2 ) dg , (7.6) and the fol lowing diagr am c ommutes V ( S 3 ) G V ( S 3 ) G ⊗ V ( S 3 ) G V ( S 3 ) G ∗ V ( S 3 ) G ∗ ⊗ V ( S 3 ) G ∗ m PD G PD G ⊗ PD G c ∗ . (7.7) Pr o of. Since PD G is an isomorphism, we may deﬁne m b y ( 7.7 ). Let us c heck that ( 7.6 ) is satisﬁed. Let ϕ 1 , ϕ 2 ∈ CF( S 3 ) and µ ∈ V ( S 3 ) G . The generalized v aluations ζ i := R G [ g ∗ ϕ i ] dg are G -in v ariant, hence smo oth, i.e. ζ i ∈ V ( S 3 ) G . Since µ and ζ i are G -inv arian t we ha v e ⟨ PD G ( µ ) , ζ i ⟩ = ⟨ PD( µ ) , ζ i ⟩ =  µ, Z G [ g ∗ ϕ i ] dg  = Z G µ ( g ∗ ϕ i ) dg = µ ( ϕ i ) . (7.8) W e no w compute: Z G µ ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = Z G × G µ (( g 1 ) ∗ ϕ 1 ∗ ( g 2 ) ∗ ϕ 2 ) dg 1 dg 2 ( µ is left-in v ariant) = Z G × G ⟨ µ, [( g 1 ) ∗ ϕ 1 ∗ ( g 2 ) ∗ ϕ 2 ] ⟩ dg 1 dg 2 = Z G × G ⟨ µ, [( g 1 ) ∗ ϕ 1 ] ∗ [( g 2 ) ∗ ϕ 2 ] ⟩ dg 1 dg 2 (Cor. 4 , Prop. 7.13 ) =  µ, Z G [( g 1 ) ∗ ϕ 1 ] dg 1 ∗ Z G [( g 2 ) ∗ ϕ 2 ] dg 2  = ⟨ PD G ( µ ) , ζ 1 ∗ ζ 2 ⟩ ( µ and ζ 1 ∗ ζ 2 are G -inv arian t) = ⟨ c ∗ ◦ PD G ( µ ) , ζ 1 ⊗ ζ 2 ⟩ = ⟨ (PD G ⊗ PD G ) ◦ m ( µ ) , ζ 1 ⊗ ζ 2 ⟩ = m ( µ )( ϕ 1 , ϕ 2 ) b y ( 7.8 ) , OPERA TIONS ON V ALUA TIONS AND CONSTRUCTIBLE FUNCTIONS 45 whic h prov es ( 7.6 ). □ There are tw o wa ys to obtain the explicit form ulas giv en in Theorem 6 . The algebra of inv arian t v aluations on S 3 with resp ect to the conv olution pro duct was computed in [ 18 ], whic h can b e translated in to kinematic for- m ulas by using Theorem 7.14 . Here w e follow a more direct approach. Pr o of of The or em 6 . Let f i ( r ) := ν i ( B (1 , r )), where B (1 , r ) is a closed geo- desic ball of radius r < π 2 and cen ter the iden tit y 1 ∈ S 3 . Then f i ( r ) is the normalized v olume of the r -tub e around a totally geo desic submanifold of dimension 3 − i , and using normal co ordinates around the submanifold, one ﬁnds that f 0 ( r ) = 1 , f 1 ( r ) = 4 π Z r 0 cos 2 ( t ) dt = 2(cos( r ) sin( r ) + r ) π , f 2 ( r ) = 2 Z r 0 sin( t ) cos( t ) dt = sin 2 ( r ) , f 3 ( r ) = 2 π Z r 0 sin( t ) 2 dt = r − cos( r ) sin( r ) π . By Theorem 7.14 we kno w that there are constants d i k,l suc h that Z G ν i ( ϕ 1 ∗ g ∗ ϕ 2 ) dg = X d i k,l ν k ( ϕ 1 ) ν l ( ϕ 2 ) , ϕ 1 , ϕ 2 ∈ CF( S 3 ) . W e take ϕ 1 = 1 B (1 ,r ) , ϕ 2 = 1 B (1 ,s ) with r + s < π 2 . Then for ev ery g ∈ G w e ha v e ϕ 1 ∗ g ∗ ϕ 2 = 1 B ( g ,r + s ) . W e thus m ust hav e f i ( r + s ) = X d i k,l f k ( r ) f l ( s ) . 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Institut f ¨ ur Ma thema tik, Goethe-Universit ¨ at Frankfur t, Rober t-Ma yer- Str. 10, 60054 Frankfur t, Germany Email address : bernig@math.uni-frankfurt.de Sorbonne Universit ´ e, Universit ´ e P aris Cit ´ e, CNRS, IMJ-PRG, F-75005 P aris, France Email address : lebovici@imj-prg.fr

Operations on constructible functions and generalized valuations

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