K-duality for stratified pseudomanifolds

G eometry & T opology XX (20X X) 1001– 999 1001 K -duality f o r stratiﬁed pseudomanif olds C LAIRE D EBORD J EAN -M ARIE L ESCURE This paper continu es the p roject started in [ 13 ] where Poincar ´ e d uality in K -theory was studied fo r sin gular manifo lds with isolated conical singu larities. Here, w e extend the study and the results to g eneral stratiﬁed pseud omanif olds. W e revie w the axiomatic deﬁnition of a smooth stratiﬁcation S of a topolog ical space X and we deﬁne a groupoid T S X , called the S -tangent space. This gro upoid is made o f different pieces enco ding the tang ent spaces of strata, and these pieces are glued into th e smooth n oncom mutative g roupo id T S X using the familiar proced ure introdu ced by A. Connes for the tangent grou poid of a manifo ld. The main resu lt is that C ∗ ( T S X ) is Poincar ´ e dual to C ( X ) , in other words, the S -tangent spa ce plays the role in K -th eory of a tangent space for X . 58B34, 46L80 , 19K35, 58H05, 57N80; 19K33, 19K56 , 58A35 Introd uction This pap er takes place in a longs tandin g pro ject aiming to stud y index theory and related questi ons on stratiﬁed pseu domanif olds using tools and conce pts from nonc ommutativ e geometry . The key observ ation at the beginnin g of this project is that in its K -theori tic form, the Atiyah-Singer index theorem [ 2 ] in volv es ingredients that should surv ive to the singul arities allo wed in a stratiﬁed pseudo m anifol d. This is possible, fro m our opinion , as soo n as one accept s reasonabl e generalizati ons and new p resent ation of certa in classic al objects on smooth manifolds, making sense on stratiﬁed pseudomanif olds. The ﬁ rst instance of these classic al objects that need to be adapted to singulari ties is the notion o f tangen t space . S ince index maps in [ 2 ] are deﬁned on the K -theory of t he tangen t spaces of smooth manifolds, on e must ha ve a similar sp ace a dapted t o str atiﬁed pseud omanifol ds. Moreo ver , such a space shou ld satisfy natura l requir ements. It should coin cide w ith the usual notion on the regu lar part of the pseudoman ifold and incorp orate in some way copies of usual tangent spaces of strata, while keep ing eno ugh smoothn ess to allo w inte resting computat ions. Moreo ver , it should be Poincar ´ e dual Published: XX Xxxember 20XX DOI: 10.214 0/gt.20 XX.XX.100 1 1002 Clair e Debord and Jean-Marie Lescur e in K -theory (shortly , K -dua l) to the pseudomanifo ld itself. This K -theo ritic prope rty in v olves biv ariant K -theory and was prov ed between smooth manifolds and their tangen t spaces by G. Kasparov [ 20 ] and A. Connes-G. Skandali s [ 8 ]. In [ 13 ], we introduc ed a can didate to be the tang ent space of a pseud omanifold with isolate d conical singul arities. It appeared to be a smooth groupo id, leading to a nonco mmutati ve C ∗ -algeb ra, and we pro ved that it fulﬁlls the expec ted K -duality . In [ 22 ], the second author interpreted the duality prove d in [ 13 ] as a principal symbol map, thus recov ering the classical picture of Poincar ´ e duality in K -theory for smooth manifold s. This interpretat ion used a no tion of nonco m mutati ve elliptic symbol s, which appeared to be the c ycles of the K -theory of the noncommuta ti ve tangen t space. In [ 14 ], the noncommuta ti ve tangen t space together with other deformation groupoids was used to construct analytical and topologica l in dex maps, and their equality was pro ved . As ex pected, th ese index m aps are straig ht gen eralizations of those of [ 2 ] for manifold s. The presen t paper is dev oted to the constructio n of the nonco m mutati ve tangent space for a genera l stratiﬁed pseudomanif old and the proof of the K -duality . It is thus a sequel of [ 13 ], b ut can be read independe ntly . At ﬁrst glance, one shoul d ha ve exp ected that the te chnics o f [ 13 ] itera te eas ily to gi ve the general resu lt. In f act, although the deﬁnitio n of the groupoid gi ving the noncommutati ve tangent space itself is natural and intuiti ve in the general case, its smooth ness is quite intricate and brings issues that did not exist in the conical case. W e hav e gi ven here a detailed treatment of this point, since we believ e that this material will be usefull in further studies about the geometry of stratiﬁed s paces. Another d ifferen ce with [ 13 ] is that we ha ve giv en u p th e exp licit constructio n of a dual Dirac element. Instead, we use an easily deﬁned Dirac element and then prov e the Poincar ´ e duality by an inductio n, b ased on an operati on called unfolding which consist s in remo ving the minimal strat a in a pseudomanifo ld and then “doub ling” it to get a new pseudomanif old, less singula r . The dif ﬁ culty in this approa ch is m ov ed to the proof of the commutati vity of certain diagrams in K -theory , necess ary to apply the ﬁve lemma and to cont inue the induction. The interpr etation of this K -duality in terms of noncommuta ti ve symbo ls and pseu- dodif ferentia l operators, as well as the construc tion of index maps togeth er w ith the statemen t of an index theorem, is postpon ed to fo rthcoming papers. This approach of in dex t heory on sin gular spac es in the fra m e work of noncommutati ve geometry takes place in a long history of past and prese nt resarch works. But the speciﬁc issue s about Poincar ´ e duality , bi varian t K -theory , topo logica l inde x m aps and G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1003 statemen t o f Atiy ah-Singer lik e theorems are q uite re cent and attract an increasin g interes t [ 28 , 26 , 16 , 15 , 32 , 30 ]. Acknowledgeme nts W e would lik e to thank the referee for his useful comments and remark s. In particula r he suggest ed to us Lemm as 2 and 3 which enable u s to sh orten and clarify signiﬁcant ly the proof of Theorem 4 . 1 Basic deﬁnition s 1.1 Ar ound Lie groupoids W e refer to [ 31 , 5 , 23 , 12 ] for the classi cal deﬁnitions and con structi ons related to group oids, their Lie algebroi ds and C ∗ -algeb ras of group oids. In this section , we ﬁx the notati ons and recall the less clas sical deﬁnitions and resu lts need ed in the sequel. Some material prese nted here is already in [ 13 , 14 ]. 1.1.1 P ull back gr oupoids Let G ⇒ M be a locally co mpact H ausdor ff grou poid w ith source s and ran ge r . If f : N → M is a surj ecti ve map, the pull back groupo id ∗ f ∗ ( G ) ⇒ N of G by f is by deﬁnitio n the set ∗ f ∗ ( G ) : = { ( x , γ , y ) ∈ N × G × N | r ( γ ) = f ( x ) , s ( γ ) = f ( y ) } with the struct ural m orphis m s giv en by (1) the unit map x 7→ ( x , f ( x ) , x ) , (2) the source map ( x , γ , y ) 7→ y and range map ( x , γ , y ) 7→ x , (3) the product ( x , γ , y )( y , η , z ) = ( x , γ η, z ) and in verse ( x , γ , y ) − 1 = ( y , γ − 1 , x ) . The resu lts of [ 29 ] apply to sho w that the groupoi ds G and ∗ f ∗ ( G ) are Morita equi va lent when f is surjecti ve and open. Let us ass ume for the r est of thi s s ubsection that G is a smooth grou poid and that f is a surjec ti ve submersio n, then ∗ f ∗ ( G ) is also a Lie groupoid . L et ( A ( G ) , q , [ , ]) be the Lie algebr oid of G . Recall that q : A ( G ) → TM is the anchor map. Let ( A ( ∗ f ∗ ( G )) , p , [ , ]) G eometry & T opology XX (20X X) 1004 Clair e Debord and Jean-Marie Lescur e be th e L ie algebroid of ∗ f ∗ ( G ) and Tf : TN → T M be the dif ferential of f . Then there exi sts an isomorphis m A ( ∗ f ∗ ( G )) ≃ { ( V , U ) ∈ TN × A ( G ) | Tf ( V ) = q ( U ) ∈ TM } under which the anch or map p : A ( ∗ f ∗ ( G )) → TN ide ntiﬁes with the projecti on TN × A ( G ) → TN . (In particu lar , if ( V , U ) ∈ A ( ∗ f ∗ ( G )) w ith V ∈ T x N and U ∈ A y ( G ) , the n y = f ( x ) .) 1.1.2 S ubalgebras and exa ct sequences of gr oup oid C ∗ -algebra s T o any smooth groupo id G are assoc iated two C ∗ -algeb ras corresp onding to two dif ferent completion s of the in vol uti ve con v olution algebra C ∞ c ( G ) , namel y the reduc ed and maximal C ∗ -algeb ras [ 6 , 7 , 3 1 ]. W e w ill denote respecti vely the se C ∗ -algeb ras by C ∗ r ( G ) and C ∗ ( G ) . Recall that the identit y on C ∞ c ( G ) induce s a surjecti ve morph ism from C ∗ ( G ) onto C ∗ r ( G ) which is an isomorphis m if the groupoid G is amenable . Moreo ver in this case the C ∗ algebr a of G is nuclear [ 1 ] . W e will use the follo wing usual notations: Let G s ⇒ r G (0) be a smooth group oid w ith source s and range r . If U is any subset of G (0) , we let: G U : = s − 1 ( U ) , G U : = r − 1 ( U ) and G U U = G | U : = G U ∩ G U . T o an open subset O of G (0) corres ponds an inclusion i O of C ∞ c ( G | O ) in to C ∞ c ( G ) which induces an injecti ve morp hism, ag ain d enoted by i O , fro m C ∗ ( G | O ) into C ∗ ( G ) . When O is satu rated, C ∗ ( G | O ) is an ideal of C ∗ ( G ) . In this case, F : = G (0) \ O i s a saturated closed subset of G (0) and the restrict ion of func tions induces a sur jecti ve morphism r F from C ∗ ( G ) to C ∗ ( G | F ) . Moreo ver , according to [ 18 ], the follo wing sequen ce of C ∗ -algeb ras is exact: 0 − − − − → C ∗ ( G | O ) i O − − − − → C ∗ ( G ) r F − − − − → C ∗ ( G | F ) − − − − → 0 . 1.1.3 KK -elements a ssociated to def ormation groupoids A smooth groupo id G is called a deformation gr oupoid if: G = G 1 × { 0 } ∪ G 2 × ]0 , 1] ⇒ G (0) = M × [0 , 1] , G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1005 where G 1 and G 2 are smooth group oids with unit space M . That is, G is obtained by gluing G 2 × ]0 , 1] ⇒ M × ]0 , 1] , which is the cart esian product of the groupo id G 2 ⇒ M with the space ]0 , 1] , with the groupo id G 1 × { 0 } ⇒ M × { 0 } . In this situation one can consider the saturated open subse t M × ]0 , 1] of G (0) . Using the isomorphisms C ∗ ( G | M × ]0 , 1] ) ≃ C ∗ ( G 2 ) ⊗ C 0 (]0 , 1]) and C ∗ ( G | M ×{ 0 } ) ≃ C ∗ ( G 1 ) , we obtain the follo wing exact sequence of C ∗ -algeb ras: 0 − − − − → C ∗ ( G 2 ) ⊗ C 0 (]0 , 1]) i M × ]0 , 1] − − − − → C ∗ ( G ) ev 0 − − − − → C ∗ ( G 1 ) − − − − → 0 where i M × ]0 , 1] is the inclusio n map and ev 0 is the evalu ation map at 0 , that is ev 0 is the map coming from the restrict ion of function s to G | M ×{ 0 } . W e assume now that C ∗ ( G 1 ) is nuclear . Since the C ∗ -algeb ra C ∗ ( G 2 ) ⊗ C 0 (]0 , 1]) is contra ctible, the long exac t sequence in KK -theory sho ws that the group homomor - phism ( ev 0 ) ∗ = ·⊗ [ ev 0 ] : K K ( A , C ∗ ( G )) → KK ( A , C ∗ ( G 1 )) is an isomorph ism for each C ∗ -algeb ra A [ 20 , 10 ]. In particular with A = C ∗ ( G 1 ) and A = C ∗ ( G ) we get that [ ev 0 ] is in v ertible in KK - theory : th ere is an element [ ev 0 ] − 1 in KK ( C ∗ ( G 1 ) , C ∗ ( G )) such that [ e v 0 ] − 1 ⊗ [ ev 0 ] = 1 C ∗ ( G 1 ) and [ e v 0 ] ⊗ [ ev 0 ] − 1 = 1 C ∗ ( G ) . Let ev 1 : C ∗ ( G ) → C ∗ ( G 2 ) be the e valu ation map at 1 an d [ ev 1 ] the correspo nding element of KK ( C ∗ ( G ) , C ∗ ( G 2 )) . The KK -element a ssociated to the defo rmation gr oupoi d G is deﬁned by: δ = [ ev 0 ] − 1 ⊗ [ ev 1 ] ∈ KK ( C ∗ ( G 1 ) , C ∗ ( G 2 )) . One can ﬁnd ex amples of such elements related to index the ory in [ 7 , 18 , 13 , 14 , 12 ]. 1.2 Generalities about K -duality W e giv e in this paragra ph some general fact s abo ut Poincar ´ e duality in biv ariant K- theory . Most of them are well kno wn and proofs are only added w hen no self contained proof could be fou nd in the literatu re. All C ∗ -algeb ras are assumed to be separable and σ -unital. Let us ﬁrst recal l what means the Poincar ´ e duality in K -theo ry [ 21 , 8 , 7 ]: G eometry & T opology XX (20X X) 1006 Clair e Debord and Jean-Marie Lescur e Deﬁnition 1 L et A , B be two C ∗ -algeb ras. One says that A and B are Poincar ´ e dual, or sho rtly K -dual , when there ex ists α ∈ K 0 ( A ⊗ B ) = KK ( A ⊗ B , C ) and β ∈ KK ( C , A ⊗ B ) ≃ K 0 ( A ⊗ B ) such that β ⊗ B α = 1 ∈ KK ( A , A ) and β ⊗ A α = 1 ∈ KK ( B , B ) Such elemen ts are then called D irac and dual-Dira c elements. It follo ws that for A , B tw o K -dual C ∗ -algeb ras and for any C ∗ -algeb ras C , D , the follo wing isomorphisms hold: β ⊗ B · : KK ( B ⊗ C , D ) − → K K ( C , A ⊗ D ); β ⊗ A · : KK ( A ⊗ C , D ) − → K K ( C , B ⊗ D ); with in v erses giv en respecti vely by · ⊗ A α and · ⊗ B α . Example 1 A basic exa mple is A = C ( V ) and B = C 0 ( T ∗ V ) where V is a closed smooth manifol d ([ 21 , 8 ], see also [ 13 ] for a de scription of the Dirac eleme nt in terms of groupoi ds). This duality allo ws to recove r that the u sual qu antiﬁcati on and princip al symbol maps are mutually in ver se isomorph isms in K -theory: ∆ V = ( · ⊗ C 0 ( T ∗ V ) α ) : K 0 ( C 0 ( T ∗ V )) ≃ − → K 0 ( C ( V )) Σ V = ( β ⊗ C ( V ) · ) : K 0 ( C ( V )) ≃ − → K 0 ( C 0 ( T ∗ V )) W e obser ve that: Lemma 1 Let A , B be two C ∗ -algeb ras. Assume that th ere exists α ∈ KK ( A ⊗ B , C ) and β , β ′ ∈ KK ( C , A ⊗ B ) satisfy ing β ⊗ B α = 1 ∈ KK ( A , A ) and β ′ ⊗ A α = 1 ∈ KK ( B , B ) Then β = β ′ so A , B are K -dual. Pro of A simple calculat ion shows tha t for all x ∈ KK ( C , A ⊗ D ) we ha ve: β ⊗ B ( x ⊗ A α ) = x ⊗ A ⊗ B ( β ⊗ B α ) . Applying this to C = C , D = A and x = β ′ we get: β ′ = β ⊗ B ( β ′ ⊗ A α ) = β ⊗ B 1 = β G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1007 Cor ollary 1 1) Giv en two K -dual C ∗ -algeb ras a nd a D irac element α , the d ual-Dirac element β satisfy ing the deﬁnition 1 is unique. 2) If there exi sts α ∈ KK ( A ⊗ B , C ) such that · ⊗ B α : K K ( C , A ⊗ B ) − → K K ( A , A ) and · ⊗ A α : KK ( C , A ⊗ B ) − → KK ( B , B ) are onto, then A , B are K -dual and α is a Dirac element. The two lemmas belo w ha ve been comunica ted to us by the referee. Lemma 2 Let J 1 and J 2 be two closed two sided ideal s in a nuclear C ∗ -algeb ra A such that J 1 ∩ J 2 = { 0 } and set B = A / ( J 1 + J 2 ) . Denote by ∂ k ∈ KK 1 ( B , J k ) , k = 1 , 2 , th e KK -element s associated respecti vel y with the exact seque nces 0 − → J 1 − → A / J 2 − → B − → 0 and 0 − → J 2 − → A / J 1 − → B − → 0 . Let also i k : J k − → A denote the inclus ions. Then the follo wing equality hold s: ( i 1 ) ∗ ( ∂ 1 ) + ( i 2 ) ∗ ( ∂ 2 ) = 0 . Pro of Let ∂ ∈ KK 1 ( B , J 1 + J 2 ) denote the K K -element asso ciated with the exact sequen ce 0 − → J 1 + J 2 − → A − → B − → 0 . D enote by j k : J k − → J 1 + J 2 and i : J 1 + J 2 − → A the inclusions and by p k : J 1 + J 2 − → J k the projection s, k = 1 , 2 . Since the diag rams ( k = 1 , 2) (1–1) 0 / / J 1 + J 2 p k   / / A   / / B / / =   0 0 / / J k / / A / J 3 − k / / B / / 0 commute, it follo ws that ( p k ) ∗ ( ∂ ) = ∂ k . Moreo ver , ( p 1 ) ∗ × ( p 2 ) ∗ : KK 1 ( B , J 1 + J 2 ) − → KK 1 ( B , J 1 ) × KK 1 ( B , J 2 ) is an iso- morphism w hose in verse is ( j 1 ) ∗ + ( j 2 ) ∗ . It follo ws that ∂ = ( j 1 ) ∗ ( ∂ 1 ) + ( j 2 ) ∗ ( ∂ 2 ) . Moreo ver t he six-term exact sequenc e associate d to 0 − → J 1 + J 2 − → A − → B − → 0 leads to i ∗ ( ∂ ) = 0 . The result follows now from the equalit ies i k = i ◦ j k , k = 1 , 2 . Lemma 3 Let X be a compact space and A be a nucle ar C ( X ) -algeb ra. Let U 1 and U 2 be disjoi nt open subsets of X . Set X 1 = X \ U 2 and J k = C 0 ( U k ) A , k = 1 , 2 . Let Ψ : C ( X ) ⊗ A − → A be the homomorp hism deﬁned by Ψ ( f ⊗ a ) = fa and let ϕ : C ( X 1 ) ⊗ J 1 − → A , ψ : C 0 ( U 2 ) ⊗ A / J 1 − → A be the homomorphisms induc ed by Ψ . Denote by ∂ 1 ∈ KK 1 ( A / J 1 , J 1 ) and e ∂ 2 ∈ K K 1 ( C ( X 1 ) , C 0 ( U 2 )) the KK -element s as- sociat ed resp ectiv ely with the exact sequences 0 − → J 1 − → A − → A / J 1 − → 0 and 0 − → C 0 ( U 2 ) − → C ( X ) − → C ( X 1 ) − → 0 . Then the followin g equality holds: ( ϕ ) ∗ ( ∂ 1 ⊗ 1 C ( X 1 ) ) + ( ψ ) ∗ (1 A / J 1 ⊗ e ∂ 2 ) = 0 . G eometry & T opology XX (20X X) 1008 Clair e Debord and Jean-Marie Lescur e Pro of W e use the notation of Lemma 2 . W e hav e commuting diagra m s (1–2) 0 / / J 1 ⊗ C ( X 1 ) ϕ 1   / / A ⊗ C ( X 1 )   / / A / J 1 ⊗ C ( X 1 ) / / χ   0 0 / / J 1 / / A / J 2 / / B / / 0 and (1–3) 0 / / A / J 1 ⊗ C 0 ( U 2 ) ψ 2   / / A / J 1 ⊗ C ( X )   / / A / J 1 ⊗ C ( X 1 ) / / χ   0 0 / / J 2 / / A / J 1 / / B / / 0 where vert ical arro ws are induced by Ψ . It follo ws that ( ϕ 1 ) ∗ ( ∂ 1 ⊗ 1 C ( X 1 ) ) = χ ∗ ( ∂ 1 ) and ( ψ 2 ) ∗ (1 A / J 1 ⊗ e ∂ 2 ) = χ ∗ ( ∂ 2 ) . W e then use the equalitie s ϕ = i 1 ◦ ϕ 1 and ψ = i 2 ◦ ψ 2 and apply Lemma 2 to conclud e. It yield s the follo wing, with the notation of Lemma 3 : Lemma 4 Let δ be in K 0 ( A ) and set D = Ψ ∗ ( δ ) , D 1 = ϕ ∗ ( δ ) , D 2 = ψ ∗ ( δ ) . Then for any C ∗ -algeb ras C and D , th e two follo wing long diagr ams commute: (1–4) · · · KK i ( C , D ⊗ C 0 ( U 2 )) / / ⊗ C 0 ( U 2 ) c i D 2   KK i ( C , D ⊗ C ( X )) / / ⊗ C ( X ) c i D   KK i ( C , D ⊗ C ( X 1 )) / / ⊗ C ( X 1 ) c i D 1   KK i + 1 ( C , D ⊗ C 0 ( U 2 )) · · · ⊗ C 0 ( U 2 ) c i + 1 D 2   · · · KK i ( C ⊗ A / J 1 , D ) / / KK i ( C ⊗ A , D ) / / KK i ( C ⊗ J 1 , D ) / / KK i + 1 ( C ⊗ A / J 1 , D ) · · · (1–5) · · · KK i ( C , D ⊗ J 1 ) / / ⊗ J 1 c i D 1   KK i ( C , D ⊗ A ) / / ⊗ A c i D   KK i ( C , D ⊗ A / J 1 ) / / ⊗ A / J 1 c i D 2   KK i + 1 ( C , D ⊗ J 1 ) · · · ⊗ J 1 c i + 1 D 1   · · · KK i ( C ⊗ C ( X 1 ) , D ) / / KK i ( C ⊗ C ( X ) , D ) / / KK i ( C ⊗ C 0 ( U 2 ) , D ) / / KK i + 1 ( C ⊗ C ( X 1 ) , D ) · · · where the c i belong to {− 1 , 1 } and are chos en such that c i = ( − 1) i + 1 c i + 1 . In particular , if two o f t hree element s D 1 , D 2 , D are Dirac eleme nts, so is the third one . Pro of Observ e ﬁrst that Lemma 3 reads: ∂ 1 ⊗ J 1 [ ϕ ] = − e ∂ 2 ⊗ C 0 ( U 2 ) [ ψ ] , which giv es: ∂ 1 ⊗ J 1 D 1 = ∂ 1 ⊗ J 1 ([ ϕ ] ⊗ δ ) = ( ∂ 1 ⊗ J 1 [ ϕ ]) ⊗ δ = ( − e ∂ 2 ⊗ C 0 ( U 2 ) [ ψ ]) ⊗ δ = − e ∂ 2 ⊗ C 0 ( U 2 ) D 2 G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1009 No w , us ing the ske w-commutati vity of the product ⊗ C , we hav e for any x ∈ KK i ( C , D ⊗ C ( X 1 )) : ∂ 1 ⊗ J 1 ( x ⊗ C ( X 1 ) D 1 ) = ( ∂ 1 ⊗ C x ) ⊗ J 1 ⊗ C ( X 1 ) D 1 = ( − 1) i ( x ⊗ C ∂ 1 ) ⊗ J 1 ⊗ C ( X 1 ) D 1 = ( − 1) i x ⊗ C ( X 1 ) ( ∂ 1 ⊗ J 1 D 1 ) = ( − 1) i x ⊗ C ( X 1 ) ( − e ∂ 2 ⊗ C 0 ( U 2 ) D 2 ) = ( − 1) i + 1 ( x ⊗ C ( X 1 ) e ∂ 2 ) ⊗ C 0 ( U 2 ) D 2 This yields , thanks to the choice of the sign c i , the commutati vity for the sq uares in v olving boundary homomorph isms in Diagra m ( 1–4 ). The other sq uares in Diagram ( 1–4 ) commute by deﬁniti on of D 1 , D 2 , D and by functorialit y of K K -theory . T he commutati vity of D iagram ( 1–5 ) is prov ed by the same argu m ents. The last asserti on is then a conseq uence of Corollary 1 and the ﬁve lemma. 2 Stratiﬁed pseudomanif olds W e are interest ed in studying strati ﬁed pseudomani folds [ 34 , 24 , 17 ]. W e w ill use the notation s and equi val ent descrip tions giv en by A. V eron a in [ 33 ] or used by J.P . Brasselet , G. Hector and M. Sarale gi in [ 3 ]. The reader shoul d also look at [ 19 ] for a hepful l surve y of the subject. 2.1 Deﬁnitions Let X be a locally compact separable metrizable space. Deﬁnition 2 A C ∞ -strati ﬁcation of X is a pair ( S , N ) such that: (1) S = { s i } is a locally ﬁni te partition of X into locally closed subsets of X , ca lled the strata, which are smooth manifolds and which satisﬁes: s 0 ∩ ¯ s 1 6 = ∅ if and only if s 0 ⊂ ¯ s 1 . In that case we will write s 0 ≤ s 1 and s 0 < s 1 if moreo ver s 0 6 = s 1 . G eometry & T opology XX (20X X) 1010 Clair e Debord and Jean-Marie Lescur e (2) N = {N s , π s , ρ s } s ∈ S is the set of contr ol data or tube system: N s is an open neighb orhood of s in X , π s : N s → s is a continuo us retractio n and ρ s : N s → [0 , + ∞ [ is a con tinuous m ap such t hat s = ρ − 1 s (0) . The map ρ s is either surjecti ve or constant equal to 0 . Moreo ver if N s 0 ∩ s 1 6 = ∅ then the map ( π s 0 , ρ s 0 ) : N s 0 ∩ s 1 → s 0 × ]0 , + ∞ [ is a smooth proper submersi on. (3) For an y strata s , t such that s < t , the inclus ion π t ( N s ∩ N t ) ⊂ N s is true and the equalit ies: π s ◦ π t = π s and ρ s ◦ π t = ρ s hold on N s ∩ N t . (4) For any tw o strata s 0 and s 1 the follo wing equiv alence s hold: s 0 ∩ ¯ s 1 6 = ∅ if and only if N s 0 ∩ s 1 6 = ∅ , N s 0 ∩ N s 1 6 = ∅ if and only if s 0 ⊂ ¯ s 1 , s 0 = s 1 or s 1 ⊂ ¯ s 0 . A str atiﬁcation gi ves rise t o a ﬁltrat ion: let X j be th e union of strata of di mension ≤ j , then: ∅ ⊂ X 0 ⊂ · · · ⊂ X n = X . W e call n the dimension of X and X ◦ : = X \ X n − 1 the r e gular part of X . The strata includ ed in X ◦ are called re gular w hile strata included in X \ X ◦ are called singular . The set of singu lar (resp. re gular) strata is denoted S sing (resp. S re g ). For a ny subset A of X , A ◦ will denote A ∩ X ◦ . A cruci al notion for our purpose will be the notion of depth . Observe that the binary relatio n s 0 ≤ s 1 is a parti al ordering on S . Deﬁnition 3 The depth d ( s ) of a stratum s is the biggest k such that on e can ﬁnd k dif ferent strata s 0 , · · · , s k − 1 such that s 0 < s 1 < · · · < s k − 1 < s k : = s . The depth of the stratiﬁca tion ( S , N ) of X is: d ( X ) : = sup { d ( s ) , s ∈ S } . A stratu m whose depth is 0 will be called minimal. G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1011 W e hav e follo wed the terminology of [ 3 ], but remark that the opposite con venti on fo r the depth also exis ts [ 33 ]. Finally we can deﬁne stratiﬁed pseudo m anifold s: Deﬁnition 4 A stratiﬁed pseud omanifo ld is a triple ( X , S , N ) where X is a locally compact separab le metrizable space, ( S , N ) is a C ∞ -strati ﬁcation on X and t he re gular part X ◦ is a dens e open subset of X . If ( X , S X , N X ) and ( Y , S Y , N Y ) are two st ratiﬁed pseudoman ifolds an homeomorphism f : X → Y is an isomo rphism of stratiﬁed pseudomani fold if: (1) S Y = { f ( s ) , s ∈ S X } and the restrictio n of f to each stratum is a dif feomorphism onto its image. (2) π f ( s ) ◦ f = f ◦ π s and ρ s = ρ f ( s ) ◦ f for any stratu m s of X . Let us make some basi c remark on the prev ious deﬁniti ons. Remark 1 (1) At a ﬁrst sight, the deﬁnition of a stratiﬁcation giv en here seems more restric tiv e than the usu al one. In fact acc ording to [ 33 ] these deﬁnition s are equi vale nt. (2) Usually , for ex ample in [ 17 ], the extr a assumption X n − 1 = X n − 2 is requi red in the deﬁnition of stratiﬁed pseudoman ifold. Our constru ctions remain without this extra assu mption. (3) A stratum s is regular if and only if N s = s and then ρ s = 0 . (4) Pseudomanifo lds of depth 0 are smooth m anifol ds, and the stra ta are t hen union of connect ed components. The follo wing simple consequ ence of the axioms will be usefull enough in the sequel to be pointe d out: Pro p osition 1 Let ( X , S , N ) be a stratiﬁed pseud omanifold. A ny subset { s i } I of distin ct elements of S is total ly orde red by < as soon as the inters ection ∩ i ∈ I N s i is non empty . In particular if the strata s 0 and s 1 are such that N s 0 ∩ N s 1 6 = ∅ then d ( s 0 ) 6 = d ( s 1 ) or s 0 = s 1 . By a slight ab use of langu age we will sometime talk ab out a strati ﬁ ed pseudoman ifold X while we on ly hav e a partition S on the spa ce X . This m eans that one can ﬁ nd at least one contr ol data N such that ( X , S , N ) is a strat iﬁ ed pseudomanif old in the sense of our deﬁnitio n 4 . G eometry & T opology XX (20X X) 1012 Clair e Debord and Jean-Marie Lescur e 2.2 Examples (1) Smooth manifolds are, without other mention, pseudo manifold s of depth 0 and with a singl e stratum. (2) Stratiﬁed pseudo m anifold s of depth one are wedg es and are obtaine d as follo ws. T ake M to be a manifold with a compact bound ary L and let π be a surjecti ve submersi on of L onto a manifold s . Consider the mapping cone of ( L , π ) : c π L : = L × [0 , 1] / ∼ π where ( z , t ) ∼ π ( z ′ , t ′ ) if and onl y if ( z , t ) = ( z ′ , t ′ ) or t = t ′ = 0 and π ( z ) = π ( z ′ ) . The image of L × { 0 } identiﬁes with s and by a slight abuse of notat ion w e will denote it s . N o w glue c π L and M along their bo undary in order to g et X . T he sp ace X with t he partiti on { s , X \ s } is a stratiﬁed pseudoman ifold. T wo extre m e examples are obtained by cons iderin g π either equa l to identity , with s = L or equal to the proj ection on one point c . In the ﬁrst case X is a manifold with bound ary L isomor phic to M and the stratiﬁcati on corresp onds to the partiti on of X by { L , X \ L } . In the second case X is a coni cal m anifold and the strati ﬁ cation corres ponds to the partition of X by { c , X \ c } , where c is the singular point. (3) Manifolds with corner s with their partitio n into faces are stratiﬁed pseudo m anifold s [ 25 , 27 ]. (4) If ( X , S , N ) is a pseudoman ifold and M is a smooth manifold then X × M is natura lly e ndo wed with a structure of ps eudomanifold of same depth a s X whose strata are { s × M , s ∈ S } . (5) If ( X , S , N ) is a pseudoman ifold of depth k then CrX : = X × S 1 / X × { p } is natura lly endo wed with a stru cture of pseudomani fold of d epth k + 1 , whos e strata are { s × ]0 , 1[ , s ∈ S } ∪ { [ p ] } . Here we ha ve identiﬁed S 1 \ { p } with ]0 , 1[ and we ha ve denote d by [ p ] the image of X × { p } in CrX . For e xample, if X is the square we get the follo wing picture: 2.3 The unf olding process Let ( X , S , N ) be a strat iﬁ ed pse udomanifold. If s is a singul ar stratum, we let L s : = ρ − 1 s (1) . Then L s inheri ts from X a structure of stratiﬁed pseudoman ifold. G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1013 One can then deﬁne the open mappin g cone of ( L s , π s ) : c π s L s : = L s × [0 , + ∞ [ / ∼ π s where ∼ π s is as befo re. Accordin g to [ 33 ], see also [ 3 ] the open mapping cone is naturally endo wed with a structu re of stratiﬁed pseudomanifold whose strata are { ( t ∩ L s ) × ]0 , + ∞ [ , t ∈ S } ∪ { s } . Here we identify s with the image of L s × { 0 } in c π s L s . Moreove r , up to isomorphism, the control data on X can be chosen such that one can ﬁnd a contin uous retrac tion f s : N s \ s → L s for which the map (2–1) Ψ s : N s → c π s L s z 7→  [ f s ( z ) , ρ s ( z )] if z / ∈ s z else where is an i somorph ism of st ratiﬁed pseudomanifo lds. Here [ y , t ] deno tes the cl ass in c π s L s of ( y , t ) ∈ L s × [0 , + ∞ [ . This res ult of local tr i viality around str ata will be cru cial for our p urpose. In p articular it enables on e to make the unf olding process [ 3 ] which consis ts in replacing eac h minimal stratum s by L s . Precisely suppose that d ( X ) = k > 0 and let S 0 be the set of strata of depth 0 . Deﬁne O 0 : = ∪ s ∈ S 0 { z ∈ N s | ρ s ( z ) < 1 } , X b = X \ O 0 and L : = ∪ s ∈ S 0 { z ∈ N s | ρ s ( z ) = 1 } ⊂ X b . Notice that it follo ws from remark 1 that the L s ’ s where s ∈ S 0 are disj oint and thus L = ⊔ s ∈ S 0 L s . W e let 2 X = X − b ∪ L × [ − 1 , 1] ∪ X + b where X ± b = X b and X − b (respe cti vely X + b ) is glue d along L with L × {− 1 } ⊂ L × [ − 1 , 1] (respec tive ly L × { 1 } ⊂ L × [ − 1 , 1] ). Let s be a stratum of X which is not minimal and which intersects O 0 . W e deﬁne the follo wing subset of 2 X : ˜ s : = ( s ∩ X − b ) ∪ ( s ∩ L ) × [ − 1 , 1] ∪ ( s ∩ X + b ) W e then deﬁne S 2 X : = { ˜ s ; s ∈ S and s ∩ O 0 6 = ∅} ∪ { s − , s + ; s ± = s ∈ S and s ∩ O 0 = ∅} . The space 2 X inherits from X a structure of stratiﬁed pseudomanifo ld of depth k − 1 whose set of strata is S 2 X . Notice that there is a nat ural m ap p fro m 2 X onto X . The restrictio n p to an y copy of X b is identity and for ( z , t ) ∈ L s × [ − 1 , 1] , p ( z , t ) = Ψ − 1 s ([ z , | t | ]) . The strata of 2 X are the conn ected components of the pre-images by p of the strata of X . G eometry & T opology XX (20X X) 1014 Clair e Debord and Jean-Marie Lescur e The interested reader can ﬁnd all the detai ls related to the unfold ing p rocess in [ 3 ] and [ 33 ] where it is called decompo sition. In partic ular starting with a compact pseud omanifol d X of depth k , one can iterate this process k times and obtain a compact smooth manifold 2 k X togeth er with a continuou s surjecti ve map π : 2 k X → X whose restric tion to π − 1 ( X ◦ ) is a tri vial 2 k -fold cov ering. Example 2 Look at the square C w ith stratiﬁcatio n giv en by its vertic es, edges and its interior . It can be endo wed with a structure of stratiﬁed pseud omanifol d of depth 2 . Applyin g onc e the unfolding process gi ves a sphere with 4 holes: S : = S 2 \ { D 1 , D 2 , D 3 , D 4 } w here the D i ’ s are disjoi nt and homeomorphic to open disks . The set of strata of S 2 is then { ◦ S , S 1 , S 2 , S 3 , S 4 } where S i is the boundar y of D i and ◦ S the inte rior of S . Applying the unfol ding process once more gi ves the to rus w ith three holes. 3 The tangent grou poid and S -tangent space of a compact stratiﬁed pseudo-manif old 3.1 The set construction W e begin by the descripti on at the set le vel of the S -tang ent gr oupoid and the S -tangen t space of a compact stratiﬁed pseudo -manifold. W e keep the notation of t he previous section: X is a compact stratiﬁed pseudo-manifold, S the se t of strata, X ◦ the re gular part and N = {N s , π s , ρ s } s ∈ S the set of c ontrol data. For e ach s ∈ S we let O s : = { z ∈ N s | ρ s ( z ) < 1 } and F s : = O s \ [ s 0 < s O s 0 . Note that F s = O s if and only if s is a minimal st ratum and O s = s when s is re gular . Lemma 5 The set { F s } s ∈S form a partitio n of X . G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1015 Pro of If z belong s to X , let R z : = { s ∈ S | z ∈ N s and ρ s ( z ) < 1 } . It follo ws from propo sition 1 that R z is a ﬁnite set totally ordere d by < . S ince the set R z contai ns the stratum pa ssing through z , i t is no nempty . Let s z 0 be the minimal element of R z . Then z belongs to F s z 0 . Moreov er , for all stratu m s ∈ S , if s 6 = s z 0 and z ∈ O s , then s ∈ R z , whence s z 0 < s , so that s / ∈ F s . Recall that O ◦ s = O s ∩ X ◦ . W e denote again by π s : O ◦ s → s the projectio n. When s is a stratum, π s is a proper sub m ersion and one can consider the pu ll-back groupo id ∗ π ∗ s ( Ts ) ⇒ O ◦ s of the usual tangen t space Ts ⇒ s by π s . It is naturall y end o w ed with a structure of smooth groupo id. When s is a regular stratu m , s = O s = O ◦ s and π s is the identit y map, thus ∗ π ∗ s ( Ts ) ≃ TO ◦ s in a canonic al way . At the set le vel, the S - tang ent space of X is the groupoid: T S X = [ s ∈ S ∗ π ∗ s ( TS ) | F ◦ s ⇒ X ◦ where F ◦ s = F s ∩ X ◦ . Follo wing the cases of smooth manifolds [ 7 ] and isolated conical singul arities [ 13 ], the S - tang ent gr oupoid of X is deﬁned to be a deformatio n of the pair group oid of the regular part of X onto its S -tang ent space: G t X : = T S X × { 0 } ∪ X ◦ × X ◦ × ]0 , 1] ⇒ X ◦ × [0 , 1] . Examples 1 (1) When X has depth 0 , we rec over the u sual tangent space and tangen t groupoid. (2) Suppose that X is a tri vial w edge (see exa m ple 2.2 ): X = c π L ∪ M where M is a manifold w ith bounda ry L and L is the product of two manifolds L = s × Q w ith π : L → s being the ﬁrst projection . W e ha ve denoted by c π L = L × [0 , 1] / ∼ π the mappi ng cone o f ( L , π ) . In other wo rd c π L = s × cQ where cQ : = Q × [0 , 1] / Q × { 0 } is the cone over Q . W e denote again by s the image of L × { 0 } in X . Then X admits two strata: s and X ◦ = X \ s , F s = O s = L × ]0 , 1[ and F X ◦ = X ◦ \ O s = M . The tange nt space is T S X = Ts × ( Q × ]0 , 1[) × ( Q × ]0 , 1[) ⊔ TM ⇒ X ◦ where Ts × ( Q × ]0 , 1[) × ( Q × ]0 , 1[) is the produc t of the tangent space Ts ⇒ s with the pair groupo id ov er Q × ]0 , 1[ and TM den otes the restriction of the usual tangen t bundl e T X ◦ to the sub-man ifold with boundary M . G eometry & T opology XX (20X X) 1016 Clair e Debord and Jean-Marie Lescur e Remark 2 For an y stratu m s , the restrict ion of G t X to F ◦ s is equal to ∗ π ∗ s ( TS ) | F ◦ s × { 0 } ∪ F ◦ s × F ◦ s × ]0 , 1] ⇒ F ◦ s × [0 , 1] which is also th e restrictio n to F ◦ s of ∗ ( π s × Id) ∗ ( G t s ) , the pull -back by π s × Id : O ◦ s × [0 , 1] → S × [0 , 1] of the (usual) tangent groupoid of s : G t s = Ts × { 0 } ∪ s × s × ]0 , 1] ⇒ s × [0 , 1] . In the follo wing, w e will denote by A t π s × Id the Lie algebro id of ∗ ( π s × Id) ∗ ( G t s ) . 3.2 The Recursiv e construction. Thanks to the unfoldi ng process descri bed in 2.3 , one can also con struct the S -tange nt spaces of strati ﬁ ed pseudomanifo lds by an indu ction on the depth. If X is of depth 0 , it i s a smooth man ifold an d the S -tangen t space is the usual tange nt space TX viewed as a grou poid on X . Let k be an integer and assume that the S -tangent space of any pseudo m anifold of depth smaller than k is deﬁned. Let X be a stratiﬁed pseudomanifo ld of depth k + 1 and let 2 X be the stratiﬁed pseudomanif old of depth k obtaine d from X by applying 2.3 . W ith the notations of 2.3 w e deﬁne T S X = T S 2 X | 2 X ◦ ∩ X + b ∪ s ∈ S 0 ∗ π ∗ s ( Ts ) | O ◦ s ⇒ X ◦ where T S 2 X is the S -tangent space of the stratiﬁed pseudomanifo ld 2 X . Here we ha ve identiﬁed 2 X ◦ ∩ X + b with the subs et X ◦ \ O 0 = X b ∩ X ◦ of X ◦ . It is a simple ex ercise to see that this cons tructio n leads to the same deﬁnition of S -ta ngent spac e as the pre vious one. 3.3 The smooth structur e In this su bsection we prove th at the S -tangent space of a st ratiﬁed pseud omanifold, as well as its S -tange nt groupoid, can be en dowed with a smooth stru cture w hich r eﬂects the local struct ure of the pseudoman ifold itse lf. Let ( X , S , N ) be a stratiﬁed pseudoma nifold . T he smooth structure of T S X will depend on the stratiﬁcatio n and a smooth, decr easing , positi ve f unctio n τ : R → R such that τ ([0 , + ∞ [) = [0 , 1] , τ − 1 (0) = [1 , + ∞ [ and τ ′ does not van ish on ]0 , 1[ . The G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1017 functi on τ will be cal led a glu ing functio n . W e will also use func tions associated with τ and deﬁned on N s for any s ingula r stratum s by: for each singula r stratum: τ s = τ ◦ ρ s Observ e that τ s = 0 outsi de O ◦ s . Before coming int o the detai ls of the smoot h stru cture of T S X , let us describe its conseq uences for the con ver gence of sequ ences: A seque nce ( x n , V n , y n ) ∈ ∗ π ∗ s n ( Ts n ) | F s n where n belongs to N , goes to ( x , V , y ) ∈ ∗ π ∗ s ( Ts ) | F s if and only if: (3–1) x n → x , y n → y , V n + π s ( x n ) − π s ( y n ) τ s n ( x n ) → V The ﬁrst two con ve r gences ha ve an obv ious meaning, and they imply that for n big enoug h, s n ≤ s . The third o ne n eeds some expl anatio ns. L et us note z = π s ( x ) = π s ( y ) and z n = π s n ( x n ) = π s n ( y n ) . Since π s ( x n ) and π s ( y n ) become clo se to z , we can interp ret w n = π s ( x n ) − π s ( y n ) as a vecto r in T π s ( y n ) s (use any local chart of s around z ). Moreov er , using π s n ◦ π s = π s n , we see that this vector w n is vertic al for π s n , that is, belongs to the kernel K n of the dif ferential of π s n (suitab ly restri cted to s n ). Now , the meaning of last c on ver gence in ( 3–1 ) is T π s n ( V − w n /τ s n ( x n )) − V n → 0 which has to be interpr eted for each subseque nces of ( x n , V n , y n ) n with s n = s n 0 for all n ≥ n 0 big enoug h. The smooth struct ure of T S X will be obtain ed by an induction on the depth of the stratiﬁca tion, and a c oncrete atlas will b e giv en. For the sake o f completene ss, we also exp licit a L ie algebroid whos e inte gration gi ves the tange nt groupoi d G t X . W e be gin by descri bing the l ocal str ucture of X ◦ around its strata, then we will prov e ind uctiv ely the exi stence of a smooth structure on the S -tangent space. Next, an atlas of the resulti ng smooth structure is giv en by brut comp utatio ns. A similar constru ction is e asy to guess for the tan gent grou poid G t X . In the last part the pre vious smooth structure is recov ered in a more abstra ct approach using an integrabl e Lie algebroi d. These parts are quite technica l and can be left out as soon as you belie ve that the tangent space and the tange nt groupoi d can be endo wed with a smooth structure compatibl e with the topol ogy described above . Before goin g into the detai ls, we shou ld point out that the constructio ns describe d above depen d on the set of control data toget her with the choic e of τ , the gluing func tion. As far as we kno w , there is no way to get rid of these extra data. Nev ertheless, a conseq uence of the last chapter is that up to K -theo ry the S -tange nt space T S X only depen ds on X . G eometry & T opology XX (20X X) 1018 Clair e Debord and Jean-Marie Lescur e 3.3.1 T he local struct ure of X ◦ . W e no w descri be loc al charts of X ◦ adapte d to the stra tiﬁcation , called distin guishe d charts . Let z ∈ X ◦ and consid er the set S z : = { s ∈ S | z ∈ N s and ρ s ( z ) ≤ 1 } It is a non empty ﬁnite set, t otally o rdered a ccording to propos ition 1 , thus we can write S z = { s 0 , · · · , s κ } , s 0 < s 1 < · · · < s κ where s κ ⊂ X ◦ must be regu lar . Let n i be the dimens ion of s i , i ∈ { 0 , 1 , . . . , κ } and n = n κ = dim X ◦ . Let U z be an open neighbo rhood of z in X ◦ such that the follo wing hold: (3–2) U z ⊂ \ s ∈ S z N s and ∀ s ∈ S sing , U z ∩ O s 6 = ∅ ⇔ s ∈ S z In parti cular , the foll owing hold on U z : (3–3) for 0 ≤ i ≤ j ≤ κ : π s i ◦ π s j = π s i and ρ s i ◦ π s j = ρ s i . W ithout loss of generality , we can also assume that U z is the domain of a local chart of X ◦ . If κ = 0 , any local chart of X ◦ with domain U z will be calle d disting uished. When κ ≥ 1 , we can take successi vely cano nical forms of the submersi ons π s 0 , π s 1 . . . , π s κ a vailab le on a possibl y smaller U z , that is, one can shrink U z enoug h and ﬁnd dif feo- morphisms : (3–4) φ i : π s i ( U z ) → R n i for all i ∈ { 0 , 1 , . . . , κ } such that the diagr am: (3–5) π s i ( U z ) π s j   φ i / / R n i σ n j   π s j ( U z ) φ j / / R n j commutes for all i , j ∈ { 0 , 1 , . . . , κ } such that i ≥ j . Above , for any integers p ≥ d , the map σ d : R p → R d denote s the canonica l proje ction onto the last d coordinates. Remember that s κ is regular so π s κ is the ide ntity map and φ : = φ κ is a loc al chart around z of X ◦ . Now we set: G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1019 Deﬁnition 5 A distin guished chart of X ◦ around z ∈ X ◦ is a local chart ( U z , φ ) around z such that U z satisﬁes ( 3–2 ) together with dif feomorphis m s ( 3–4 ) satisfying ( 3–5 ) and φ = φ κ . From now on, a riemannian metric is chosen on X ◦ (an y adapted m etric in th e sense of [ 4 ] is s uitabl e for our purpose ). Recall that for an y stratum s , the map π s : N ◦ s → s is a smooth submersion. T hus, if K s ⊂ T N ◦ s denote s the kernel o f the d ifferen tial map T π s and q s : T N ◦ s → T N ◦ s the orthog onal proje ction on K s , the map (3–6) ( q s , T π s ) : T N ◦ s → K s ⊕ π ∗ s ( Ts ) is an isomorp hism and the vector bu ndle π ∗ s ( Ts ) can be identiﬁed with the othogona l complemen t of K s into T N s = TX ◦ | N s . No w , let ( U z , φ ) be a distingu ished chart around some z ∈ X ◦ . Set S z = { s 0 , s 1 , . . . , s κ } with s 0 < s 1 < . . . < s κ , and set K i = K s i | U z , U i = π s i ( U z ) for all i = 0 , 1 , . . . , κ . By 3–3 we ha ve: (3–7) U z × { 0 } = K κ ⊂ K κ − 1 ⊂ · · · ⊂ K 1 ⊂ K 0 ⊂ TU z . Rewri ting the diagra m ( 3–5 ) for the diff erentia l maps and i = κ , we get for all j ≤ κ : (3–8) TU z T π s j   T φ / / R n × R n σ n j × σ n j   TU j T φ j / / R n j × R n j and we see that T φ sen ds the ﬁltration ( 3–7 ) to the follo wing ﬁ ltration : (3–9) R n × { 0 } ⊂ R n × R n − n κ − 1 ⊂ · · · ⊂ R n × R n − n 0 ⊂ R n × R n , where R n − n i is included in R n by the map v 7→ ( v , 0) ∈ R n − n i × R n i ≃ R n . This proper ty can be reformula ted in terms of natural gr aduations associ ated w ith ( 3–7 ) an d ( 3–9 ) (and will be used in this latte r form). Indeed, let T i be the orth ogona l complemen t of K i into K i − 1 for all i = 0 , . . . , κ (with the con ven tion K − 1 = TU z ). More over , on the euclide an side, let us embed R n i − n i − 1 into R n by the map: v ∈ R n i − n i − 1 7− → (0 , v , 0) ∈ R n − n i × R n i − n i − 1 × R n i − 1 ≃ R n for all i = 0 , 1 , . . . , κ (by con v ention n − 1 = 0 ). W ith these no tations an d con ven tions, the ﬁltrations ( 3–7 ) and ( 3–9 ) gi ve rise to the follo wing decomp ositio ns: (3–10) TU z = T κ ⊕ T κ − 1 ⊕ · · · ⊕ T 0 G eometry & T opology XX (20X X) 1020 Clair e Debord and Jean-Marie Lescur e and (3–11) R n × R n = R n × ( R n − n κ ⊕ R n κ − n κ − 1 ⊕ · · · ⊕ R n 1 − n 0 ⊕ R n 0 ) No w , that T φ respec ts the ﬁltrations ( 3–7 ) and ( 3–9 ) means that for all x ∈ U z the linear map T φ x is uppe r triangular w ith resp ect to the decomposition s ( 3–10 ) and ( 3–11 ). The diagon al blocks of T φ are the maps: (3–12) δ j φ : T j − → R n × R n j − n j − 1 ; j = 0 , 1 , . . . , κ, obtain ed by composin g T φ on the left a nd on the right respecti vely by th e projections : TU z = T κ ⊕ T κ − 1 ⊕ · · · ⊕ T 0 − → T j and R n × ( R n − n κ ⊕ R n κ − n κ − 1 ⊕ · · · ⊕ R n 1 − n 0 ⊕ R n 0 ) − → R n × R n j − n j − 1 . The diagon al part of T φ will be deﬁned by ∆ φ = ( δ κ φ, δ κ − 1 φ, . . . , δ 0 φ ) . Of course , the in verse of T φ is also uppe r triangu lar with diago nal block s gi ven by ( δ j φ ) − 1 , j = 0 , 1 , . . . , κ . W e ha ve similar pr operties for all the unde rlying m aps φ i , i = 0 , 1 , . . . , κ − 1 coming with the disti nguish ed chart. T o ﬁx notati ons and for future reference s, let U i denote π s i ( U z ) , and T j i denote T π s i ( T j ) for all j ≤ i < κ . Applying no w T π s i to ( 3–10 ) yields: (3–13) TU i = T i i ⊕ T i − 1 i ⊕ · · · ⊕ T 0 i , It follo ws that the differ ential maps: (3–14) T φ i : T i i ⊕ T i − 1 i ⊕ · · · ⊕ T 0 i − → R n i × ( R n i − n i − 1 ⊕ · · · ⊕ R n 1 − n 0 ⊕ R n 0 ) for all i = 0 , 1 , . . . , κ − 1 are upper triang ular with diago nal blocks δ j φ i deﬁned as abo ve. Note that for all j ≤ i ≤ k ≤ κ , ( T π s i )( T j k ) = T j i and that apply ing the correct restric tions and projections in ( 3–8 ) gi ves the follo wing commutativ e diagr am: (3–15) T j T π s i   δ j φ k / / R n k × R n j − n j − 1 σ n i × Id   T j i δ j φ i / / R n i × R n j − n j − 1 G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1021 3.3.2 T he smooth struct ure by induction W e sho w that T S X can be provid ed with a smooth structu re by a simple recursi ve ar gument. Let us ﬁrst intro duce the s -e xpone ntial maps . Let s be a stratum. The corresp onding s -e xponent ial m ap will be an exponent ial along the ﬁbers of π s . Precisely , recall that the map π s : N ◦ s → s is a smooth submersion, K s ⊂ T N ◦ s denote s the kernel of the dif ferential map T π s and q s : T N ◦ s → T N ◦ s the orth ogona l proj ection on K s . The subb undle K s of TX ◦ inheri ts from TX ◦ a riemannian m etric whose associated riemanni an connec tion is ∇ s = q s ◦ ∇ , where ∇ is the riemannian connection of the metric on X ◦ . The associat ed exp onenti al map Exp s : V s ⊂ K s → N ◦ s is smooth and deﬁned on an open neighbor hood V s of the zero section of K s . Moreove r it satisﬁes : - π s ◦ E xp s = π s . - For any ﬁ ber L s of π s , the restric tion of Exp s to L s is the usual expone ntial map for the submanifo ld L s of X ◦ with the induc ed riemannian structure. If X is a stratiﬁed pseudomanifo ld of de pth 0 it is smoo th and its S -tan gent space is the usual tange nt space TX equi pped with its usual smooth structure . Suppose that the S -tang ent spac e of any stratiﬁed pseudomanifol d of depth strictly smaller than k is equipped w ith a smooth structure for some integer k > 0 . Let X be a stratiﬁed pseu domanifold of dept h k and tak e 2 X be the stratiﬁed pseud omanifold of depth k − 1 obtained from X by the unfold ing proc ess 2.3 . A ccordi ng to 3 , with the notati ons of 2.3 we ha ve T S X = T S 2 X | 2 X ◦ ∩ X + b ∪ s ∈ S 0 ∗ π ∗ s ( Ts ) | O ◦ s ⇒ X ◦ . Let L ◦ be the boundary of 2 X ◦ ∩ X + b in X ◦ . W e equip the restriction of T S X to 2 X ◦ ∩ X + b \ L ◦ with the smoo th structur e coming from T S 2 X and its restric tion to any O s 0 , s 0 ∈ S 0 , with the usual smooth structure. It remains to describe the gluing over L ◦ . One can ﬁ nd an open subset W of T S 2 X which con tains the restric tion of T S 2 X to L ◦ such that the follo wing map is deﬁned: Θ : W − → T S X ( x , u , y ) 7→  ( x , T π s 0 ( u ) , Exp s 0 ( y , − τ s 0 ( x ) q s 0 ( u ))) if x ∈ O ◦ s 0 , s 0 ∈ S 0 ( x , u , y ) elsewhe re G eometry & T opology XX (20X X) 1022 Clair e Debord and Jean-Marie Lescur e Here, if s denotes the unique stratum such that x , y ∈ F s , the vector bund le π ∗ s ( Ts ) is identi ﬁed with the ort hogon al complement o f K s into T N ◦ s , in other words q s 0 ( y , u ) = q s 0 ( W − q s ( W )) w here W ∈ T y X ◦ satisﬁes T π s ( W ) = u . Then, we eq uip T S X with the unique smooth stru cture compatible with the one pre- viousl y deﬁned on T S X | X ◦ \ L ◦ and such that the map Θ is a smooth dif feomorph ism onto its image. The non tri vial point is to check that the restriction of the map Θ ov er O ◦ s 0 is a dif feomorphism onto its image for any s 0 ∈ S 0 . This will follo ws from the follo wing lemma. Lemma 6 If s 0 < s , for any x 0 ∈ s 0 and x ∈ s with π s 0 ( x ) = x 0 . The follo wing asserti ons hold: (1) E : = q s 0 ( π ∗ s ( Ts )) is a sub-b undle of K s 0 of dimension dim( s ) − dim( s 0 ) . (2) Let E x : = q s 0 ( π ∗ s ( Ts )) | π − 1 s ( x ) be the restric tion of E to the submani fold π − 1 s ( x ) . There exist s a neighborh ood W of the zero section of E x such that the restrictio n of Exp s 0 to W is a dif feomorphism onto a neighborho od of π − 1 s ( x ) in π − 1 s 0 ( x 0 ) . Pro of 1. The ﬁrst assertion follows from the inclusion : π ∗ s 0 ( Ts 0 ) = K ⊥ s 0 ⊂ K ⊥ s = π ∗ s ( Ts ) which ensures that the dimension of the ﬁbers of q s 0 ( π ∗ s ( Ts )) is constant equal to dim( s ) − dim( s 0 ) . The same ar gument shows that K s 0 = K s ⊕ E . 2. If Ψ d enotes the restriction of Exp s 0 to E x then T Ψ ( z , 0)( U , V ) = U + V where ( z , U ) ∈ K s and V ∈ E z . Since K s ∩ E is the tri vial bun dle we get that T Ψ is injecti ve and since E x and π − 1 s 0 ( x 0 ) ha ve same dimension, it is bijecti ve. W e conlude w ith the local in v ersion theore m . 3.3.3 A n atla s fo r T S X The atlas will contai n two kinds of local charts. The kind of these charts will depend on the fact that their domains mee t or not a glu ing between the dif ferent pieces co m posing the tangen t space T S X , that is the bounda ry of some F s . The ﬁrst kind of chart s, called reg ular chart s are charts whose domain is contain ed in T S X | ◦ F s for a gi ven stratum s of the strati ﬁcation. W e obs erve that T S X | ◦ F s is a smooth groupoi d as an o pen sub groupoid of ∗ π ∗ s ( Ts ) ⇒ N ◦ s . T hus, regular charts hav e domains contain ed in ⊔ s ∈S ◦ F s G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1023 and coincide with the us ual local charts o f the (disjoi nt) unio n of th e smooth groupo ids ⊔ s ∈S ∗ π ∗ s ( Ts ) . The second kind of charts, called deformat ion charts (adapt ed to a stratum s ), are charts whose domain meets T S X | ∂ F s for a gi ven stratum s , that is, charts around points in [ s ∈ S T S X | ∂ F s . Their description is more in v olved. Let ( p , u , q ) ∈ T S X . Thus there is a stratum s such that p a nd q belong to F s with π s ( p ) = π s ( q ) and u ∈ T π s ( p ) s . Assu me that p ∈ ∂ F s . This m eans that ρ s ( p ) < 1 , that ρ t ( p ) ≥ 1 for all stra ta t < s and that the set of strata t such that t < s and ρ t ( p ) = 1 is not empty . Using again the axioms of the stratiﬁcation , we see that this set is totall y ordere d and w e denote s 0 , s 1 , . . . , s l − 1 its elemen ts listed by increasing order . W e als o set s l = s . Observ e that: (3–16) { s 0 , s 1 , . . . , s l } = S p ∩ { t ∈ S | t ≤ s } and that, than ks to the compa tibilit y condition s ( 3–3 ), this set only depends on π s ( p ) and thus is equa l with the correspo nding set associated with q . Let us take distin guished ch arts φ : U p → R n around p and φ ′ : U q → R n around q . Since π s ( p ) = π s ( q ) , w e can also assume withou t loss of generality that: (3–17) π s i ( U p ) = π s i ( U q ) and φ i = φ ′ i for i = 0 , . . . , l . W e w ill use the same notations as in parag raph 3.3.1 : n i = dim s i , U i = π s i ( U p ) , K i = ker( T π s i ) | U p , T i = K ⊥ K i − 1 i for all i = 0 , 1 , . . . , l (here again K − 1 = TU p ). The main dif ference w ith the settings of the paragrap h 3.3.1 is that we for get the strata bigger tha n s in S p and S q to co ncentrate on the lo w er (an d common) st rata in S p and S q . It amounts to forg et the tail of the ﬁltration ( 3–7 ) up to the term K l : (3–18) K l ⊂ K l − 1 ⊂ · · · ⊂ K 0 ⊂ TU p and this leads to a less ﬁne gradu ation: (3–19) TU p = K l ⊕ T l ⊕ T l − 1 ⊕ · · · ⊕ T 0 Let us also intro duce the positi ve smooth functions: t i = i X j = 0 τ ◦ ρ s j , i = 0 , 1 , . . . , l ; θ i = l Y j = i − 1 t j , i = 1 , . . . , l G eometry & T opology XX (20X X) 1024 Clair e Debord and Jean-Marie Lescur e Note that t j (resp. θ j ) is strictly positi ve on F s i if j ≥ i (res p. j > i ) and v anishes identi cally if j < i (resp. j ≤ i ). Finally we will write: ∀ x ∈ U p , φ ( x ) = ( x l + 1 , x l , . . . , x 1 , x 0 ) ∈ R n − n l × R n l − n l − 1 × · · · × R n 1 − n 0 × R n 0 , and for all j = 0 , 1 , . . . , l , π s j ( x ) = x j , thus φ j ( x j ) = ( x j , x j − 1 , . . . , x 0 ) ∈ R n j ; and we adop t similar notati ons for φ ′ and y ∈ U q . W e are ready to deﬁne a deformati on chart around the point ( p , u , q ) . T he domain will be: (3–20) e U = T S X | U p U q and the chart itself : (3–21) e φ : e U → R 2 n is deﬁned as follo w s. Up to a shrinking of U p and U q , the follo wing is true: for all ( x , v , y ) ∈ e U , there exist s a unique i ∈ { 0 , 1 , . . . , l } such that x ∈ F s i . Then ( x , v ) ∈ π ∗ s i ( TU i ) , an d we set: (3–22) e φ ( x , v , y ) =  φ ( x ) , x l + 1 − y l + 1 θ l + 1 ( x ) , . . . , x i + 1 − y i + 1 θ i + 1 ( x ) , ∆ φ i ( x i , v )  The map e φ is clea rly injecti ve with in verse deﬁned as follows. For ( x , w ) ∈ e φ ( e U ) and i such that φ − 1 ( x ) ∈ F s i : e φ − 1 ( x , w ) =  φ − 1 ( x ) , ( ∆ φ i ) − 1 ( x i , w ) , φ ′− 1 ( x − Θ [ i + 1] ( φ − 1 ( x )) · w )  where x i = σ n i ( x ) and, using the decompos ition w = ( w l + 1 , w l · · · , w 0 ) ∈ R n − n l × R n l − n l − 1 × · · · × R n 1 − n 0 × R n 0 , we ha ve set Θ [ i + 1] ( x ) · w = θ l + 1 ( x ) w l + 1 + · · · + θ i + 1 ( x ) w i + 1 ∈ R n − n i × { 0 } ⊂ R n . T o ensure that ( e φ, e U ) is a local chart, it remains to check that e φ ( e U ) is an open subset of R 2 n . It is easy to see that e φ ( ◦ F s i ) is open for e very i ∈ { 0 , . . . , l } so we consid er ( p , u , q ) ∈ e U such that p ∈ ∂ F s i for some integer i . L et J = { i 0 , . . . , i k } ⊂ { 0 , 1 , . . . , i − 1 } such that: ∀ j ∈ J , ρ s j ( p ) = 1 . G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1025 Thus we ha ve: (3–23) ρ s i ( p ) < 1; ∀ j ∈ J , ρ s j ( p ) = 1; ∀ j 6∈ J and j < i , ρ s j ( p ) > 1 by construc tion, q sat isﬁes the same relations. Set e φ ( p , u , q ) = ( x 0 , v 0 ) . Using the T aylor formula and the f act that θ j + 1 is ne gligib le with respect to 1 − ρ s j at the re gion ρ s j = 1 , noting also the in va riance of ρ s k with respect to perturbat ions of points along the ﬁbers of π s k + 1 , π s k + 2 , . . . ; we prove that ther e exist an open ball B 1 of R n center ed at x 0 and an open ball B 2 of R n center ed at 0 and conta ining v 0 such that for all ( x , v ) ∈ B 1 × B 2 , if x = φ − 1 ( x ) ∈ F s j for j ∈ J or j = i , then y = φ ′− 1 ( x − Θ [ j + 1] ( x ) · v ) ∈ F s j . This pro ves that ( x , v ) ∈ Im e φ , thus e φ ( p , u , q ) ∈ B 1 × B 2 ⊂ Im e φ and the require d assertion is prov ed. W e e nd with: Theor em 1 The collecti on of regu lar and deformation charts provides T S X with a structu re of smooth groupoid. Pro of The comp atibili ty between a re gular and a defo rmation chart contain s no issue and is ommitted. W e need only to check the compati bility between a deformat ion chart adapted to a stratu m s and a deformati on chart adapted t o a str atum t , whe n their domains ov erlap, which implies automatical ly that s < t or s > t or s = t . Let us work out only the case s = t , since the other case is similar . W e ha ve he re to compare two charts e φ and e ψ with common domain e U and in v olving the same chai n of strata s = s l > s l − 1 > · · · > s 0 . The w hole notat ions are as be fore and ψ , ψ ′ are the underly ing c harts of X ◦ allo wing the deﬁnitio n of e ψ . W e note, for the sak e of concis ion, u k (resp. u ′ k ), k = l + 1 , . . . , 0 , the coordinate functions of u : = ψ ◦ φ − 1 (resp. ψ ′ ◦ φ ′− 1 ) with respect to the decompo sition ( 3–11 ) of R n . Observe , thank s to the par ticula r assumptions made on φ, φ ′ , ψ , ψ ′ (cf.( 3–5 ), ( 3–17 )), that u k ( x ) only depen ds on x k : = ( x k , x k − 1 , . . . , x 0 ) ∈ R n k and that u k = u ′ k for all k < l + 1 . Let ( x , v ) ∈ Im e φ and i such that x = φ − 1 ( x ) ∈ F s i . T hen: (3–24) e ψ ◦ e φ − 1 ( x , v ) =  u ( x ) , u l + 1 ( x ) − u ′ l + 1 ( x − Θ [ i + 1] · v ) θ l + 1 , u l ( x ) − u l ( x − Θ [ i + 1] · v ) θ l , . . . . . . , u i + 1 ( x ) − u i + 1 ( x − Θ [ i + 1] · v ) θ i + 1 , ( ∆ ψ i ) ◦ ( ∆ φ i ) − 1 ( v )  W e need to che ck that the abov e expres sion matches smoothl y w ith the corre spond ing exp ression for an inte ger k ∈ [ i , l ] when θ k ( x ) (and thus θ k − 1 , . . . , θ i + 1 ) goes to zero. G eometry & T opology XX (20X X) 1026 Clair e Debord and Jean-Marie Lescur e For tha t, the T aylor formula applied to u r , k ≥ r ≥ i + 1 , shows that the map deﬁned belo w is smooth in ( x , v , t ) where ( x , v ) are as before and t = ( t l , t l − 1 , . . . , t 0 ) ∈ R l + 1 is this time an arbitr ary ( l + 1) -uple close to 0 : ( u r ( x ) − u r ( x − Θ [ i + 1] · v ) θ r if θ r = Π l r − 1 t j 6 = 0 d ( u r ) x ( v r + t r − 2 v r − 1 + · · · + t i v i + 1 ) if ∃ j ∈ { r − 1 , r , . . . , l } such that t j = 0 . In our case , t j = t j ( x ) and t k − 1 , . . . , t i go to zero, so the seco nd line in the pre vious exp ression is just : d ( u r ) x ( v r ) and for obv ious matricial reasons: d ( u r ) x ( v r ) = ( ∆ ψ k ) ◦ ( ∆ φ k ) − 1 ( v r ) Summing up these relat ions for r = i + 1 , . . . , k , we arri ve at the desired identity . Thus, T S X is endo wed with a str ucture of smooth man ifold. Changing the riemannian metric on X ◦ modiﬁes the choices of the T i j ’ s, but giv es rise to compat ible char ts. Moreo ver , the smoothnes s of all algebraic operation s associate d with this groupo id is easy to check in these local charts. 3.3.4 T he Lie algebr oid of the tangent groupoid W e desc ribe here the smooth structure of t he tangent space v ia i ts inﬁnitesim al structure, namely its Lie algebr oid. Precisely , we deﬁne Q s : TX ◦ − → TX ◦ ( z , V ) 7→  ( z , τ s ( z ) q s ( z , V )) if z ∈ N ◦ s 0 else where By a slight ab use of notation, we will keep the notations q s and Q s for the correspond ing maps indu ced on the set of local tangent vecto r ﬁelds on X ◦ . Let A be the smooth vector b undle A : = TX ◦ × [0 , 1] ov er X ◦ × [0 , 1] . W e deﬁne t he follo wing m orphis m of vector b undle : Φ : A = TX ◦ × [0 , 1] − → TX ◦ × T [0 , 1] ( z , V , t ) 7→ ( z , tV + P s ∈ S sing Q s ( z , V ); t , 0) In the seq uel we will gi ve an id ea of how one ca n sho w that there is a uniq ue structu re of L ie algeb roid on A such that Φ is its anchor map. The Lie algebroid A is almost G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1027 injecti ve and so it is integra ble, moreov er w e w ill see that at a set le vel G t X must be a group oid which integ rates it [ 9 , 11 ]. In particu lar G t X can be equipped w ith a unique smooth struc ture such that it integrat es the Lie algebroid A . No w we can state the follo wing: Theor em 2 There exists a unique struc ture of Lie algebroid on the smooth vec tor b undle A = T X ◦ × [0 , 1] ov er X ◦ × [0 , 1] with Φ as anch or . T o pro ve this theorem w e will need se veral lemmas: Lemma 7 Let s 0 and s 1 be two strat a such that d ( s 0 ) ≤ d ( s 1 ) . (1) For an y tangent vector ﬁeld W on X ◦ , Q s 1 ( W )( τ s 0 ) = 0 . (2) For an y ( z , V ) ∈ TX ◦ , the follo wing equali ty holds: Q s 1 ◦ Q s 0 ( z , V ) = Q s 0 ◦ Q s 1 ( z , V ) = τ s 0 ( z ) Q s 1 ( z , V ) . Pro of First notice that outside O s 0 ∩ O s 1 either Q s 1 hence Q s 1 ( W ) or τ s 0 and Q s 0 v anish thus the equalities in (1) and (2) are simply 0 = 0 . (1) According to the compatibility conditio ns 3–3 we ha ve ρ s 0 ◦ π s 1 = ρ s 0 on O s 0 ∩ O s 1 . Thus ρ s 0 is co nstant o n the ﬁbers of π s 1 and s ince τ s 0 = τ ◦ ρ s 0 , τ s 0 is also cons tant on the ﬁbers of π s 1 . For any tangent vector ﬁeld W , a nd an y z ∈ O ◦ s 1 the vector Q s 1 ( W )( z ) is tange nt to the ﬁbers of π s 1 thus Q s 1 ( V )( τ s 0 ) = 0 on O s 0 ∩ O s 1 . (2) The result follo ws from the ﬁrst remark and the e quality 3–7 of the part abo ve. The nex t lemma ensures that Φ is almost injecti ve, in particula r it is injecti ve in restric tion to X ◦ × ]0 , 1] . A simple calcul ation shows the follo wing: Lemma 8 For an y t ∈ ]0 , 1] the b undle m ap Φ t is bijec ti ve, moreo ver Φ − 1 t ( z ) = 1 t V − X s ∈ S sing 1 ( t + t s ( z )) · ( t + t s ( z ) − τ s ( z )) Q s ( z , V ) where for an y singu lar stratu m s the map t s is deﬁned as follo ws: t s : X ◦ → R , t s ( z ) = l X s 0 ≤ s τ s 0 ( z ) . Thus in ord er to prov e the theorem 2 it is eno ugh to sho w that locally the image of the map induced by Φ fr om the set of smooth local section s of A to the set of smooth local tange nt vector ﬁelds on X ◦ × [0 , 1] is stable under the Lie brack et. G eometry & T opology XX (20X X) 1028 Clair e Debord and Jean-Marie Lescur e Idea of the pro of of Theor em 2 First notice that outs ide the closure of ∪ s i ∈ S sing O ◦ s i the image under Φ of loca l tangent vector ﬁelds is clearly stable under Lie Bracke t. Thus using decomposition of the form 3–10 describe d in the last part and standard ar guments it remains to sho w that if s a and s b are strat a of depth respecti vely a and b with s a ≤ s b , if U is an open subset of X ◦ , as small as we want contained in N s a ∩ N s b , and if W ⊥ , V ⊥ , V a and W b are tang ent vector ﬁelds on U , satis fying: V ⊥ and V a can be porje cted by π s b , Q s ( W ⊥ ) = Q s ( V ⊥ ) = 0 for any s ∈ S , Q s ( V a ) =  τ s V a when s ≤ s a 0 else where and Q s ( W a ) =  τ s W a when s ≤ s b 0 else where , then [ Φ ( W ⊥ + W b ) , Φ ( V ⊥ )] and [ Φ ( W b ) , Φ ( V a )] are in the image of Φ . In oth er word, we ha ve to show that the maps ( z , t ) ∈ X ◦ × ]0 , 1] 7→ ( Φ − 1 t ([ Φ ( W ⊥ + W b ) , Φ ( V ⊥ )]( z )) , t ) and ( z , t ) ∈ X ◦ × ]0 , 1] 7→ ( Φ − 1 t ([ Φ ( W b ) , Φ ( V a )]( z )) , t ) can be exte nded into smooth local section of A . The result follo ws from our prece ding lemmas and usual calcula- tions. No w we can state: Theor em 3 The groupoid G t X can be equi pped with a smooth struc ture such tha t its Lie algebr oid is A with Φ as anch or . Pro of Accordin g to proposition 2 and lemma 8 , the Lie algebroid A is almost injecti ve. Thus acc ording to [ 11 ] there is a unique s -connected qua si-grap hoid G ( A ) ⇒ X ◦ × [0 , 1] which inte grates A . Suppose for simpli city that for each stratum s , O ◦ s is connect ed (which w ill ensu re that G t X | F ◦ s × [0 , 1] is a s -conn ected quasi-grap hoid). Moreo ver the map Φ satisﬁes: (i) Φ induces an isomorphis m from A ]0 , 1] : = A | X ◦ × ]0 , 1] to TX ◦ × ]0 , 1] , (ii) for an y stratum s , the Lie al gebroi d A restricted ov er F ◦ s × [0 , 1] to a L ie algebr oid A s : = A| F ◦ s × [0 , 1] which is isomorphic to the restri ction of A t π s × Id ov er F ◦ s × [0 , 1] . Thus, again b y usi ng th e un iquene ss of s -connecte d quasi -graph oid inte grating a gi ven almost inject i ve L ie algebr oid, we obtain: (i) the restr iction of th e groupoid G ( A ) over X ◦ × ]0 , 1] is isomorphic to X ◦ × X ◦ × ]0 , 1] ⇒ X ◦ × ]0 , 1] , the pair groupoid on X ◦ parametr ized by ]0 , 1] , G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1029 (ii) for each stratum s the restrictio n ov er F ◦ s × [0 , 1] is equal to G t X | F ◦ s × [0 , 1] . Finally G ( A ) = G t X and there is a u nique smooth stru cture on G t X such that A is i ts L ie algebr oid. If some O ◦ s is not c onnec ted, we rep lace in the cons truction of the tangent space the groupoi d ∗ π ∗ s ( Ts ) | F s by its s -connect ed compon ent. L et CT S X and C G t X be the corres pondin g groupoids. The previo us ar guments apply and the groupoid C G t X admits a unique smooth stru cture such that A is its Lie algebroid . One can then sho w that there is a uniq ue smooth structu re on G t X such th at C G t X is its s -con nected compone nt. Precisely , a ccordi ng to [ 11 ] there is a quasi-graph oid G I ( A ) ⇒ X ◦ × [0 , 1] which inte grates A and is m aximal for the inclusion among quasi-graph oids which integrate A . The groupoi d C G t X is then the s -connect ed component of G I ( A ) . In particu lar it is open in G I ( A ) . Let X r : = X ◦ \ ∂ s ∈ S F s . T he restriction of G t X to X r × [0 , 1] is a quasi-graph oid which integ rates the restr iction of A to X r × [0 , 1] and is then clearly an open sub-groupoi d of G I ( A ) . Now we hav e G t X = { γ · η | γ ∈ C G t X , η ∈ G t X | X r × [0 , 1] , s ( γ ) = r ( η ) } which is open in G I ( A ) and so G t X inheri ts the required smooth struc ture. Thus T S X , w hich is the restrictio n of G t X to the satur ated set X ◦ × { 0 } , inherits from G t X a smooth str ucture which i s equiv alent to the on e described in prev ious par agraphs. 3.3.5 S tandard pr ojection from the tangent space onto the space The spa ce of orbits of X ◦ / T S X is equi vale nt to X in the s ense that there is a can onical isomorph ism C 0 ( X ◦ / T S X ) ≃ C ( X ) . Deﬁnition 6 L et r , s : T S X → X ◦ be the targ et and source maps of the S -tangent space of X . A continuou s map p : X → X is a stan dar d pr ojection for T S X on X if: (1) p ◦ r = p ◦ s . (2) p is homotopic to the identity map of X . A stand ard projectio n p for T S X on X is surjec ti ve if p | X ◦ : X ◦ → X is onto. This deﬁnitio n leads to the follo wing: Lemma 9 (1) There e xists a standar d surjec ti ve projection for T S X on X . (2) T wo standar d projections are homotopic and the homotopy can be done within the set of standa rd projections . G eometry & T opology XX (20X X) 1030 Clair e Debord and Jean-Marie Lescur e Pro of 1) If X has depth 0 , X ◦ = X and we just tak e p = id . Let us consi der X with depth k > 0 . Choo se a smooth non decreasin g function f : R + → R + such that f ([0 , 1]) = 0 and f | [2 , + ∞ [ = Id . R ecall tha t there exists for e ach singular stratum s an isomorph ism 2–1 : Ψ s : N s → c π s L s = L s × [0 , + ∞ [ / ∼ s . we deﬁne the map p s : N s − → N s by the formula : Ψ s ◦ p s ◦ Ψ − 1 s [ x , t ] = [ x , f ( t )] . For e ach inte ger i ∈ [0 , k − 1] , we deﬁne a continu ous m ap: p i : X − → X by setting p i ( z ) = p s ( z ) if z belon gs to N s for some singu lar stratum of depth i and p i ( z ) = z else where. In particular , p i | O s = π s for e very stratum s of depth i . Finally we set: p = p 0 ◦ p 1 ◦ · · · ◦ p k − 1 . This is the map we look ed for . Indeed: Let γ ∈ T S X . There exists a unique stratum s such that γ ∈ ∗ π ∗ s ( TS ) . If s is regu lar , then r ( γ ) = s ( γ ) so the result is tri vial here. Let us assume that s is singu lar and let i < k be its depth. B y deﬁnition, r ( γ ) and s ( γ ) belong to O s . For each stratum t ≥ s of depth j ≥ i , we ha ve ev erywhere it makes sense: π t ◦ p t = π t , π s ◦ π t = π s , ρ s ◦ π t = ρ s thus: ρ s ◦ p t = ρ s ◦ π t ◦ p t = ρ s ◦ π t = ρ s which pro ves that p j ( O s ) = O s , and moreo ver: π s ◦ p t = π s ◦ π t ◦ p t = π s ◦ π t = π s Recalling that p i | O s = π s | O s , this last relation implies: p i ◦ · · · ◦ p k − 1 | O s = π s ◦ p i + 1 ◦ · · · ◦ p k − 1 | O s = π s | O s Since by deﬁnitio n we also ha ve π s ( r ( γ )) = π s ( s ( γ )) , we conclude that: p ( r ( γ )) = p 0 ◦ · · · p i − 1 ◦ π s ( r ( γ )) = p 0 ◦ · · · p i − 1 ◦ π s ( s ( γ )) = p ( s ( γ )) If in the deﬁnition of p , w e replace the function f by t Id R + + (1 − t ) f , we get a homotop y between p and Id X . G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1031 Finally , p has the required surjecti vity property: p k − 1 ( X ◦ ) = X ◦ S d ( s ) = k − 1 s and for all j we ha ve the equality p j − 1 ( X ◦ S d ( s ) ≥ j s ) = X ◦ S d ( s ) ≥ j − 1 s . 2) Let q be a standard proj ection and p be the standard project ion b uilt in 1). Let also q t be a homo top y between q and Id X and p t the homoto py built in 1) between p and Id X . Observ e that q t ◦ p is a standard projection, pro viding a path of standard projec tions between q ◦ p and p . Moreo ver , by constr uction of p t , the inclusi on Im( p t ◦ r , p t ◦ s ) ⊂ Im( r , s ) holds for any 1 ≥ t > 0 , thus q ◦ p t is a sta ndard projecti on, pro viding a path of standard projec tions between q ◦ p and q . Thus, an y standard projec tion q is homotopic to p within the set of standa rd projection s and the result is pro ved . Remark 3 L et p be the surjecti ve standard projection built in the proof of the last propo sition. The map p ◦ r : T S X → X pr ovi des T S X w ith a structur e of continuous ﬁeld of groupoid s. Follo wing the arg uments of ([ 13 ], remark 5), it can be sho wn that each ﬁber of this ﬁeld is amenable, thus T S X is amenable and C ∗ ( T S X ) = C ∗ r ( T S X ) is nuclea r . T he same holds for G t X and all othe r deformation groupoids used below . 4 Poin car ´ e duality f o r stratiﬁed pseudo-manif olds Let X be a compact stratiﬁed pseudomanifo ld of depth k ≥ 0 . The tangent group oid G t X is a deformation groupoid, thus it pro vides us with a K - homolog y class, called a pr e-Dirac element: (4–1) δ X = [ e 0 ] − 1 ⊗ [ e 1 ] ∈ KK ( C ∗ ( T S X ) , C ) . Here e 0 : C ∗ ( G t X ) → C ∗ ( T S X ) and e 1 : C ∗ ( G t X ) → K ( L 2 ( X ◦ )) are the usual e v aluation homomorph isms. No w we need: Lemma 10 1) Let p : X ◦ → X be a surje cti ve standar d projecti on for T S X . The formula: ∀ a ∈ C ∗ ( T S X ) , f ∈ C ( X ) , γ ∈ T S X , ( a · f )( γ ) = f ( p ◦ r ( γ )) . a ( γ ) deﬁnes a C ( X ) -algeb ra structure on C ∗ ( T S X ) . 2) For an y standard projectio n p for T S X , the formula : ∀ a ∈ C ∗ ( T S X ) , f ∈ C ( X ) , γ ∈ T S X , Ψ X ( a · f )( γ ) = f ( p ◦ r ( γ )) . a ( γ ) deﬁnes a homomorp hism Ψ X : C ∗ ( T S X ) ⊗ C ( X ) → C ∗ ( T S X ) whose class [ Ψ X ] ∈ KK ( C ∗ ( T S X ) ⊗ C ( X ) , C ∗ ( T S X )) does not depend on the choice of p . G eometry & T opology XX (20X X) 1032 Clair e Debord and Jean-Marie Lescur e The last assertion uses Lemma 9 . Note that if k = 0 , X is smooth and we can choo se p = Id , thus: (4–2) Ψ X ( a ⊗ b )( V ) = b ( x ) . a ( x , V ) for all V ∈ T x X , x ∈ X , a ∈ C ( X ) and b ∈ C ∗ ( TX ) . From now on, we choos e a surjecti ve standard projection and denot e by Ψ X : C ( X ) ⊗ C ∗ ( T S X ) → C ∗ ( T S X ) the homomorp hism deﬁned in the pre vious lemma. W e set: (4–3) D X = Ψ ∗ X ( δ X ) = [ Ψ X ] ⊗ δ X ∈ KK ( C ∗ ( T S X ) ⊗ C ( X ) , C ) . This sectio n is dev oted to the proof of the main theorem: Theor em 4 Let X be a comp act stratiﬁed ps eudoman ifold. The K -homolo gy class D X is a Dirac element, that is, it provi des a Poincar ´ e duality between the algeb ras C ∗ ( T S X ) and C ( X ) . W e need some notatio ns. If W is an open set of the stratiﬁed pseudoman ifold X and W its closure, we set: T S W = T S X | W ◦ ; T S W = T S X | W ◦ and G t W = G t X | W ◦ × [0 , 1] . The g roupoid G t W is a deformat ion grou poid w hich deﬁnes the K -homolog y class δ W ∈ K 0 ( C ∗ ( T S W )) . W e deﬁne the homomorphi sms induc ed by Ψ X : ˆ Ψ W : C ∗ ( T S W ) ⊗ C 0 ( W ) → C ∗ ( T S W ) and ˆ Ψ W : C ∗ ( T S W ) ⊗ C ( W ) → C ∗ ( T S W ) and w e set Ψ W = i W ◦ ˆ Ψ W and Ψ W = i W ◦ ˆ Ψ W where i w : C ∗ ( T S W ) → C ∗ ( T S X ) is the natural homomorph ism. Fina lly we let : D W = ( ˆ Ψ W ) ∗ ( δ W ) = ( Ψ W ) ∗ ( δ X ) ∈ KK ( C ∗ ( T S W ) ⊗ C 0 ( W ) , C ) and D W = ( ˆ Ψ W ) ∗ ( δ W ) = ( Ψ W ) ∗ ( δ X ) ∈ K K ( C ∗ ( T S W ) ⊗ C ( W ) , C ) . In the seque l, we will be intereste d in the disjoint open sets: (4–4) O − = [ s ∈ S 0 { z ∈ N s | ρ s ( z ) < 2 } and O + = X \ O − , as well as in the inter section of their closures: (4–5) L = O + ∩ O − = [ s ∈ S 0 { z ∈ X | ρ s ( z ) = 2 } . W e recall from Paragrap h ( 2.3 ) that S 0 denote s the set of minimal strata. G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1033 Pro of of Th eor em 4 It will be prov ed by induction on the depth of the stratiﬁcati on and the unfold ing process will be used to reduce the depth. If depth( X ) = 0 the conten t of the theo rem is well known, and that D X is a Dirac element is a conseq uence of [ 13 ]. Let k ≥ 0 , ass ume that the theorem 4 holds for all compact stratiﬁed pseudomanifo lds with depth ≤ k and let X be a compact stratiﬁed pseud omanifol d of depth k + 1 . The proof of the inductio n is divi ded in two part s. First part of the proof . W e co nsider two natural “re stricti ons” of D X , na m ely D O + ∈ K 0 ( C ∗ ( T S O + ) ⊗ C 0 ( O + )) and D O − ∈ K 0 ( C ∗ ( T S O − ) ⊗ C ( O − )) . T hen, we re duce the proof of the theore m to checking that D O + is a Dirac element. Let O 0 be the open set of X obtain ed by replaci ng the condi tion ρ s < 2 by ρ s < 1 in the deﬁnitio n of O − in ( 4–4 ). T he C ∗ -algeb ra C ∗ ( T S O 0 ) is a closed two-sided ideal in C ∗ ( T S O − ) and the quot ient C ∗ ( T S O − ) / C ∗ ( T S O 0 ) ≃ C 0 ([1 , 2[) ⊗ C ∗ ( T S L × R ) is con tractible in K -theory . It fo llo ws that the in clusion C ∗ ( T S O 0 ) ⊂ C ∗ ( T S O − ) is a KK -equi valenc e w hich s ends δ O 0 to δ O − . This is obvio us once we consider the correspo nding tangen t groupo ids G t O 0 , G t O − . As already noted there is a natural Morita equiv alence betwee n the groupoid T S O 0 = S s ∈ S 0 ∗ π ∗ s ( Ts ) | O s and the tang ent space TS = S s ∈ S 0 Ts of the closed smooth m anifold S = ∪ s ∈ S 0 s . Under thi s Morita equi val ence, δ O 0 corres ponds to δ S : this follo ws from the ext ension of the pre vious Morita equiv alence to the tangent group oids G t O 0 and G t S . Moreo ver the control data pro vide a homotopy equi valence betwen O − and S and w e ﬁ nally get a K K -equiv alence between C ∗ ( T S O − ) ⊗ C ( O − ) and C ∗ ( T S S ) ⊗ C ( S ) under which the class [ Ψ O − ] coinci des w ith the class [ Ψ S ] . W e ha ve prov ed: Lemma 11 There is a KK -equi va lence between C ∗ ( T S O − ) ⊗ C ( O − ) and C ∗ ( T S S ) ⊗ C ( S ) under which the D irac element D S corres ponds to D O − . In particula r , D O − is a Dirac elemen t. W e no w apply Lemma 4 to th e nuclea r C ( X ) -alge bra C ∗ ( T S X ) , the disjoint open subset s O − and O + and the K -homology class δ X . Since by Lemma 11 D O − is a Dirac elemen t, we immediately get: D X is a Dirac element if and only if D O + is. Second part of the pro of. W e chec k that D O + is a Dirac element. Let us go back to the compact pseudo manifold of dep th k coming from the unfo lding process : 2 X . G eometry & T opology XX (20X X) 1034 Clair e Debord and Jean-Marie Lescur e Modifyin g slightly the deﬁnition of Paragraph 2.3 , we set: 2 X = O + ∪ L L × [ − 2 , + 2] ∪ L O + W e consider this time the disjoint open subse ts U = L × ] − 2 , + 2[ and V = 2 X \ L × [ − 2 , + 2] = O + ⊔ O + of the pseudoma nifold 2 X . Let us introd uce as before the homomorph isms induced by Ψ 2 X : (4–6) Ψ U : C ∗ ( T S U ) ⊗ C ( U ) − → C ∗ ( T S 2 X ) and (4–7) Ψ V : C ∗ ( T S V ) ⊗ C 0 ( V ) − → C ∗ ( T S 2 X ) where T S U = T S 2 X | U ◦ and T S V = T S 2 X | V ◦ . Note that under the natural identiﬁca- tion C ∗ ( T S V ) ⊗ C 0 ( V ) ≃ M 2 ( C ∗ ( T S O + ) ⊗ C 0 ( O + )) , the homomorphism Ψ V has the follo wing diagonal form: Ψ V = diag( Ψ O + , Ψ O + ) . W e shall conside r three K -homology classe s: D 2 X = Ψ ∗ 2 X ( δ 2 X ) , D U = Ψ ∗ U ( δ 2 X ) , D V = Ψ ∗ V ( δ 2 X ) . Since 2 X is a compact stratiﬁed pseudomanifo ld of depth k , we kno w by induction hypot hesis that D 2 X is a Dirac element. The space L with the stratiﬁcation induced by X is also a compact stratiﬁed pseu- domanifo ld of depth k . So it has a Dirac element D L deﬁned as before. Observ e that Ψ U has range in the ideal C ∗ ( T S U ) of C ∗ ( T S 2 X ) . W e note ˆ Ψ U the induced homomorph ism, i U : C ∗ ( T S U ) → C ∗ ( T S 2 X ) the inclusion and δ U the KK -element associ ated with the deformation gro upoid G t U : = G t 2 X | U ◦ . W e ha ve δ U = ( i U ) ∗ ( δ 2 X ) , hence D U = ( ˆ Ψ ) ∗ U ( δ U ) . On the other hand, let δ be the KK -elemen t associ ated with the deformat ion groupo id G t ] − 2 , 2[ . It is clear that δ is a generat or of K 0 ( C ∗ ( T ] − 2 , 2[)) ≃ Z and its pull-back ∆ under the homotop y equi v alence C ([ − 2 , 2]) → C is a Dirac ele- ment. Now , und er the groupoid isomor phism T S U ≃ T S L × T ] − 2 , 2[ , the element δ U corres ponds to δ L ⊗ C δ and D U to D L ⊗ C ∆ . It follo ws that D U is a Dirac element. Since D 2 X and D U are Dirac elements, we get from Lemma 4 applied to the nuclear C (2 X ) -algebra C ∗ ( T S 2 X ) , to the open sets U , V and to the K -homology class δ 2 X , that D V is a Dirac element. S ince Ψ ∗ V has diagon al form, we hav e: (4–8) D V = D O + ⊕ D O + ∈ K 0 ( C ∗ ( T S O + ) ⊗ C 0 ( O + )) ⊕ 2 ⊂ K 0 ( C ∗ ( T S V ) ⊗ C 0 ( V )) . It is clear from this formula that D V is a Dirac element if and only if D O + is, so we ha ve prov ed that D O + is a Dirac element, which ends the proo f of the theorem. G eometry & T opology XX (20X X) K -duality for stratiﬁed pseudomanifolds 1035 The fol lo wing remark collect some tech nical facts which were in t he main body of the proof of Theorem 4 before we took into accou nt the Referee’ s sugges tions. Remark 4 Let us replace ρ s ( z ) < 2 by ρ s ( z ) < 1 in the deﬁnitio n of O − in ( 4–4 ) and modify accord ingly the subsequ ent sets in ( 4–4 ). Let ∂ ± ∈ KK 1 ( C ∗ ( T S L × R ) , C ∗ ( T S O ± )) be the KK -elements associat ed with the exact sequences of C ∗ - algebr as: (4–9) 0 − → C ∗ ( T S O ± ) − → C ∗ ( T S O ± ) − → C ∗ ( T S L × R ) − → 0 . W e can apply Lemma 2 to A = C ∗ ( T S X ) , J 1 = C ∗ ( T S O + ) and J 2 = C ∗ ( T S O − ) . T his gi ves: (4–10) ∂ + ⊗ δ O + = − ∂ − ⊗ δ O − ∈ K 1 ( C ∗ ( T S L × R )) . Moreo ver , one can sho w that this element is, modulo sign and Bott perio dicity K 1 ( C ∗ ( T S L × R )) ≃ K 0 ( C ∗ ( T S L )) , the Dirac element D L associ ated w ith L . The idea to pro ve this is to bui ld a smooth groupoid: ˆ G t O − : = G t O − ⊔ ( G t L × R ) ⇒ O − × [0 , 1] . such that the follo wing (smooth) isomorphisms hold: • ˆ G t O − | O − ×{ 0 } ≃ T S O − , • ˆ G t O − | O − ×{ 1 } ≃ ( L ◦ × L ◦ ) × ( R ⋊ φ R | ]0 , 1] ) , where R ⋊ φ R ⇒ R is the groupoi d of the ac tion of R onto itself by the complete ﬂo w of the vector ﬁ eld τ ( h ) ∂ h and τ is the gluing function used in Paragraph 3.3 (thi s is exa ctly the tang ent space of [1 , + ∞ [ with { 1 } as a conica l point). Since ˆ G t O − | O − ×{ 1 } has v anishing K -theo ry , hence the KK -element α asso ciated with the exac t sequen ce: (4–11) 0 − → C ∗ ( O ◦ − × O ◦ − ) − → C ∗ ( ˆ G t O − | O − ×{ 1 } ) − → C ∗ ( L ◦ × L ◦ × R ) − → 0 is in vert ible in KK -theory , thus correspond s to Bott periodicit y modulo a sign and the Morita equiv alences bet ween C ∗ ( O ◦ − × O ◦ − ) , C ∗ ( L ◦ × L ◦ ) and C . Finally , we con sider the commutati ve diagra m: (4–12) 0 / / C ∗ ( T S O − ) / / C ∗ ( T S O − ) / / C ∗ ( T S L × R ) / / 0 0 / / C ∗ ( G t O − ) e O − 0 O O e O − 1   / / C ∗ ( ˆ G t O − ) e O − 0 O O e O − 1   / / C ∗ ( G t L × R ) e L 0 ⊗ 1 O O e L 1 ⊗ 1   / / 0 0 / / C ∗ ( O ◦ − × O ◦ − ) / / C ∗ ( ˆ G t O − | O − ×{ 1 } ) / / C ∗ ( L ◦ × L ◦ × R ) / / 0 G eometry & T opology XX (20X X) 1036 Clair e Debord and Jean-Marie Lescur e It gi ves by functoria lity: ∂ − ⊗ δ O − = δ L ⊗ C α which prov es the claim. 4.0.6 S tratiﬁed pseudomanif old w ith boundary . As a byprodu ct of the proof of Theorem 4 , we hav e prov ed that Poincar ´ e duality also holds for compact strat iﬁed pseudomanifo lds with boundary . Precisely a strati ﬁed pseud omanifol d with bound ary is ( X b , L , S b , N b ) where: (1) X b is a compact separable metrizab le space and L is a co m pact subsp ace of X b . (2) S b = { s i } is a ﬁnite partition of X b into locally closed subset of X b , which are smooth manifolds possib ly w ith boun dary . Moreo ver for each s i we ha ve s i ∩ L = ∂ s i . (3) N b = {N s , π s , ρ s } s ∈ S b , where N s is an open neighborh ood of s in X , π s : N s → s is a continuo us retractio n and ρ s : N s → [0 , + ∞ [ is a continuo us map such that s = ρ − 1 s (0) . (4) The double : X = X b ∪ L X b obtain ed by gluing two copies of X b along L togethe r with the partit ion S : = { s i | ∂ s i = ∅} ∪ { s i ∪ ∂ s i s i } ∪ { s i | ∂ s i = ∅} and the set of co ntrol data N = { ˜ N s , ˜ π s , ˜ ρ s } s ∈ S where N s = N s i , π s = π s i , ρ s = ρ s i if s = s i with ∂ s i = ∅ and N s = N s i ∪ N s i ∩ L N s i , π s | N s i \ L = π s i , ρ s | N s i \ L = ρ s i else where is a stratiﬁed pseudoman ifold. W e let O b : = X b \ L . 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Press, Princeton, N. J., 1965. Laborato ire de Math ´ ematiques, Universit ´ e Blaise Pascal, Com plexe universitaire dse C ´ ezeaux, 24 A v . des Lan dais, 63177 Aubi ` ere cedex, France Laborato ire de Math ´ ematiques, Universit ´ e Blaise Pascal, Com plexe universitaire dse C ´ ezeaux, 24 A v . des Lan dais, 63177 Aubi ` ere cedex, France debord @math .univ-bpclermont.fr , lescu re@ma th.univ-bpclermont.fr http:/ /math .univ-bpclermont.fr/ ~ debord / , http:/ /math .univ-bpclermont.fr/ ~ lescur e/ G eometry & T opology XX (20X X)

K-duality for stratified pseudomanifolds

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