Torsion Invariants for Families

TORSION INV ARIANTS F OR F AMILIES SEBASTIAN GOETTE De dic ate d to Je an-Michel Bismut on the o c c asion of his 60th birthday Abstra ct. W e giv e an ov erview o ver the higher torsion inv ariants of Bism ut-Lott, Igusa-Klein and Dwyer-W eiss-Willia ms, including some more or less recent dev elopmen ts. The classical F ranz-Reidemeister torsion τ FR is an inv ariant of manifolds with acyclic unitarily ﬂat v ector b undles [63], [34]. In con trast to most other algebraic-to p ological inv ariants kno wn at that time, it is in v arian t un der h ome- omorphisms and simple-homotop y equiv alences, b ut not under general homo- top y equ iv alences. In particular, it can d istinguish homeomorphism t yp es of homotop y-equiv alen t lens spaces. Hatc her and W ag oner suggested in [40] to extend τ FR to families of manifolds p : E → B using ps eudoisotopies and Morse theory . A construction of such a higher F ranz-Reidemeister torsion τ w as ﬁrst prop osed b y John Klein in [49] using a v ariation of W aldhausen’s A -theory . Other d escriptions of τ were later giv en by Igusa and Klein in [46], [47]. In this o v erview, we will r efer to the co nstru ction in [43]. Let p : E → B b e a family of smo oth manifolds, and let F → E b e a un itarily ﬂat complex v ector bun dle of rank r suc h that the ﬁbrewise cohomology with co eﬃcien ts in F forms a unip oten t bundle o ver B . Using a function h : E → R that has only Morse and b irth-death singularities along eac h ﬁbre of p , and with trivialised ﬁ brewise unstable ta ngen t b undle, one co nstructs a homotop y class of maps ξ h ( M /B ; F ) f rom B to a classifying space W h h ( M r ( C ) , U ( r )). No w, the higher torsion τ ( E /B ; F ) ∈ H 4 • ( B ; R ) is deﬁn ed as the pu ll-bac k of a certain unive rsal cohomolog y class τ ∈ H 4 •  W h h ( M r ( C ) , U ( r )); R  . On the other hand , Ra y and Sin ger deﬁned an analytic torsion T RS of unitar- ily ﬂat complex v ector bundles o n compact manifolds in [62] and conjectured that T RS = τ FR . This conjecture wa s established indep endent ly b y Cheeger [27] and M ¨ uller [60]. Th e most general comparison result was giv en by Bism ut and Zhang in [18] and [19]. In [65], W agoner pred icted the existence of a “higher analytic torsion” that dete cts h omotop y classes in the diﬀeomorphism groups of sm o oth closed manifolds. Such an in v ariant was deﬁn ed later b y Bism u t an d Lott in [16]. Kam b er and T ond eur constructed c haracteristic classes c h o ( F ) ∈ H odd ( M ; R ) of ﬂat v ector b undles F → M in [48] that p ro vide obstructions 2000 Mathe matics Subje ct Classiﬁc ation. 58J52 (57R22 55R40). Key wor ds and phr ases. Bism ut-Lott torsion, Igusa-K lein torsion, Dwyer-W eiss-Williams torsion, higher analytic torsion, higher F ranz-R eidemeister torsion. Supp orted in part by DF G sp ecial programme “Global Diﬀeren tial Geometry”. 1 2 SEBASTIAN GOETTE to wards ﬁnding a parallel m etric. If p : E → B is a smo oth bund le of compact manifolds and F → E is ﬂat, Bismut and Lott pro v ed a Grothendiec k-Riemann- Ro c h theorem rela ting the c haracteristic classes of F to those of the ﬁbr ewise co- homology H ( E /B ; F ) → B . The higher analytic torsion form T ( T H E , g T X , g F ) app ears in a reﬁnement of this theorem to the lev el of diﬀerential forms . Its comp onen t in degree 0 equ als th e Ray-Singer analyt ic torsion of the ﬁbres, and the reﬁned Grothendiec k-Riemann-Ro c h theorem implies a v ariation form ula for the Ra y-Singer torsion that w as already disco v er ed in [18]. In [3 1], Dwy er , W eiss and Williams ga v e y et another approac h to h igher tor- sion. They d eﬁned thr ee g eneralised Euler characte ristics for bundles p : E → B of h omotop y ﬁ nitely d ominated sp aces, to p ological manifolds, and smo oth manifolds, resp ectiv ely , with v alues in certain b undles o ver B . A ﬂat com- plex v ector bundle F → E deﬁnes a h omotop y class of maps from E to the algebraic K -theory space K ( C ). The Euler charact eristics ab o ve giv e analo- gous maps B → K ( C ) for the ﬁbrewise cohomol ogy H ( E /B ; F ) → B . If F is ﬁbrewise acyclic, these maps lift to three d iﬀeren t generalisations of Reidemeis- ter torsion, giv en again as sections in certain bundles o v er B . By comparing the three charact eristics for smo oth manifold bund les, Dwy er, W eiss and Williams also sho w ed that the Grothendieck- Riemann-Ro c h theorem in [16] h olds already on the lev el of classifying maps to K ( C ). Bism ut-Lott torsion T ( E /B ; F ) and Igusa-Klein torsion τ ( E /B ; F ) are v ery closely related. F or particularly nice bundles, this was pro ved by Bi smut and the auth or in [12] an d [37], [38]. W e will establish the general case in [39]. Igusa al so ga v e a set of axio ms in [45 ] that c haracterise τ ( E /B ; F ) and hop e- fully also T ( E /B ; F ) when F is trivial. Badzioch, Dorabia la and Williams recen tly ga ve a cohomologic al v ersion of th e smo oth Dwy er-W eiss-Willia ms tor- sion in [3]. T og ether with Kle in, they pro ved in [2] that it satisﬁes Igusa’s axioms as w ell. On the other hand, the other t wo torsions in [31] are deﬁnitely coa rser than Bismut-Lo tt and Igusa-Klein torsion, b ecause they do not d ep end on the diﬀeren tiable structur e. They migh t ho w ev er b e related to the Bism ut-Lott or Igusa-Klein torsion of a virtual ﬂat v ector bundle F of rank zero, s ee Remark 7 .5 b elo w. Let us no w recall one of the most imp ort applications of higher torsion in- v arian ts. I t is p ossible to construct t wo smo oth manifold bun dles p i : E i → B for i = 0, 1 w ith d iﬀeomorphic ﬁb res, suc h that there exists a homeomor- phism ϕ : E 0 → E 1 with p 0 = p 1 ◦ ϕ and with a lift to an isomo rph ism of vertica l tangen t bu ndles, but n o such diﬀeomorphism. Th e ﬁr st example o f suc h bun- dles p i w as constructed by Ha tc h er, and it w as later p ro v ed b y B¨ okstedt that p 0 and p 1 are not diﬀeomorphic in the sense ab o v e [20]. Igusa sho w ed in [43] that the higher torsion in v arian ts τ ( E i /B ; C ) diﬀer, an d b y [37], the Bism u t-Lott torsions T ( E i /B ; C ) d iﬀer as w ell. Hatc her’s example can b e generalised to construct many d iﬀeren t smo oth str uctures on bu ndles p : E → B . W e exp ect that higher torsion in v arian ts distinguish man y of t hese diﬀeren t structures, but not all of them. One m a y w onder w h y on e wan ts to consid er so man y diﬀeren t higher torsion in v arian ts, in particular, if some of th em a re conjectured to pro vide the same information. W e will see that diﬀeren t constructions of these inv arian ts giv e TORSION INV ARIANTS FOR F AMILIES 3 rise to diﬀerent a pplications. Since Hatc her’s example and its generalisations come with natural ﬁbrewise Morse functions, the diﬀerence of the Igusa-Klein torsions of d iﬀeren t smo oth s tructures is someti mes easy to c ompute. D ue to Igusa’s axiomati c app roac h , one can also understand the to p ological meaning of Igusa-Klein torsion. O n the o ther hand, one can cl assify smo oth structur es on a top ologic al manifold bundle p : E → B in a more abstract wa y as classes of sections i n a certain bu ndle of classifying space s o ver B . These se ction spaces ﬁ t w ell into the framew ork of generalised E uler c haracteristics and Dwy er-W eiss- Williams torsion. But some extra work is necessary to reco v er co homological information f rom this app roac h . Finally , Bism u t-Lott torsion is deﬁned using th e language of lo cal index the- ory . The pro ofs of some int eresting prop erties of Bism u t-Lott torsion w ere inspired b y parallel results in the setting of the classical A tiy ah-Singer family index theorem or th e Grothendiec k-Riemann-Ro c h th eorem in Arake lo v geom- etry . Bismut-Lo tt torsion is deﬁned for any ﬂ at v ector bu ndle F → E , whereas Igusa-Klein torsion and Dwy er-W eiss-Williams torsion can only b e deﬁned if the ﬁbrewise cohomology is of a sp ecial t yp e. T his make s Bismut-Lot t torsion useful for other applications, for example in th e deﬁn ition of a secondary K - theory b y Lott [53]. H eitsc h and Lazaro v generalised Bismut-Lott torsion to foliations [41], so one ma y try to use it to detect d iﬀeren t s mo oth stru ctures on a giv en foliation, which ind uce the same structur es on the space of lea v es. Finally , Bismut and Leb eau recen tly deﬁned higher torsion in v ariants using a h yp o elliptic Laplacian on the cota ngen t bun dle [8], [15]. Conjecturally , this torsion can giv e some information ab out the ﬁbrewise geo desic ﬂow. This ov erview is organised as follo ws. W e start by discussin g the ind ex the- orem for ﬂat v ector bund les b y Bism u t and Lott in Section 1. In Sectio ns 2 and 3, we in tro duce Bism ut-Lott torsion and state some prop erties and ap- plications that are inspired by lo cal ind ex th eory . In Section 4 and 5, we in tro duce Igus a-Klein torsion and relate it to Bismut-Lott torsion u sing t w o diﬀeren t approac h es. Section 6 is d ev oted to generalised Euler c haracteristics and Dwy er-W eiss-Willi ams torsion. In Section 7, we discuss smo oth stru ctures on ﬁ bre bun dles and a p ossible g eneralisation to foliations. Finally , we sk etc h the h yp o elliptic op erator on the co tangen t bun dle and its torsion due to Bism ut and Leb eau in S ection 8. W e ha v e tried to k eep the nota tion and the normalisati on of the inv ariants consisten t throughout this pap er; as a result, b oth will disagree with m ost of the references. In particular, w e use the Chern normalisatio n of [12], whic h is the only normalisation for whic h Theorem 3.7 and a few other resu lts hold. T o k eep this pap er reasonably short, o nly the most basic v ersions of some of the theorems on higher torsion will b e explained. Thus w e will not discuss some non- trivial generalisations of the th eorems b elo w to ﬁbre b undles with group a ctions. W e will also only give hints tow ards the relation with the classical A tiy ah- Singer family in dex theorem or the Grothend iec k-Riemann-Ro c h theorem in algebraic geomet ry . Finally , we will not discuss th e interesting reﬁnemen ts and generalisatio ns of classical F ranz-Reidemeister torsion and Ra y-Singer torsion for single manifolds that ha v e b een in v en ted in the last few years. 4 SEBASTIAN GOETTE A cknow le dgements. T his pap er is a somewhat extend ed v ersion of a ser ies of lectures at the Ch ern Institute at Tianjin i n 2 007, whose supp ort and hospitalit y w e highly appr eciate d. The author wa s supp orted in part by th e DF G sp ecial programme “Global Diﬀeren tial Ge ometry”. W e are g rateful to J.-M. Bismut for in tro du cing us to higher torsion, and also to U. Bunk e, W. Dorabia la, K. I gusa, K . K¨ ohler, X. Ma, B. Willia ms and W. Zhang, from whom w e learned man y d iﬀeren t asp ects of th is int riguing su b ject. W e also thank the anonymous referee for her or h is helpfu l commen ts. 1. An Index Theor em for Fla t Vector Bundles There exists a theory of charac teristic cl asses of ﬂat vect or bundles that is parallel to the theory of Ch ern c lasses an d Chern-W eil diﬀeren tial forms. These classes h a ve b een constructed b y Kamber and T ond eur [48], and are closely related to the classes us ed by Borel [21] to study the algebraic K -theory of n umber ﬁ elds. Analytic torsion forms made their ﬁrst app earance in a lo cal index theorem for these Kamb er-T ondeur classes by Bism ut and Lott [16]. Reﬁnemen ts of this theorem hav e later been giv en by Dwy er, W eiss and Williams [31] and by Bism ut [7 ] and Ma and Z hang [57]. 1.1. Characteristic classes for ﬂat v ector bundles. Before w e int ro- duce Kam b er-T ondeur forms , let us ﬁrst recall classical C hern-W eil theory . Let V → M b e a complex ve ctor bundle, and let ∇ V b e a connection on V with cu rv ature ( ∇ V ) 2 ∈ Ω 2 ( M ; E nd V ). Then o ne deﬁnes the Ch ern c h aracter form (1.1) c h  V , ∇ V  = tr V  e − ( ∇ V ) 2 2 π i  ∈ Ω ev en ( M ; C ) . This form is closed b ecause the co v arian t deriv ativ e [ ∇ V , ( ∇ V ) 2 ] of the curv ature v anishes by the Bianchi iden tit y , so (1.2) d c h  V , ∇ V  = tr V  ∇ V , e − ( ∇ V ) 2 2 π i  = 0 . If ∇ V , 0 and ∇ V , 1 are t wo connectio ns on V , one can c ho ose a connection ∇ ˜ V on the n atural extension ˜ V of V to M × [0 , 1] with ∇ ˜ V | M ×{ i } = ∇ V ,i for i = 0, 1. Stoke s’ theorem then imp lies (1.3) c h  V , ∇ V , 1  − ch  V , ∇ V , 0  = d e c h  V , ∇ V , 0 , ∇ V , 1  , with e c h  V , ∇ V , 0 , ∇ V , 1  = Z 1 0 ι ∂ ∂ t c h  ˜ V , ∇ ˜ V  dt. Th us, the class c h ( V ) of ch( V , ∇ V ) in de Rham cohomo logy is in dep endent of ∇ V . Moreo v er, e c h ( V , ∇ V , 0 , ∇ V , 1 ) is indep endent of the choice of ∇ ˜ V up to an exact form. No w let F → M b e a ﬂ at v ector bundle, so F comes with a ﬁxed connec- tion ∇ F suc h that ( ∇ F ) 2 = 0. W e c ho ose a metric g F on F and deﬁne the TORSION INV ARIANTS FOR F AMILIES 5 adjoin t connection ∇ F , ∗ with resp ect to g F suc h that (1.4) dg ( v , w ) = g  ∇ F v , w  + g  v , ∇ F , ∗ w  for all s ections v , w of F . Th en the f orm (1.5) c h o  F , g F  = π i e c h  F , ∇ F , ∇ F , ∗  ∈ Ω odd ( M ; R ) is real, o dd and also closed, b ecause (1.6) d c h o  F , g F  = π i c h  F , ∇ F , ∗  − π i c h  F , ∇ F  = 0 . Clearly , if g F is p arallel with r esp ect to ∇ F , then c h o ( F , g F ) = 0. Let g F , 0 , g F , 1 b e tw o m etrics on F . Pro ceeding as in (1.3), one constructs a form e c h o ( F , g F , 0 , g F , 1 ) ∈ Ω ev en ( M ) such that (1.7) c h o  F , g F , 1  − ch o  F , g F , 0  = d e c h o  F , g F , 0 , g F , 1  . So aga in, the de Rham cohomology class c h o ( F ) of c h o ( F , g F ) do es not d ep end on the c hoice of metric g F — bu t of course, it d ep ends on the ﬂat connection ∇ F . Note that the form e c h o ( F , g F , 0 , g F , 1 ) is again naturally w ell-deﬁned up to an exact form. 1.1. Deﬁnition. T he forms c h o k ( F , g F ) = c h o ( F , g F ) ∈ Ω 2 k − 1 ( M ) are calle d Kamb er-T ondeur forms , and th eir classes c h o k ( F ) ∈ H 2 k − 1 ( M ; R ) are calle d Kamb er-T ondeur classes or Bor el classes . Note that in the literature, there are at least three diﬀeren t normalisations of these classes. There are how ev er goo d reasons to stic k to the norm alisatio n here, see section 3.4. F or later referen ce, w e giv e a more explicit construction of the K am b er- T ondeur forms. If w e deﬁne a connection ∇ ˜ F o ver p : M × [0 , 1] → M that in terp olates b et w een ∇ F and ∇ F , ∗ b y (1.8) ∇ ˜ F = (1 − t ) p ∗ ∇ F + tp ∗ ∇ F , ∗ , then by ﬂatness of ∇ F and ∇ F , ∗ , (1.9)  ∇ ˜ F  2 = − t (1 − t ) p ∗  ∇ F , ∗ − ∇ F  2 − p ∗  ∇ F , ∗ − ∇ F  dt . F rom this formula and (1.3 ), (1.5) one deduces th at there exist rational multi- ples c k of (2 π i ) k suc h that (1.10) c h o  F , g F  = ∞ X k =0 c k tr F  ω ( F , g F ) 2 k + 1  , with ω  F , g F  = ∇ F , ∗ − ∇ F = ( g F ) − 1 [ ∇ F , g F ] ∈ Ω 1 ( M ; E nd V ) . Bism ut and Lott u se the real, o dd and closed diﬀeren tial forms (1.11) tr F  ω  F , g F  e ω ( F,g F ) 2 2 π i  and th eir cohomolog y classes instead of c h o , whic h is m ore conv enient for some of th e foll o wing constructions. It is not h ard to see that th ese fo rms a re giv en b y a similar form ula a s (1.10) , but w ith diﬀerent constants c k ∈ (2 π i ) k Q . W e prefer the C hern normalisation given b y (1.5) for reasons explained in Remark 3.8. 6 SEBASTIAN GOETTE The Chern-W eil classes like ch( V ) v anish w henev er V adm its a ﬂat connec- tion. Similarly , the classes ch o ( F ) v anish whenev er F a dmits a ∇ F -parallel metric. W e w ill see that there are m ore analogies b et ween these constructions. A go o d o v erview can b e found in the introd uction to [53 ]. 1.2. The cohomological index theorem. Th e central theme in [1 6] is a fam- ily index theorem f or ﬂ at vecto r bu ndles in terms of their Kam b er-T ondeur classes. The analytic ind ex in question is give n by ﬁb rewise cohomology . More pr ecisely , let p : E → B b e a smo oth pr op er submersion, in other w ords, a smo oth ﬁ bre bund le with n -dimensional compact ﬁbres, to b e de- noted M . Let ( F , ∇ F ) b e a ﬂat v ector bundle then w e consider the vect or bund les H k ( E /B ; F ) → B , whose ﬁbres o v er x ∈ B a re giv en as the t wisted de Rham cohomology (1.12) H k ( E /B ; F ) x = H k  Ω • ( E x ; F | E x ) , ∇ F  . The b undles H k ( E /B ; F ) naturally carry the Gauß-Manin connection ∇ H , whic h is ag ain ﬂat. The analytic index is thus give n by the virtual ﬂat vec- tor b undle (1.13) H ( E /B ; F ) = dim M M k =0 ( − 1) k H k ( E /B ; F ) . The top ological index is giv en b y the Bec ker-Go ttlieb transf er of [4]. Re- call that the Bec ker-Go ttlieb transfer is given as a stable homotop y class of maps tr E /B : S • B + → S • E + . It acts on de Rh am cohomology b y (1.14) tr ∗ E /B α = Z E /B e ( T M ) α ∈ H k ( B ; R ) for all α ∈ H k ( E ; R ), where e ( T M ) ∈ H n ( E ; o ( T M ) ⊗ R ) denote s the Chern - W eil th eoretic Eu ler cla ss of the v ertical tangen t bundle T M = ke r dp ⊂ T E , and R E /B denotes int egration o v er the ﬁb re. Here is a cohomolo gical version of the family index theorem. 1.2. Theorem (Bism ut and Lott [16]) . F or al l smo oth pr op er submersions p : E → B and al l ﬂat ve ctor b u nd les F → E , (1.15) ch o ( H ( E /B ; F )) = tr ∗ E /B c h o ( F ) ∈ H odd ( B , R ) . One notes that tr ∗ E /B preserve s th e degree of diﬀerential form s and cohomol- ogy classes. F or t his reason, an analogous result h olds for the classes constructed in (1.11), and in fact for all classes of the f orm (1.10), ind ep endent of the c hoice of the constan ts c k . The cohomolo gical index theorem can b e reﬁned as follo ws, see also Sec- tion 3.5. F ollo wing [28], to a v ector b undle V → M with connection ∇ V , one asso ciates a Cheeger-Simons d iﬀeren tial c haracter b c h ( V , ∇ V ), from whic h b oth the rational Chern c haracter c h ( V ) ∈ H ev en ( M ; Q ) and the C hern-W eil form c h( V , ∇ V ) ∈ Ω ev en ( M ) can b e read oﬀ. If ∇ V is a ﬂat conn ection, TORSION INV ARIANTS FOR F AMILIES 7 then b c h ( V , ∇ V ) b ecomes a cohomol ogy cl ass in H odd ( M ; C / Q ). It has already b een obser v ed in [16] that its imaginary part is giv en b y (1.16) Im b c h  V , ∇ V  = c h o ( V ) ∈ H odd ( M ; R ) . 1.3. Theorem (Bism ut [7], Ma and Zhang [57]) . F or al l smo oth pr op er sub- mersions and al l ﬂat ve ctor b u nd les F → E , (1.17) b c h  H ( E /B ; F ) , ∇ H  = tr ∗ E /B b c h  F , ∇ F  ∈ H odd ( B ; C / Q ) . It is natural to ask if the same th eorem holds on the lev el of ﬂat v ector bund les on B . A ﬂat v ector bu ndle F → E , or more ge nerally , a bundle of ﬁnitely generated p ro jectiv e R -mod ules for some ring R , is classiﬁed by a map from E to the classifying space B GL ( R ) × K 0 ( R ). F ollo wing Quillen, there is a natural map fr om B GL ( R ) to the algebraic K -theory space K ( R ). Thus, w e ma y asso ciate to F the corresp ondin g h omotop y class [ F ] of maps from E to K ( R ), whic h is sligh tly coarser th an the class of F in the K -theory of ﬁnitely generated pr o jective R -mo du le bun dles on E . 1.4. Theorem (Dwy er, W eiss and Williams [31]) . If p : E → B is a bund le of smo oth close d m anifolds, then (1.18) [ H ( E /B ; F )] = tr ∗ E /B [ F ] in the ho motopy cla sses of maps B → K ( R ) . Although b oth sides of (1.1 8 ) exist in a muc h more general situation, the smo oth bundle structure is needed in the proof of the th eorem, s ee section 6.1 b elo w, in particular T heorem 6.3. Theorem 1.2 can b e dedu ced from Theo- rem 1.4 b ecause the class c h o can already b e deﬁ ned on K ( R ). 1.3. A reﬁne d index t heorem. There is another p ossible r eﬁnemen t of The- orem 1.2, wh ere o ne r eplaces de Rham cohomo logy cla sses by diﬀeren tial forms . F or this, one ﬁrst c ho oses metrics g T M and g F on the bun dles T M → E and F → E , and a horizon tal complemen t T H E of T M ⊂ T E . These data giv e rise to a natural connection ∇ T M on T M b y [6]. T h us, one can co nsider the C hern-W eil theoretic Euler form e ( T M , ∇ T M ). W e also ha v e a natural decomp osition (1.19) Ω • ( E ; F ) = Ω • ( B ; Ω • ( E /B ; F )) using T E = T H E ⊕ T M , and an L 2 -metric on the inﬁnite dimensional bun- dle Ω • ( E /B ; F ) → B of vertica l f orms t wisted by F . Regarding H • ( E /B ; F ) as the subbund le of ﬁbrewise harmonic forms, w e get a metric g H L 2 on H • ( E /B ; F ). Bism ut and Lott no w construct a form T ( T H E , g T M , g F ) on B that dep ends natural on the data, the analytic torsio n form, see Section 2.2 b elo w. 1.5. Theorem (Bism ut and Lott [16]) . In the situation ab ove, (1.20) d T  T H E , g T M , g F  = Z E /B e  T M , ∇ T M  c h o  F , g F  − ch o  H , g H L 2  . 8 SEBASTIAN GOETTE In the theory of ﬂat v ector bundles, this result plays the same r ole as the η -forms in the heat k ernel pro of of the cla ssical family index theorem [6], [5], see also [9] and [29]. The holomorphic torsion forms similarly arise in a dou- ble transgression form ula [14] in th e Riemann-Ro c h-Grothendiec k theo rem for prop er holomorph ic submersions in K¨ ahler geometry . This analog y with η - forms and holomorphic torsion forms has inspired most of the constru ctions and resu lts of the follo win g t wo sections. 2. Construction of the Bismut-Lott to rsion In this s ection, we recall the construction of the torsion forms occurring in Theorem 1.5. As in [16], w e start with a ﬁ nite-dimensional to y mo del that will b e of in dep endent inte rest. W e then present the original construction of T ( T H M , g T M , g F ) b y Bism ut and Lott, and also a construction using η - forms by Ma and Zhang. 2.1. A ﬁnite-dimensional mo del. Consider ﬂat ve ctor bundles V k → M and parallel ve ctor bund le homomorphisms a k : V k → V k +1 , su c h that (2.1) 0 − − − − → V 0 a 0 − − − − → V 1 a 1 − − − − → · · · a n − 1 − − − − → V n − − − − → 0 forms a co c hain complex o v er eac h p oin t in M . Th en (2.2) A ′ = ∇ V + a is a sup erconnection, wh ic h is ﬂat b ecause (2.3) ( A ′ ) 2 = a 2 +  ∇ V , a  +  ∇ V  2 , and eac h term on the r igh t hand side v anish es b y assumption. W e w ill call the pair ( V , ∇ V + a ) a p ar al lel family of (ﬁnite-dimensional) c o c hain c omplexes . If we ﬁ x a metric g V k on eac h V k , we can consider the a djoint connection ∇ V , ∗ as in (1 .4), and let a ∗ k : V k +1 → V k b e the adjoin t of a k with resp ect to g V k and g V k +1 . Th en w e obtain another ﬂat sup erconnection (2.4) A ′′ = ∇ V , ∗ + a ∗ . As in Hodge theory , the ﬁbrewise cohomology of ( V , a ) is represented by H = k er( a + a ∗ ) ⊂ V . Pro j ection of ∇ V on to H deﬁn es a connectio n ∇ H on H . On e c h ec ks that ∇ H is indep end en t of g V , and in fact, ∇ H is the natural Gauß- Manin connection. Let g H V denote the restriction of g V to H . Bism ut and Lott then deﬁn e a diﬀeren tial form T ( ∇ V + a, g V ) ∈ Ω ev en ( M ) and obtain a ﬁn ite-dimensional analogue of Th eorem 1.5. 2.1. Theorem (Bism ut and Lott, [16]) . In the situation ab ove, (2.5) dT  ∇ V + a, g V  = c h o  V , g V  − ch o  H , g H V  . The core of the pro of is the constru ction of T ( ∇ V + a, g V ) that we no w describ e. On the p ullbac k ˜ V of V to ˜ M = M × (0 , ∞ ), w e in tro duce t wo ﬂ at TORSION INV ARIANTS FOR F AMILIES 9 sup erconn ections (2.6) ˜ A ′ = ∇ V + √ ta − N V 2 t dt , ˜ A ′′ = ∇ V , ∗ + √ ta ∗ + N V 2 t dt , where N V ∈ End V acts on V k as multiplica tion b y k . The d iﬀerence of the t wo su p erconnections ab ov e is an endomorphism (2.7) ˜ X = ˜ A ′′ − ˜ A ′ = ω  V , g V  + √ t ( a ∗ − a ) + N V t dt ∈ Ω •  ˜ M , E nd ˜ V  . W e also deﬁne th e sup er trace b y (2.8) str V = tr V ◦ ( − 1) N V : Ω • ( · , End V ) → Ω • ( · ) . F or con v enience, w e stic k to the con ven tions of [16]. In analogy with (1.1 1 ), the form (2.9) (2 π i ) 1 − N M 2 str V  ˜ X e ˜ X 2  ∈ Ω odd ( ˜ M ) is real, o dd and closed. By (2.7 ), we ha ve (2.10) lim t → 0 str V  ˜ X e ˜ X 2     M ×{ t } = str V  ω  V , g V  e ω ( V ,g V ) 2  . T o un derstand th e limit for t → ∞ , n ote th at a ∗ − a is a sk ew-adjoint op erator. In particular, the “ﬁn ite d imensional Laplac ian” − ( a ∗ − a ) 2 has n onnegativ e eigen v alues, and its k ern el is giv en b y the “harmonic elemen ts” H . In particular, the “heat op erator” e t ( a ∗ − a ) 2 con verges to the orthogonal pro jection on to H as t tends to inﬁnity . More generally , it is prov ed in [16] that (2.11) lim t →∞ str V  ˜ X e ˜ X 2     M ×{ t } = str H  ω  H , g H V  e ω ( H,g H V ) 2  . Because the form in (2. 9 ) is closed, the forms in (2.10) and (2 .11) b elong to the same cohomology class. Thus we ha v e alrea dy pro ved a ﬁn ite-dimensional v ersion of Theorem1.2. T o deﬁn e th e torsion form , w e ha v e to in tegrate the form in (2.9) o v er (0 , ∞ ). W e note that (2.12) ι ∂ ∂ t str V  ˜ X e ˜ X 2     M ×{ t } = str V  N V t (1 + 2 ˜ X 2 ) e ˜ X 2      M ×{ t } . Unfortunately , the integral ov er (2.12) dive rges b oth for t → 0 and for t → ∞ . Ho wev er, the d iv ergence can b e compen sated easily . F or an y Z -graded v ector bund le V , we deﬁne (2.13) χ ( V ) = X k ( − 1) k rk V k and χ ′ ( V ) = X k ( − 1) k k rk V k . Then it is p ro ved in [16 ] that the inte gral (2.14) Z ∞ 0  (2 π i ) − N M 2 str V  N V (1 + 2 ˜ X 2 ) e ˜ X 2  − χ ′ ( H ) − ( χ ′ ( V ) − χ ′ ( H ))(1 − 2 t ) e − t  dt t ∈ Ω ev en ( M ) 10 S EBASTIAN GOETTE con verges and gives a torsion f orm for the c haracteristic classes considered in (1.1 1). Adjusting the coeﬃcien ts c k in (1.1 0), we obtain the form T ( ∇ V + a, g V ) n eeded for Th eorem 2.1. 2.2. Deﬁnition. Th e Bismut-L ott torsion of the parallel family of cochain co m- plexes ( V , ∇ V + a ) is d eﬁned as (2.15) T  ∇ V + a, g V  = − Z 1 0  s (1 − s ) 2 π i  N M 2 Z ∞ 0  str V  N V (1 + 2 ˜ X 2 ) e ˜ X 2  − χ ′ H − ( χ ′ ( V ) − χ ′ ( H ))(1 − 2 t 2 ) e − t 2  dt 2 t ds ∈ Ω ev en ( M ) . Pr o of of The or em 2.1. Let N M act on Ω k ( M ) a s m ultiplication by k . Because the form (2.9 ) is closed, it follo ws from (2.10) and (2.11) that (2.16) dT  ∇ V + a, g V  = 1 2 Z 1 0  s (1 − s ) 2 π i  N M − 1 2  lim t → 0 str V  ˜ X e ˜ X 2     M ×{ t } − lim t →∞ str V  ˜ X e ˜ X 2     M ×{ t }  = c h o  V , g V  − ch o  H , g H V  .  2.3. R emark. The correction terms in Deﬁnition 2.2 are constant and only aﬀect the Bismut-Lott torsion in degree 0. Th ey are c hosen suc h that (2.17) T  ∇ V + a, g V  [0] x = 1 2 X k ( − 1) k k log det  − ( a ∗ − a ) 2   V k ∩ H k ⊥  = 1 2 X k ( − 1) k log d et  aa ∗   V k ∩ im a  . But this is ju st one w a y to repr esen t the F ranz-Reidemeister torsion of the co c h ain co mplex ( V x , a ) with metric g V for x ∈ M . Hence T ( ∇ V + a, g V ) is called a “higher torsion form”. 2.2. The Bism ut- L ott torsion form. As in [16], section 3, we now translate the construction of T ( ∇ V + a, g V ) to the inﬁnite-dimensional family of ﬁbrewise de Rham complexes. Let p : E → B b e a smo oth prop er submersion with t ypical ﬁbre M , and let T M = k er dp ⊂ T E . As in Section 1.3, w e ﬁx T H E ⊂ T E su c h th at T E = T M ⊕ T H E . Because T H E ∼ = p ∗ T B , we can identify vecto r ﬁelds on B with their pu llbac k to E , which w e call basic ve ctor ﬁelds. Let F → E b e a ﬂat v ector bundle, then we ma y regard the ﬂat connec- tion ∇ F as a diﬀerentia l on th e total complex Ω • ( E ; F ). Using the split- ting (1.19), we ma y also regard ∇ F as a sup erconnection on th e inﬁ nite- dimensional bundle Ω • ( E /B ; F ) → B with (2.18) A ′ = ∇ F = d M + ∇ Ω • ( E /B ; F ) + ι Ω b y [5]. Here, d M denotes the ﬁbr ewise diﬀerential on Ω • ( E /B ; F ), ∇ Ω • ( E /B ; F ) is th e connection indu ced by the Lie deriv ativ e by basic v ector ﬁ elds, and Ω is the vertica l comp onent of the Lie b rac ket of tw o basic v ector ﬁ elds on E . TORSION INV ARIANTS FOR F AMILIES 11 In analogy w ith (2.4), w e also deﬁne an adjoint sup erconnection (2.19) A ′′ = d M , ∗ + ∇ Ω • ( E /B ; F ) , ∗ + ε Ω with r esp ect to the ﬁ brewise L 2 -metric g L 2 on Ω • ( E /B ; F ). Let ˜ B = B × (0 , ∞ ), ˜ E = E × (0 , ∞ ) and ˜ F = F × (0 , ∞ ), and let t b e the coord inate of (0 , ∞ ). Then we deﬁne su p erconnections (2.20) ˜ A ′ = √ t d M + ∇ Ω • ( ˜ E / ˜ B ; ˜ F ) + 1 √ t ι Ω − N ˜ E / ˜ B 2 t dt , ˜ A ′′ = √ t d M , ∗ + ∇ Ω • ( ˜ E / ˜ B ; ˜ F ) , ∗ + 1 √ t ε Ω + N ˜ E / ˜ B 2 t dt , where now N ˜ E / ˜ B acts on Ω k ( ˜ E / ˜ B ; ˜ F ) as multiplicatio n b y k . T hen (2.21) ˜ X = ˜ A ′′ − ˜ A ′ = √ t  d M , ∗ − d M  + ω  Ω • ( ˜ E / ˜ B ; ˜ F ) , g L 2  + 1 √ t ( ε Ω − ι Ω ) + N ˜ E / ˜ B t dt ∈ Ω •  ˜ B ; End Ω • ( ˜ E / ˜ B ; ˜ F )  . Note that d M , ∗ − d M is a ske w-adjoint ﬁb rewise elliptic diﬀeren tial op erator, whereas the other terms on the righ t hand s ide inv olv e no diﬀeren tiation at all. The op erator − ˜ X 2 can b e regarded as a generalised Laplacian along the ﬁ bres of p . If the metric g F is parallel along the ﬁb res, then − ˜ X 2 is precisely the curv ature of the Bism ut sup erconnection, whic h already app eared in the h eat equation pro of of the A tiy ah-Singer families ind ex theo rem [6]. In particular, the ﬁb rewise o d d heat o p erator ˜ X e ˜ X 2 is well- deﬁned and of trace cla ss. Using Getzler rescaling, one pr o ves (2.22) lim t → 0 str Ω • ( ˜ E / ˜ B ; ˜ F )  ˜ X e ˜ X 2     B ×{ t } = Z E /B e  T M , ∇ T M  str  ω  F , g F  e ω ( F,g F ) 2  in analogy with (2.10). S imilarly , if w e iden tify H = H • ( E /B ; F ) with the ﬁbrewise harmonic d iﬀeren tial forms, equipp ed w ith the restriction g H L 2 of the L 2 -metric on Ω • ( E /B ; F ), then (2.23) lim t →∞ str Ω • ( ˜ E / ˜ B ; ˜ F )  ˜ X e ˜ X 2    B ×{ t } = str H  ω  H , g H L 2  e ω ( H,g H L 2 ) 2  as in (2. 11). T o obtain the torsion form, we ha v e to tak e care of some dive rgen t terms and of th e co eﬃcien ts in (1.10) as b efore. 2.4. Deﬁnition. The Bismut-Lot t torsion is d eﬁned as (2.24) T  T H E , g T M , g F  = − Z 1 0  s (1 − s ) 2 π i  N B 2 Z ∞ 0  str Ω • ( ˜ E / ˜ B ; ˜ F )  N ˜ E / ˜ B  1 + 2 ˜ X 2  e ˜ X 2  − χ ′ ( H ) −  χ ( M ) dim M rk F 2 − χ ′ ( H )  (1 − 2 t ) e − t  dt 2 t ds ∈ Ω ev en ( B ) . 12 S EBASTIAN GOETTE Pr o of of The or em 1.5. As in the pro of of Th eorem 2.1, this follo ws f rom (2.22) and (2.23), b ecause the form (2.25) str Ω • ( ˜ E / ˜ B ; ˜ F )  ˜ X e ˜ X 2  ∈ Ω • ( B × (0 , ∞ ); C ) is closed.  2.5. R emark. Agai n, the correction te rms in (2.24) are constant scalars. T hey are chosen suc h th at (2.26) T  T H E , g T M , g F  [0] x = 1 2 dim M X k =0 ( − 1) k k log Det  −  d M , ∗ − d M  2   Ω k ( M x ; F ) ∩ H k ⊥  , where “Det” denotes a zeta -regularised determinan t. The r igh t hand s ide is pre- cisely the Ra y-Singer analytic to rsion of the ﬁbre M x . Hence T ( T H E , g T M , g F ) is called a Bismut-Lott torsion form. 2.3. Elemen tary Propert ies. F rom Th eorem 1.5, one can derive a v ariation form ula for Bism ut-Lott torsion. If w e choose T H j E , g T M j , g F j for j = 0, 1, let ∇ T M ,j denote the corresp onding connections on T M , and let g H,j L 2 denote the corresp onding L 2 -metrics on H . As in (1.3), there exists a Chern-Simons Euler class ˜ e ( T M , ∇ T M , 0 , ∇ T M , 1 ) suc h th at (2.27) d ˜ e  T M , ∇ T M , 0 , ∇ T M , 1  = e  T M , ∇ T M , 1  − e  T M , ∇ T M , 0  . 2.6. Theorem (Bism ut and Lott [16]) . Mo dulo exact form s on B , (2.28) T  T H 1 E , g T M 1 , g F 1  − T  T H 0 E , g T M 0 , g F 0  = Z E /B  ˜ e  T M , ∇ T M , 0 , ∇ T M , 1  c h o  F , g F 0  + e  T M , ∇ T M , 1  e c h o  F , g F 0 , g F 1   − e c h o  H • ( E /B ; F ) , g H, 0 L 2 , g H, 1 L 2  . A v ariation formula lik e this has already b een pr o ved for the Ra y-Singer torsion in [18]. Theorem 2.6 is a dir ect consequence of Theorem 1.5. Similar v ariation form ulas exist for η -forms [10] and holomorphic torsion forms [14]. 2.7. Corollary (Bism ut and Lott [16]) . If the ﬁbr es of p : E → B ar e o dd- dimensional and F → E is ﬁbr ewise acyclic, then T ( T H E , g T M , g F ) deﬁnes an even c ohomo lo gy class on B that is i ndep endent of the choic es of T H E , g T M and g F . There is another situation w here T ( T H E , g T M , g F ) deﬁn es a cohomology class, at least its higher degree comp onents. Assume that g F 0 and g F 1 are b oth parallel with resp ect to ∇ F . Th en (2.29) g F t = (1 − t ) g F 0 + tg F 1 is a parallel metric on F for all t ∈ [0 , 1]. P ut the metric g ˜ F | F ×{ t } = g F t on the pullbac k ˜ F to ˜ E = E × [0 , 1], then (2.30) ω  ˜ F , g ˜ F  = ( g F t ) − 1 ∂ ∂ t g F t dt ∈ Ω 1 ( E × [0 , 1]; End ˜ F ) TORSION INV ARIANTS FOR F AMILIES 13 b ecause ω ( ˜ F , g ˜ F ) | E ×{ t } = 0 by (1.10). In particular (2.31) e c h o  F , g F 0 , g F 1  = c 0 Z 1 0 tr F  ( g F t ) − 1 ∂ ∂ t g F t  dt ∈ Ω 0 ( E ) is in f act ju st a constan t fun ction on E . 2.8. Deﬁnition. If the bun dles F → E and H ( E /B ; F ) → B adm it paral- lel metrics g F and g H , one d eﬁnes the higher analytic torsion or Bismut-L ott torsion as (2.32) T ( E /B ; F ) = T  T H E , g T M , g F  [ ≥ 2] + e c h o  H , g H , g H L 2  [ ≥ 2] ∈ Ω ≥ 2 ( B ) . It follo w s from Theorems 1.5 and 2.6 that T ( E /B ; F ) d eﬁnes a cohomology class in H ≥ 2 ( B ; R ) that is in dep endent of T H E , g T M , g F and g H , as long as g F and g H are parallel metrics. 3. Pr oper ties of Bismut-Lott tors ion Since η -forms, analytic torsion forms and h olomorphic torsion f orms are p ar- allel ob jects in three somewhat similar theories, one can try to translate an y result concerning one of th ose three ob jects into th eorems on the other tw o. In this s ection, w e presen t a few resu lts on higher torsion that where a t least partially motiv ated b y r esults on η -forms or on holomorphic torsion forms. In particular, we recall r esults b y Ma and Bunk e on torsion forms of iterated ﬁbrations, and of Bu nk e, Bism ut and th e author ab out the relation with equi- v arian t Ray-Singer torsion. Most of these theorems ha v e not y et b een pr o ved for Igusa-Klein or Dwyer-W eiss-Williams torsion. W e also discuss Ma and Z hang’s construction u sing η -inv arian ts of subsignature op erators. One should men tion at this p oin t that in the theory of ﬂat vec tor bun dles, w e are only considerin g p rop er submersions. The reason is that the direct image of a ﬂat v ector bundle u nder ot her maps like op en or closed em b edd ings is in general not giv en b y a ﬂ at ve ctor bu ndle. Another r eason is that there is no suitable analogue of the Bec ker-Go ttlieb transfer for general m aps. F or this reason, many b eautiful results f or η -inv ariant s and holomorphic torsion hav e no count erpart for Bismut-Lott torsion. 3.1. A t ra nsfer formula. C onsider a smo oth prop er sub mersion p 1 : E → B with t ypical ﬁ bre M as b efore, and assume that p 2 : D → E is another smo oth prop er submersion with ﬁb re N . Then p 3 = p 1 ◦ p 2 is ag ain a smooth prop er submersion, and its ﬁbre L maps to M w ith ﬁbre N . Let F → D b e a ﬂat v ector b undle, then we ha v e higher d irect images (3.1) K = dim N M k =0 ( − 1) k H • ( D /E ; F ) → E and H = dim L M k =0 ( − 1) k H • ( D /B ; F ) → B . Note that H is not the higher direct image of K under p 1 . Instead, there is a ﬁb rewise L era y-Ser re sp ectral sequence o v er B with E 2 -term H • ( E /B ; K ) 14 S EBASTIAN GOETTE that con verges to H . Beginning with E 2 , the higher terms in this sp ec- tral sequence are giv en by parallel families of ﬁnite-dimensional co c h ain com- plexes  E k , ∇ E k + d k  o ver B . Of course, E n = E ∞ and d n = 0 for all su ﬃ- cien tly large n . W e no w choose co mpatible complemen ts of the vertic al ta ngen t bundles for all three ﬁbr ations, ﬁb rewise Riemannian metric s, and a metric on the bu ndle F . Again, these d ata indu ce connections on the three v ertical tangen t bun dles T M , T N and T L ∼ = T N ⊕ p ∗ 2 T M . They also indu ce L 2 -metrics on the ﬂat v ector bund les H and E k o ver B for k ≥ 2 and on K → E . W e need the Chern-Simons Euler form ˜ e , whic h is constru cted in analogy w ith e c h in (1.3), and we also need another ﬁ nite-dimensional to rsion form T ( H , E ∞ , g H , g E ∞ ) relating the ﬁ ltered ﬂat vect or bu ndle H to its graded ve rsion E ∞ = E n for n suﬃcien tly large. 3.1. Theorem (T ransfer formula, Ma [56]) . M o dulo exact forms on B , we have (3.2) T  T H D , g T L , g F  = Z E /B e  T M , ∇ T M  T  H H D ⊕ T H L, g T N , g F  + T  T H E , g T M , g K  + ∞ X k =2 T  ∇ E k + d k , g E k  + T  H , E ∞ , g H , g E ∞  + Z D /B ˜ e  T L, ∇ T L , ∇ T N ⊕ p ∗ 2 ∇ T M  c h o  F , g F  . The ﬁr st t w o terms on the r igh t hand side s hould b e regarded as torsion forms o f the terms E 0 and E 1 of the Lera y-Serre sp ectral sequence. The sum of the torsions of th e remaining terms is of co urse ﬁnite. The theorem says in other w ords th at t he analyt ic torsion f orm of the total ﬁbration is the sum of the torsion forms of all terms in the Lera y-S erre sp ectral sequence and t wo natural correction terms. A similar form ula for h olomorphic torsion forms has b een pro v ed by Ma [54], [55]. F or η -inv ariant s of signature op erators, an analog ous result is due to Bunke and Ma [26]. 3.2. Lott’s Secondary K -theory of ﬂat bundles. In Arake lo v geometry , one s tudies arithmetic C ho w group s, w hic h constitute a simultaneous reﬁn emen t of classical Chow groups and of de Rham forms, see [64] for an introdu ction. The cent ral ob jects in this theory are algebraic vec tor bun dles o ver arithmetic sc h emes, toge ther with Hermitian metrics on the corresp onding holomorphic v ector bu ndles o ver the complex p oint s of those sc hemes, whic h form classical complex algebraic v arieties. T o construct the “complex alg ebraic” part of the direct image of suc h v ector bu ndles, one needs the holomorphic to rsion forms of Bism u t and K¨ ohler [14]. T o establish elemen tary prop erties of this direct image constr uction, one n eeds dee p results on holomo rph ic t orsion forms. Thus, Arak elo v geometry has b een one of the main motiv ations for the many results on holomo rph ic torsion by Bismut and others. F or this reason, it is te mpting to ha ve a similar theory for ﬂat v ector bund les o v er smo oth manifolds, where Bism ut-Lott torsion pla ys the role of h olomorphic torsion f orms. Lott’s K -theory of ﬂat vect or bu ndles with v anishing Kam b er-T ondeur classes is a ﬁr st step in this dir ection. But note that there are n o ob jects TORSION INV ARIANTS FOR F AMILIES 15 corresp onding to Chow cycles, and that we can tak e d irect images only for submersions, for reasons explained at the b eginning of this s ection. Thus w e cannot exp ect a th eory that is as ric h as arithmetic Chow theory . Nev ertheless, some nice results are motiv ated by Lott’s construction. W e consider triples ( F , g F , α ), where F → M is a ﬂat vec tor bundle, equip p ed with a metric g F , and α ∈ Ω ev en ( M ) /d Ω odd ( M ) satisﬁes (3.3) c h o  F , g F  − dα = 0 ∈ Ω odd ( M ) . A short exact sequence (3.4) 0 − − − − → F 1 a 1 − − − − → F 2 a 2 − − − − → F 3 − − − − → 0 of ﬂat v ector bun dles and parallel linear maps can b e interpreted as a parallel family of acyclic c hain complexes ( F, ∇ F + a ). Let g F 1 , g F 2 , g F 3 b e metrics on these b undles. By Theorem 2.1, the higher torsion form of this family satisﬁes (3.5) dT  ∇ F + a, g F  = c h o  F 2 , g F 2  − c h o  F 1 , g F 1  − ch o  F 3 , g F 3  . 3.2. Deﬁnition. Lott’s secondary K -group K 0 ( M ) is the ab elian group gener- ated by triples ( F , g F , α ) su b ject to (1) the condition (3.3 ), and (2) the relation T  ∇ F + a, g F  = α 2 − α 1 − α 3 ∈ Ω ev en ( M ) /d Ω odd ( M ) for eac h s hort exact sequence (3.4 ). In fact, Lott co nsiders groups K 0 R ( M ) in [53]. Here, R is a rin g s atisfying a few tec hnical assum ptions with a representat ion ρ : R → End C n , and al l ﬂat v ector b undles arise fr om lo cal systems of R -mo dules b y tensoring with C n . Similarly , relations come f rom short exact sequences of su c h lo cal systems. Let now p : E → B b e a prop er su bmersion with ﬁbre M . W e c ho ose T H E and g T M as b efore. 3.3. Deﬁnition. Let ( F, g F , α ) b e a generator of K 0 R ( M ) and let g H L 2 denote the L 2 -metric on the virtual v ector bund le H = dim M M k =0 ( − 1) k H • ( E /B ; F ) → B . Then the push -forw ard of ( F , g F , α ) is deﬁn ed as p ! ( F , g F , α ) =  H , g H L 2 , Z E /B e  T M , ∇ T M  α − T  T H E , g T M , g F   . Lott then v eriﬁes th at p ! deﬁnes a push-forward m ap (3.6) p ! : K 0 R ( E ) → K 0 R ( B ) . Moreo ve r, on the lev el of K -theory , the pu sh-forw ard is indep en den t of the c h oices of T H E and g T M . 3.4. Theorem (Bunke [25]) . L ott’s se c ondary K - gr oups to gether with the push- forwar d deﬁne a functor fr om the c ate gory of smo oth pr op er su bmersions to the c ate gory of ab elian gr oups. 16 S EBASTIAN GOETTE The pro of is b ased on Ma ’s Theorem 3.1. Bunke sh o ws that if p 1 : E → B and p 2 : D → E are smo oth p rop er sub mersions, then (3.7) ( p 1 ◦ p 2 ) ! = p 1! ◦ p 2! : K 0 R ( D ) → K 0 R ( B ) . A similar p ush-forward in secondary L -theo ry has been deﬁned by Bu nk e and Ma [26], correcting an older deﬁnition by Lott [53]. 3.3. Rigidit y of Kam b er-T ondeur classes. In this sec tion, we discuss the dep enden ce of the Kamber -T ondeur forms and the torsion forms on th e ﬂat structure on th e bundle F . Let V → M b e a vec tor bund le and assume that ( ∇ V ,t ) t ∈ [0 , 1] is a family of ﬂ at conn ections on V . I f we deﬁne a connec- tion ∇ ˜ V on the pull-bac k ˜ V of V to M × [0 , 1] su c h that ∇ ˜ V | M ×{ t } = ∇ V ,t , then the connection ∇ ˜ V will in general not b e ﬂat. In particular, the arguments in (1.3 ) and (1.7) are not applicable here. If w e ﬁx a f amily of metrics ( g V t ) on V , we ha ve a family ( ∇ V ,t, ∗ ) t ∈ [0 , 1] of adjoin t connections that are again ﬂat. Let n o w ˜ V denote the p ullbac k of V to M × [0 , 1] 2 and construct ∇ ˜ V suc h that (3.8) ∇ ˜ V   M ×{ s }× [0 , 1] = (1 − s ) ∇ V ,t + s ∇ V ,t, ∗ . W e d eﬁne forms L  ( ∇ V ,t , g V t ) t  ∈ Ω ev en ( M ) by (3.9) L  ( ∇ V ,t , g V t ) t  = π i Z 1 0 Z 1 0 ι ∂ ∂ s ι ∂ ∂ t c h  ˜ V , ∇ ˜ V  dt ds. Because ∇ V ,t and ∇ V ,t, ∗ are ﬂat, for s ∈ { 0 , 1 } , w e hav e (3.10) c h  ˜ V , ∇ ˜ V    M ×{ 0 , 1 }× [0 , 1] = ( 1 2 tr V  ∂ ∂ t ∇ V ,t  dt s = 0 , 1 2 tr V  ∂ ∂ t ∇ V ,t, ∗  dt s = 1 . Hence it follo ws from Stok es’ theorem that (3.11) dL  ( ∇ V ,t ) t , g V  [ ≥ 2] = c h o  V 1 , g V 1  [ ≥ 3] − ch o  V 0 , g V 0  [ ≥ 3] , where V t denotes the ﬂat vect or bu ndle ( V , ∇ V ,t ). One can sh o w that L (( ∇ V ,t , g V t ) t ) c hanges by exact forms if one re- places ( ∇ V ,t ) t b y a h omotopic path of ﬂat connections. On the other hand , if ∇ V , 1 = ∇ V , 0 , then the cohomology class of L (( ∇ V ,t , g V t ) t ) d ep ends on the homotop y class of th e lo op ( ∇ V ,t ) t in the sp ace of ﬂat connections. No w assume that V is Z -graded and that ( V • , ∇ V ,t + a t ) is a p arallel family of coc hain complexes on M suc h th at the ﬁbr ewise cohomology H • ( V , a t ) has the same ran k for all t . Then w e obtain a family of ﬂat Ga uß-Manin connec- tions ( ∇ H,t ) t and a family of metrics g H V ,t on a ﬁxed vec tor bun dle H → M . 3.5. Theorem (Rigidit y , Bism ut and G. [12]) . Under these assumptions, (3.12) T  ∇ V , 1 + a 1 , g V  [ ≥ 2] − T  ∇ V , 0 + a 0 , g V  [ ≥ 2] = L  ( ∇ V ,t , g V t ) t  [ ≥ 2] − L  ( ∇ H,t , g H V ,t ) t  [ ≥ 2] . Similarly , let ( ∇ F ,t ) t b e a family of ﬂat connecti ons on F → E s uc h that the ﬁbrewise cohomology H • ( E /B ; F t ) has the same rank for all t . Then w e again ha v e a family ( H t , g H L 2 ,t ) of ﬂat vec tor bun dles o ver B . TORSION INV ARIANTS FOR F AMILIES 17 3.6. Theorem (Rigidit y , Bism ut and G. [12]) . Under these assumptions, (3.13) T  T H E , g T M , g F 1  [ ≥ 2] − T  T H E , g T M , g F 0  [ ≥ 2] = Z M /B e  T M , ∇ T M  L  ( ∇ F ,t , g F t ) t  [ ≥ 2] − L  ( ∇ H,t , g H L 2 ,t ) t  [ ≥ 2] . Because T ( T H E , g T M , g F ) [0] equals the Ray-Singer analytic torsion, w e can- not expect Theorems 3.5 and 3.6 to hold f or the scalar part of the Bismut-Lott torsion, too. In fact th ese theorems as w ell as the construction of T ( E /B ; F ) indicate that th e “higher” Bism ut-Lott torsion h as a diﬀeren t topological mean- ing than the Ra y-Singer torsion. 3.4. Equiv a ria n t analytic torsions. Let u s assume that p : E → B is asso ci- ated to a G -principal bundle P → B for some compact, connected Lie group G . In particular, G acts by isometries on the ﬁbre ( M , g T M ). Let F → M b e a G -equiv ariant ﬂat v ector bundle such that elemen ts X of the Lie algebra g of G act by ∇ F X M , where X M is the corresp ond ing Killing ﬁeld on M . T hen the induced ve ctor bun dle (3.14) P × G F − → E = P × G M , whic h w e will again call F , is al so ﬂat. A G -equiv ariant ﬁbre bundle connec- tion T H P deﬁn es T H E , and w e also ﬁx a G -in v arian t m etric on F . Let Ω ∈ Ω 2 ( B ; g ) d enote the cur v ature of T H P . It was already observed in [1 6] and [51] that in this situation, the Bismut-Lott torsion is gi v en by an Ad-in v arian t formal p o w er series T g ( g T M , g F ) ∈ C [ [ g ∗ ] ] on g , suc h th at (3.15) T  T H E , g T M , g F  = T Ω 2 π i  g T M , g F  ∈ Ω ev en ( B ) . On the other hand , there is a G - equiv ariant ge neralisation of the Ra y-Singer analytic torsion. If g ∈ G acts b y isometries on M and p reserv es ∇ F , put (3.16) ϑ g  g T M , g F  ( s ) = − str  N M g ( d M + d ∗ M ) − 2 s  and T g  g T M , g F  = ∂ ∂ s ϑ g  g T M , g F  ( s ) . Inspired by results of Bismut, Berline and V ergne ab out th e equalit y of t wo notions of the equiv ariant index [5], one c an ask if the inﬁnitesimal equi- v arian t Bismut-Lott torsion T g ( g T M , g F ) is r elated to the equiv arian t tor- sion T G ( g T M , g F ). Bunke prov ed in [23] and [24] that b oth equiv arian t torsions can b e compu ted from th e G -equiv arian t Euler characte ristic of M up to a con- stan t when G is connected and F satisﬁes some technical assumptions. F rom Bunk e’s r esults, on e can deduce a relation b et ween b oth equiv ariant torsions in some interesting sp ecial cases. T o state a more general relation b et w een b oth equiv arian t torsions, we need the inﬁnitesimal Euler form e g ( T M , ∇ T M ) ∈ Ω • ( M )[ [ g ∗ ] ] and an equiv arian t Mathai-Quillen cu rren t ψ X ( T M , ∇ T M ) on M such that (3.17) dψ X  T M , ∇ T M  = e X  T M , ∇ T M  − e  T M X , ∇ T M X  δ M X , 18 S EBASTIAN GOETTE where δ M X is the Dirac current of in tegration o ver the ﬁxp oint set M X of the K illing ﬁ eld X M , for X ∈ g . Finally , for a pr op er submers ion p : E → B with typical ﬁb re M and a ﬁb rewise G -action, there exists an ev en clo sed form V X ( E /S, T H E , g T M ) that is lo cally compu table on E , v anishes for ev en- dimensional ﬁbres, and satisﬁes (3.18) V r X  E /S, T H E , g T M  = 1 | r | r − N B 2 V X  E /S, T H E , g T M  for all r ∈ R \{ 0 } . In particular, the class V X ( E /S ) ∈ H ev en ( B ; R ) is indep en- den t of T H E and g T M . Let V X ( M ) = V X ( E /S ) [0] denote the scalar p art. 3.7. Theorem (Bism ut and G. [13]) . F or X ∈ g , th e e qu ivariant torsions ar e r elate d by (3.19) T X  g T M , g F  − T e X  g T M , g F  = Z M ψ X  T M , ∇ T M  c h o  F , g F  + V X ( M ) rk F . Similar results for equiv arian t η -inv ariants ha v e b een prov ed in [36], and f or equiv arian t h olomorphic torsion by Bism ut and the author in [11]. 3.8. R emark. In T e X ( g T M , g F ), all p o we rs of X o ccur sim ultaneously . Thus, Theorem 3.7 can only hold for one c hoice of constan ts c k in (1.10), and this is precisely the so-called Chern normalisation introdu ced in [12] and also u sed in this o ve rview. The C hern normalisatio n is also needed f or Lott’ s noncomm uta- tiv e h igher torsion classes in [52], see Remark 7.6 b elo w. The theorem ab ov e is of course compatible with Bu nk e’s computations. As a simple application, we ca n use K¨ ohler’s computatio n of the equiv ariant analytic torsion on compact symmetric sp aces [50] to c ompute th e Bism ut-Lott torsion o f bund les with c ompact sy mmetric ﬁbr es an d compact structure groups . The ca se of sphere b undles will b e imp ortan t late r. Let ζ denote the Riema nn ζ -function. W e deﬁn e an additiv e c haracteristic class 0 J ( W ) for a v ector bundle W → M b y (3.20) 0 J ( W ) = 1 2 ∞ X k =0 ζ ′ ( − 2 k ) c h ( W ) [4 k ] ∈ H • ( M ; R ) . 3.9. C orollary (Sph ere bun dles, Bunk e [24], Bism ut and G. [12]) . L et E → B b e the unit n -spher e bund le of an oriente d r e al ve ctor bund le W → B . Then T ( E /B ; C ) = χ ( S n ) 0 J ( W ) . The meaning of the class V X ( M ) is not quite clear from Theorem 3.7. As Bism ut explains in [8], the Bismut-Lo tt torsion of a smo oth prop er submer- sion p : E → B is formally giv en by ev aluating V on the generator of the n atu- ral S 1 action on the ﬁbrewise free lo op space L B E , view ed as a bundle o ver B . Although the ﬂat v ector bu ndle F and its cohomology are not visible in this approac h , man y prop erties of V X ( E /S ) prov ed in [13] mirror w ell-known prop- erties of Bism ut-Lott torsion, including the b eha viour under iterate d ﬁb rations in Section 3.1 and under Witten deformation in S ection 5.1. TORSION INV ARIANTS FOR F AMILIES 19 3.5. The Ma-Zhang subsignature op erator. In section 1.1, w e ha ve con- structed K am b er-T ondeur forms b y lifting the Chern c haracter to ﬂat v ector bund les. In section 2.2, we ha ve co nstructed th e torsion form as a correction term in a family index theorem. T h us, Bismut-Lo tt torsion is a double trans- gression of the Chern charact er. Ma and Zhang ﬁrst pr o duce an η -inv ariant, whic h can b e regarded as a transgression of the Ch ern c h aracter. Then th ey deriv e Bismut-Lott torsion fr om a transgression of η -forms in [57]. In other w ords, they get torsion forms b y a diﬀerent doub le transgression. On the wa y , they gi v e a new analytic pr o of of Theorems 1.2 and 1.3. Dai and Zhang ha ve recen tly giv en a rela ted construction in [30], wh ere Bism ut-Lott torsion app ears in the adiabatic limit of a Bismut-F r eed connection f orm that is related to Ma and Zh ang’s η -inv arian t. Let p : E → B b e a prop er submersion of closed manifolds, wh ere B is orien ted, and let F → E b e a ﬂ at vect or bundle, then F is rationally trivial in the top ologic al K -theory of E . Th us there exists an isomorphism q F ∼ = E × C q rk F for some p ositiv e in teger q . Let ∇ 0 denote the trivial ﬂat connection on E × C q rk F , then (3.21) b c h  F , ∇ F  = 1 q e c h  ∇ 0 , ∇ q F  ∈ H • ( E ; C / Q ) . Cho ose T H E , g T M , g F as b efore. W e also c ho ose a metric g T B on B and put g T E = g T M ⊕ p ∗ g T B using th e sp litting T E = T M ⊕ T H E . Let W → B b e a Hermitian v ector bundle with metric g W and connection ∇ W . Ma and Zhang consider tw o op erators D W,F sig and ˆ D W,F sig on Ω • ( E ; p ∗ W ⊕ F ). Wherea s D W,F sig is an honest Dirac- op erator if g F is p arallel, the op erator ˆ D W,F sig diﬀeren tiates only in the directions of the ﬁbres. T hese op erators sh ould b e view ed as “quantisa tions” in the sense of [5], applied to the Bism ut t yp e sup erconnection ˜ A = 1 2 ( ˜ A ′ + ˜ A ′′ ) and the op erator ˜ X of (2.21). If B is o d d-dimensional, then (3.22) D W,F sig ( r ) = D W,F sig + ir ˆ D W,F sig is a selfadjoint op er ator on Ω ev en ( B ; Ω • ( E /B ; p ∗ W ⊕ F )) for all r ∈ R . The r e duc e d η - invariant of D W,F sig ( r ) is as u sual deﬁn ed as (3.23) η  D W,F sig ( r )  = 1 2  η  D W,F sig ( r )  + d im k er  D W,F sig ( r )   ∈ R / Z . F or the virtual bu ndle H ( E /B ; F ) → B , one d eﬁnes similarly (3.24) η  D W,H sig ( r )  = X k ( − 1) k η  D W,H k ( E /B ; F ) sig ( r )  ∈ R / Z . F or ε > 0, let D W,F sig ,ε ( r ) denote the analogous op erator, where the metric g T B has b een r eplaced by 1 ε g T B . T he red uced η -in v arian ts are related in the adia- batic limit ε → 0. 3.10. Theorem (Ma and Zhang [68], [57]) . One has (3.25) lim ε → 0 η  D W,F sig ,ε ( r )  = η  D W,H sig ( r )  ∈ R / Z . 20 S EBASTIAN GOETTE Pr o of of The or em 1.3. The p ro of for the imaginary part of b c h uses th e identitie s (3.26) ∂ ∂ r     r =0 η  D W,F sig ,ε ( r )  = Z B L ( T B ) c h( W ) tr ∗ E /B ∞ X k =0 c ′ k Im b c h ( F ) [2 k +1] and ∂ ∂ r     r =0 η  D W,H sig ( r )  = Z B L ( T B ) c h( W ) ∞ X k =0 c ′ k Im b c h ( H ) [2 k +1] for some constan ts c ′ k 6 = 0, where L ( T B ) denotes the Hirzebruc h L -class. Be- cause H ev en ( B ; R ) is s panned b y the v alues of L ( T B ) ch( W ) for all complex v ector bu ndles W , o ne g ets the imaginary part of (1.17) in Th eorem 1.3 from Theorem 3.10 b y comparison of co eﬃcient s in (3.26). The real part also follo ws from Theorem 3.10 b ecause (3.27) η  D W,F sig ,ε  − r k F η  D W sig ,ε  = Z B L ( T B ) c h( W ) tr ∗ E /B Re b c h ( F ) ∈ R / Q and a similar equation holds for the t w o virtual bun dles H ( E /B ; F ) → B and rk F · H ( E /B ; C ) → B . T o complete the pro of, one n eeds that (3.28) b c h ( H ( E /B ; C )) = 0 ∈ H • ( B ; C / Q ) , whic h was already p ro v ed for ev endimensional M by Bism ut in [7].  T o reco ver Bismut-Lott torsion and Theorem 1.5 from this approac h, one considers a generalised η -form (3.29) ˆ η r = (2 π i ) − N B +1 2 Z ∞ 0 tr s  ∂ ∂ t  ˜ A t + ir 2 ˜ X t   e − ( ˜ A t + ir 2 ˜ X t ) 2  − ir · a √ 1 + r 2 t − 3 2 ! dt for some lo cally compu table fun ction a : B → R . 3.11. Theorem (Ma and Zhang [57]) . F or c e rtain c onstants c ′′ k 6 = 0 , one ha s (3.30) ∂ ˆ η r ∂ r    r =0 = ∞ X k =0 c ′′ k d T ( T H E , g T M , g F ) . Dai and Zhang will giv e a more exp licit construction in [30]. These last results seem to indicate a strong relat ion b et w een Bism ut-Lott torsion and η - forms that still h as to b e explored. A similar rela tion h as b een established b y Bra verman and Kapp eler in a deﬁn ition of complex-v alued Ra y-Singer torsion in [22] for single manifolds. 4. Igusa-Kl ein tor sion W e hav e seen in sectio ns 1–3 h o w to establish an ind ex theorem for ﬂat v ector b undles u sing metho ds from lo cal index th eory for families, and ho w to disco v er Bismut-Lott torsion in a natural reﬁ nemen t of this ind ex theorem. It is some what surp rising that homotop y theoretical metho ds from diﬀeren tial top ology lea d to an in v ariant that is ve ry closely related to Bismut-Lott torsion. TORSION INV ARIANTS FOR F AMILIES 21 There are s ev eral sligh tly diﬀerent approac hes to this top ologic al higher torsion b y Igusa and Klein [49], [43], [46], [47]. In th is sect ion, w e fo cus on Igusa-Klein torsion as describ ed in [43]. In section 6, we discuss the approac h b y Dwye r, W eiss and Williams [31]. 4.1. Generalised Morse functions and ﬁltered complexes. It is wel l- kno wn that smo oth manifolds admit Morse functions. If p : E → B is a smo oth prop er submers ion, then in general, there is no function h : E → R t hat is a Morse fu nction on every ﬁbr e of p . Ho we v er, by resu lts of Igusa [42] and Eliash b erg and Mishac hev [32 ], there alw ays exist generalised Morse fu nctions. By a b irth-death singularit y of h : E → R , w e mean a ﬁb rewise critica l p oint of t yp e A2 th at is unf olded ov er B . In other words, there exist k , a f unction h 0 on B and coord inates u 1 , . . . on B and x 1 , . . . , x n along the ﬁbr es such that lo cally , h ( x, u ) = h 0 ( u ) + x 3 n 3 − u 1 x n − x 2 1 + · · · + x 2 k 2 + x 2 k +1 + · · · + x 2 n − 1 2 . Birth-death singularities o ccur ov er a tw o-sided immersed submanifold B 0 ⊂ B giv en b y u 1 = 0 in the coordinates a b ov e. Tw o ﬁbr ewise Morse critical p oin ts of adjacen t ind ices o ver the “p ositiv e” side of B come together in a ﬁbrewise cubical s ingularit y . In a neigh b ourho o d o v er the “negativ e” side, th e fun ction is regular. Let C = C M ∪ C bd ⊂ E d enote the su bmanifold of ﬁbrewise critical p oin ts of h . Note that th e submanifold C M of Morse ﬁ brewise critical p oin ts of h lo cally co v ers B , and that the submanifold C bd of birth-death critical p oin ts lo cally b ounds tw o comp onen ts of C M . After ﬁ xing a ﬁbrewise metric g T M , the negativ e ei genspaces of the Hessian of h form a v ector b undle T u M ⊂ T M | C M o ver C M , whose rank is giv en by the Morse index ind h . A t the birth-death singularities C bd , the n atural extension of T u M o f the tw o adjacen t comp onents of C M diﬀer by an orien ted trivial line bu ndle, the “cubical direction”. 4.1. Deﬁnition. A gener alise d ﬁbr ewise Morse fu nction on p : E → B is a fun c- tion h : E → R that has o nly Morse and birth-death t yp e ﬁbrewise singularities. A fr ame d function is a generalised ﬁbr ewise Morse function together w ith triv- ialisatio ns of T u M o ver eac h connected comp onen t of C M that ext end up to the b oun dary , suc h that the tw o frames at eac h p oint of C bd diﬀer only by the preferred generator of the cubical direction. 4.2. Theorem (Igusa [42]) . L et p : E → B b e a smo oth ﬁbr e bund le with typic al ﬁbr e M . If dim M ≥ dim B , ther e exists a fr ame d function, and if dim M > dim B , i t is unique up to ho motopy. Here, u niqueness up to homotop y means that if h 0 , h 1 : E → R are tw o f ramed functions, t hen there exists a f ramed fu nction h : E × [0 , 1] → R that r estricts to h j at E × { j } for j = 0, 1. If the dimension of the ﬁbres is to o small to app ly Theorem 4.2, one can tak e cr oss pro ducts with manifolds of Euler n umb er 1, for exa mple R P 2 n . One can c heck that this will n ot alte r th e torsion cla sses of Igusa and Klein that w e are go ing to in tro duce, so that the follo w ing constructions are v alid for ﬁbre bund les of arbitrary dimen sions. 22 S EBASTIAN GOETTE 4.2. Filtered c ha in complexes and the Whitehead space. W e assume that w e are giv en a smo oth ﬁ bre bund le p : E → B and a prop er framed fu nc- tion h : E → R with ﬁnitely man y ﬁb rewise critical p oints o v er small subsets of B . Let F → E be a ﬂat v ector bu ndle. Ov er a small op en sub set U ⊂ B , one can use h to ﬁlter the singular chain complexes of the ﬁ bres o v er U . The ﬁltered chain complexes are quasiisomorphic to a ﬁltered c hain complex on the v ector sp ace (4.1) V ′ x = M C ∈ C ′ M | x F c . Here C ′ M | x is a subset of C M | x , where some pairs of comp onent s of C M near birth-death singularities are omitted. Both the ﬁltration and the qu asiisomor- phism are natural and unique up to contract ible choice . Moreo ve r, the t wo lea v es of C M near a birth -death singularity generate a direct su mmand isomorphic to (4.2) 0 − − − − → F id − − − − → F − − − − → 0 after applying another quasiisomorphism that is again unique up to con tractible c h oice. Add ing or deleting a sub complex of the form (4.2) is called an elemen- tary exp ansion or elementary c ol lapse . Supp ose no w that the ﬂ at b undle F is ﬁbrewise acyclic and comes with an R -structure for a suitable ring R as in Section 3.2 ab o v e. Also assume that the holonom y of F if con tained in some group G ⊂ GL r ( R ), with r = rk F . A t ypical c hoice w ould b e R = M r ( C ) and G = U ( r ) with r ∈ N . In [43], Igusa constructs a classifying space for acyclic locally ﬁltered ﬁnite dimens ional c h ain complexes o v er R with holonom y G , up to ﬁltered quasiiso morphism s and elemen tary expansions and collapses. This space is called the acyclic Whitehe ad sp ac e W h h ( R, G ). W e giv e a slightly more explicit description in Section 5.2. 4.3. Theorem (Igusa [43]) . Each gener alise d ﬁbr ewise Morse function h : E → R gives rise to a classifying map (4.3) ξ h ( E /B ; F ) : B − → W h h ( R, G ) that is uni q ue up to hom otopy. T ogether with Th eorem 4.2 , one can asso ciate to a s mo oth ﬁbre bu n- dle p : E → B as ab o v e and a ﬂ at, ﬁbrewise acycli c vect or bu ndle F → E with R -structure and holonom y group G a unique homotop y class of m aps (4.4) ξ ( E /B ; F ) = ξ h ( E /B ; F ) : B − → W h h ( R, G ) , b y c ho osing h to b e a framed function. Assume that G pr eserv es a Hermitian metric, in other wo rds, that F carries a parallel metric. Then Igusa constructs cohomology classes (4.5) τ = ∞ X k =1 τ 2 k with τ 2 k ∈ H 4 k ( W h h ( R, G ); R ) that are related to the Kamber-T ondeur classes of Section 1.1. These classes are natural u nder pai rs of compatible ring and group homomorphisms ( R, G ) → TORSION INV ARIANTS FOR F AMILIES 23 ( S, H ). In particular, it is enough to construct th em for R = M r ( C ) and G = U ( n ). 4.4. Deﬁnition. The Igusa-Klein torsion o f a smo oth ﬁbre bund le p : E → B as ab ov e and a ﬂ at ﬁbr ewise acyclic vecto r bund le F → E with a parallel Hermitian metric is deﬁned as (4.6) τ ( E /B ; F ) = ξ ( E /B ; F ) ∗ τ ∈ H 4 • ( B ; R ) . Igusa also explains h o w to deﬁne ξ ( E /B ; F ) a nd τ ( E /B ; F ) if H • ( E /B ; F ) → B is a trivial bundle, or more generally , a globally ﬁltered ﬂat vect or bundle suc h that the associated graded vec tor bund le is trivial. In other w ords , the ﬂat cohomology bund le H • ( E /B ; F ) → B is then giv en b y a unip oten t repre- sen tation of π 1 B . 4.5. R emark. Th e map ξ ( E /B ; F ) of (4.4) is a higher torsion in v arian t in its o w n r igh t. In fact, most of the prop erties of τ ( E /B ; F ) in the next subsection already hold at the lev el of ξ ( E /B ; F ). Moreo v er, ξ ( E /B ; F ) is well-deﬁned ev en if F carries no parallel metric. Ho w ev er, the cohomology class τ ( E /B ; F ) mak es it p ossible to compare Igusa-Klein torsion with Bism u t-Lott torsion. 4.3. Prop erties of t he Igusa-Klein torsion. Assume that the ﬁbre b un- dle p : E → B arises by gluing t w o familie s p i : E i → B f or i = 1, 2 a long their ﬁ- brewise b oundary ∂ B E 1 = ∂ B E 2 . Then there exist a framed function h : E → R suc h that h | E 1 ≥ 0 and h | E 2 ≤ 0. Igusa p ro v es in [43] that the corresp onding classifying m ap ξ ( E /B ; F ) : B → W h h ( R, G ) “splits” in an appropriate sense, at least if R is a ﬁeld and the cohomology b undles H • ( E i /B ; F | E i ) → B are unip otent as ab o v e. Let D E i := E i ∪ E i → B denote the ﬁbrewise double o f E i , and let F i → D E i denote the ﬂat v ector b undle induced b y F | E i . Then the splitting ab o v e h as the follo wing consequence, in a w ording su ggested by Bunk e. 4.6. Theorem (Additivit y , Igusa [45]) . If E = E 1 ∪ E 2 → B is as ab ove and the bund les H • ( E i /B ; F | E i ) → B ar e unip otent, th en (4.7) 2 τ ( E /B ; F ) = τ ( D E 1 /B ; F 1 ) + τ ( D E 2 /B ; F 2 ) . Supp ose that p : E → B is a (2 n − 1)-sphere bundle with structure group U (1) n ⊂ O (2 n ). Th en E is th e ﬁb rewise join of n circle bun dles o ver B . The Igusa-Klein torsion of circle b undles has b een computed explicitly in [43], and Theorem 4.6 giv es the Igusa-Klein torsion of p . By the splitting principle for v ector bun dles an d naturalit y of the Igusa-Klein torsion, on e can no w com- pute the h igher torsion of all unit sph ere b undles in Euclidean v ector b undles. W e use the same normalisation as f or B ismut-Lo tt torsion. Let 0 J denote the c h aracteristic class deﬁned in equation (3.20). 4.7. Theorem (Sph ere bu ndles, Igusa [43]) . L et V → B b e an oriente d Eu- clide an v e ctor bund le with unit spher e bund le p : E → B . Then (4.8) τ ( E /B ; F ) = 2 0 J ( V ) . Note that this ag rees with the computations of the Bism u t-Lott torsion in Corollary 3.9 if the ﬁbres are o dd-dimen sional. 24 S EBASTIAN GOETTE Assume that h : E → R is a generalised ﬁbr ewise Mo rse fu nction for p : E → B that is not fr amed. Because at th e b irth-death singularities C bd , the n atural extensions of T u M at the t wo adjacen t c omp onents of C M are stably isomorp hic, w e h a ve a class 0 J ( T u M ) ∈ H • ( C ; R ). Let ˆ p = p | C and let C j M denote the ﬁbrewise Morse critical p oin ts of Morse index j , then there exists a well-deﬁned push-d o wn map (4.9) ˆ p ∗ α = dim M X j = 0 ( − 1) j ( p | C j M ) ∗ ( α | C j M ) ∈ H • ( B ) for all α ∈ H • ( C ). One c an compute the Igusa-Klein torsion of p : E → B using the classifying map ξ h ( E /B ; F ) even though h is not framed. 4.8. Theorem (F raming principle, Igusa [43], [44]) . In the situation ab ove, (4.10) τ ( E /B ; F ) = ξ h ( E /B ; F ) ∗ τ − 2 ˆ p ∗ 0 J ( T u M ) rk F . As an example, supp ose that p : E → B is the ﬁ brewise susp ension of the unit sph ere bundle in a vec tor bundle V → B . Then there exists a ﬁbrewise Morse function with only t wo ﬁbrewise critical p oin ts, and the unstable tangen t bund le at the ﬁ brewise maxim ums is isomorphic to the p ullbac k of V . In this case, Th eorems 4.7 and 4.8 giv e the same Igusa-Klein torsion for E → B . 5. Bismut-Lott = Igusa-Klein The higher analytic torsion of Bismut and Lott and the h igher F ranz- Reidemeister torsion of Igusa and Klein are deﬁn ed using rather diﬀeren t meth- o ds. Nev er theless, it was noticed that b oth torsions assign sp ecial v alues of the Bloch-Wig ner dilogarithm to acyclic ﬂat line bundles o ver circle bund les o ver S 2 [46], [16]. In this section, we describ e t wo appr oac hes to pro v e that b oth torsions agree. The ﬁrs t, d ue to Bism ut and the author, is in spired b oth b y the pro of of a general C heeger-M¨ uller theorem in [18], [19] and by the con- structions of Igusa-Klein torsion u sing Morse theory [43], [49]. T he second approac h classiﬁes a ll inv ariants of smooth ﬁbre b undles satisfying t wo simple axioms [45]. It is also suitable to compare Igusa-Klein torsion with the Dwy er- W eiss-Will iams construction in [31], see [2] and Theorem 6.5 b elo w. W e also giv e some consequences of the equalit y o f b oth torsions. 5.1. The Witten deformation. L et p : E → B b e a sm o oth p rop er su bmer- sion, and let F → E b e a ﬂat v ector b undle. W e assume that there exists a ﬁbrewise Morse f unction h : E → R suc h that the ﬁbrewise gradien t ﬁ eld ∇ h satisﬁes the Thom-Smale transv ersalit y condition on ev ery ﬁ bre of p . A non- trivial example is g iv en by the ﬁ brewise susp ension of a unit sp here bundle at the end of section 4.3. Let o ( T u M ) → C denote the orienta tion bun dle of T u M → C M , whic h extends n aturally to the birth-death singularities. Recall that C M = S j C j M , where C j M is the set of ﬁbrewise critica l p oints of Mo rse index j . W e deﬁne a ﬁnite-dimensional Z -graded ve ctor bund le V = M j V j − → B , TORSION INV ARIANTS FOR F AMILIES 25 with (5.1) V j =  p | C j M  ∗ ( F ⊗ o ( T u M )) . This bundle c arries a ﬂat connection ∇ V induced b y ∇ F , and a ﬁ brewise Thom- Smale d iﬀeren tial a . Then a is parallel, so by [16], there exists a torsion form (5.2) T  ∇ V + a, g V  ∈ H • ( B ; R ) as in Theorem 2.1 for all metrics g V induced b y metrics g F on F . W e c h o ose a horizon tal subbun dle T H E and a ﬁb rewise Riemannian met- ric g T M as in sections 1.3 and 2.2. Then there exists a Mathai-Quillen cur- ren t ψ ( ∇ T M , g T M ) on the total space of T M , s uc h that (5.3) d  ( ∇ h ) ∗ ψ ( ∇ T M , g T M )  = e  T M , ∇ T M  − δ C , where δ C denotes the alternating su m of the curr en ts of inte gration o v er C j M . Recall that we ha ve deﬁned t w o metrics g H V and g H L 2 on the ﬂ at v ector bun dle (5.4) H = H • ( M /B ; F ) ∼ = H • ( V , a ) → B . 5.1. Theorem (Bism ut and G. [12]) . Mo dulo exact forms on B , (5.5) T  T H M , g T M , g F  = T  ∇ V + a, g V  + e c h o  H , g H L 2 , g H V  + Z M /B ( ∇ h ) ∗ ψ  ∇ T M , g T M  · c h o  F , g F  + ˆ p ∗ 0 J ( T s M − T u M ) rk F . This theorem is pro v ed usin g the Witten deformation of the ﬁbrewise de Rham complex by h as in [18], [19]. By Theorems 1.5 and 2.1 and (1.7) and (5.3), taking the exterior deriv ativ e in Th eorem 5.1 gives a tr ivial identit y . Th e ﬁr st three terms on the righ t hand side can b e guessed that w a y . O n the other hand, the last term con tains top ologic al in formation related to Igusa’s framing principle. In fact, if F carries a parallel metric, then T ( ∇ V + a, g V ) [ ≥ 2] = 0 by the axiomatic d escription of T in [16], and th e metric g H V is p arallel, to o. Recall the Bec k er-Gottlie b transfer tr ∗ E /B : H • ( E ) → H • ( B ) of (1.14). I n th is case, Theorem 5.1 reduces to (5.6) T ( E /B ; F ) = ˆ p ∗ 0 J ( T s M − T u M ) rk F = ˆ p ∗ 0 J ( T M | C ) rk F − 2 ˆ p ∗ 0 J ( T u M ) rk F = τ ( E /B ; F ) + t r ∗ E /B 0 J ( T M ) rk F , where we hav e us ed a families version of th e P oincar ´ e-Hopf theorem, the f raming principle of T heorem 4.8, and the trivialit y of the classifying map ξ h ( E /B ; F ) : B → W h h ( R, G ). This already explains the similarity of Corollary 3.9 and Theorem 4.7 for su sp ended un it sphere bun dles. 5.2. Analytic I gusa-Klein torsion. Let u s assume again that h : E → R is a ﬁbrewise M orse function. W e still consider the Z -graded ﬂat vect or bun dle V → B of 5.1 with co nnection ∇ V and met ric g V induced from F . The function h acts b y m u ltiplicatio n on F | C , giving rise to a selfadjoin t endomorp hism h of V . 26 S EBASTIAN GOETTE An end omorphism of V is called h -upp er triangular if it map s eac h λ - eigen vecto r of h to the sum of the µ -eigenspaces with µ > λ . F or a generic ﬁbrewise Riemannian metric g T M , the ﬁbrewise gradient ∇ h will satisfy the Smale tr ansv ersalit y condition o v er an op en den se subset of B . Ov er this sub set, the Thom-Smale co c hain diﬀerent ial is a parallel, h -up p er triangular endomorphism a of V . The v arious d iﬀeren tials a ov er d iﬀeren t p oin ts along a path in B are conju gated b y endomorphisms of V of the t yp e id + b , where b is ag ain h -upp er triangular. As on e mov es around in a small c ircle on B , these endomorphisms comp ose to an automorphism of ( V , a ) that is homotopic to the identit y by an h -upp er triangular h omotop y . These v arious h omotopies are again related by h -upp er triangular higher h omotopies, and so on. If the cohomology bun dle H is unip oten t, then all these structures are encod ed in Igusa’s m ap ξ h ( E /B ; F ) : B → W h h ( R, G ) of Theorem 4.3. One may also consider these algebraic s tructures as a sin gular su p erconnec- tion on V . If R = M r ( R ) or R = M r ( C ), t here exists a smo oth ﬂat sup ercon- nection (5.7) A ′ = ∇ V + a 0 + a 1 + . . . of total degree one with h -upp er triangular (5.8) a j ∈ Ω j  B ; End 1 − j ( V )  = Ω j  B ; M k Hom  V k , V k +1 − j   and an Ω • ( B )-linear quasiisomorph ism (5.9) I :  Ω • ( E ; F ) , ∇ F  → (Ω • ( B ; V ) , A ′ ) b y [37]. The map I arises as a modiﬁ cation of the c lassical “in tegration o v er the unstable cells” , and it maps forms supp orted on h − 1 ( λ, ∞ ) to the sum of the µ -eigenspaces of h ∈ End V with µ ≥ λ . Moreo ver, the pair ( A ′ , I ) is un iquely determined up to con tractible c hoice b y h and g T M . It is shown in [38 ] that for acyclic F , Igusa’s map ξ h ( E /B ; F ) : B → W h h ( R, G ) also classiﬁes ( A ′ , I ) up to a natural notion of h omotop y . The ﬁn ite-dimensional torsion form of Deﬁnition 2.2 is only well-deﬁned for ﬂat sup erconnections of the form ∇ V + a 0 . In [37], [38] a torsion form T ( A ′ , ∇ V , g V ) is constr ucted using the fact that A ′ − ∇ V is a form on B with v alues in a nilp oten t subalgebra of End V , which ma y v ary o ver B . W e can still construct a metric g H V on (5.10) H = H • ( E /B ; F ) = H • ( V , a 0 ) → B as in section 2.1. Then w e s till h a ve (5.11) dT  A ′ , ∇ V , g V ) = c h o  V , g V ) − ch o  H , g H V ) . 5.2. Theorem ([37], [38]) . Mo dulo exact forms on B , (5.12) T  T H E , g T M , g F  = T  A ′ , ∇ V , g V  + ch o  H , g H L 2 , g H V  + Z E /B ( ∇ h ) ∗ ψ  ∇ T M , g T M  c h o  F , g F  + ˆ p ∗ 0 J ( T u M − T s M ) rk F . TORSION INV ARIANTS FOR F AMILIES 27 If b oth F and H carry parallel m etrics, w e can co nstruct a cohomology class as in Deﬁnition 2.8. Let g V b e the indu ced parallel metric on V . 5.3. Deﬁnition. The analytic Igusa-Klein torsion is deﬁned as (5.13) T ( E /B ; F ) = T  A ′ , ∇ V , g V  [ ≥ 2] + e c h o  H , g H , g H V  ∈ H ev en , ≥ 2 ( B ; R ) . T o justify the name, assume that g F is parallel and the bun dle H → B is a trivial ﬂat bund le. Then b oth τ ( E /B ; F ) and T ( E /B ; F ) are d eﬁned. 5.4. Theorem ([38]) . Under these a ssumptions, (5.14) T ( E /B ; F ) = ξ h ( E /B ; F ) ∗ τ ∈ H 4 • , ≥ 4 ( B ; R ) . In con trast to the situation in the previous section 5.1, th ese cohomology classes will b e non trivial in general. Also n ote that T ( A ′ , ∇ V , g V ) c an stil l b e constructed f or generalised ﬁ brewise Morse fu nctions h as in Deﬁn ition 4.1. In this conte xt, Theorem 5.4 still h olds. A ge neralisation of Theorem 5.2 will be pro v ed in [39]. As in (5.6), on e can no w compare Bism ut-Lott torsion and Igusa-Klein torsion. 5.5. Theorem ([38 ], [39]) . If F c arries a p ar al lel metric and H → B is a trivial ﬂat bund le, th en (5.15) T ( E /B ; F ) = τ ( E /B ; F ) + tr ∗ E /B 0 J ( T M ) rk F . 5.3. Axioms for higher t orsions. In this section, w e consider all smooth prop er submersions p : E → B with orien ted ﬁb res, suc h th at the ﬂat cohomol- ogy bu ndle H • ( E /B ; C ) → B is unip oten t in the sens e of sections 4.2, 4.3. W e will consider c haracteristic classes τ ( E /B ) ∈ H • ( B ; R ) of suc h ﬁbre bund les that are natural u nder pullback. Su c h a class is called add itive if it satisﬁes a gluing formula as in Theorem 4.6. Let W → E b e an orien ted r eal vecto r bu ndle of rank n + 1, and let S → E b e its un it n sphere bun dle. Th en H • ( S/B ; C ) → B is still un ip oten t. A c h aracteristic class τ as ab o v e is said to satisfy the tr ansfer r elation if (5.16) τ ( S/B ) = χ ( S n ) τ ( E /B ) + tr ∗ E /B τ ( S/E ) ∈ H • ( B ; R ) . F or the analytic torsion, the analogous resu lt is a s p ecial case of Ma’s transfer theorem 3.1. 5.6. Deﬁnition. A higher torsion invariant in degree k is a charac teristic class τ k ( E /B ) ∈ H k ( B ; R ) for all p : E → B as ab o ve that is natural under pullbac k, ad ditiv e, and satisﬁes the transfer relation (5.16). 5.7. Theorem (Igusa [45]) . H igher torsion invariants exist in de gr e e 4 k f or al l k > 0 , and every higher torsion invariant is a line ar c ombination of tr ∗ E /B 0 J ( T M ) [4 k ] , (ev en) and τ 2 k ( E /B ; C ) + t r ∗ E /B 0 J ( T M ) [4 k ] . (o dd) Note that ev en higher torsion in v arian ts v anish for p : E → B if the ﬁ bres are o dd-dimensional, and vice v ersa. The even classes tr ∗ E /B 0 J ( T M ) [4 k ] are called Mil ler-Morita-Mumfor d classes in [45], b ecause they generalise the classes for 28 S EBASTIAN GOETTE surface bund les introdu ced in [58], [59], [61]. Th e odd higher torsion classes are multiples of the Bismut-Lot t torsion T ( E /B ; C ) under th e assumptions of Theorem 5.5. The follo win g result f ollo ws from the p ro of of u niqueness in T heorem 5.7. 5.8. Theorem (Igusa [45 ]) . F or ﬁbr e b u nd les p : E → B as ab ove, (5.17) τ 2 k ( E /B ; C ) ∈ H 4 k ( B ; ζ ′ ( − 2 k ) Q ) . Theorem 5.7 could in prin ciple also b e used to pro v e Theorem 5.5. Un for- tunately , additivit y of the Bismut-Lo tt torsion is on ly kno wn as a consequ ence of Th eorem 5.5. An other consequence of th is resu lt is a more general transfer form ula for Igusa-Klein torsion as in Ma’s Theorem 3.1 , including the case of ﬁbre pro ducts. By T heorems 3.7 and 5.5, Igusa-Klein torsion is also related to equiv arian t torsion in the case of ﬁbr e bundles with compact structure groups. Finally , Theorems 3.6 and 5.5 describ e the v ariation of Igusa-Klein torsion un- der change s of th e ﬂat b undle F → E . W e already ment ioned the smooth Dwy er-W eiss-Williams torsion. Its deﬁ- nition is giv en in [31], see sect ion 6.2 b elo w. In [3], corresp onding cohomology classes in H 4 k ( B ; R ) are co nstructed. Additivit y and the transfer relation ha v e recen tly b een pro ved in [2]. This imp lies that cohomological smo oth Dwy er- W eiss-Will iams torsion shares all the other p rop erties m en tioned ab o v e. It also implies a more general transfer f orm ula for Igusa-Klein torsion. 6. D wye r-Weiss-Williams torsion In this sect ion, we p resen t the homoto py theoretic al app roac h to generalised Euler c haracteristics and h igher torsion in v arian ts in [31] and [3], and we sk etc h the pro of of Theorem 1.4. Dwyer, W eiss and Williams constru ct three g ener- alised Euler c h aracteristics for ﬁbrations p : E → B , w hic h con tain information ab out the existence of a to p ological or ev en smooth bu ndle of manifolds that is ﬁbr e homotop y equ iv alen t to p . If F → E is a ﬁbrewise acyclic bun dle of R - mo dules, then these Euler c h aracteristics can b e lifted to three diﬀeren t higher torsion inv ariants. 6.1. The top ological index theorem. The W aldhausen K -theory A ( E ) of a space E is the K -theory of a certain ca tegory of retrac tiv e spaces ov er E [66]. It is a homotop y in v ariant fun ctor, but not excisiv e, so it do es not deﬁn e a generalised homology theory . One can ho w ev er deﬁne an excisiv e functor A % b y pu tting (6.1) A % ( E ) = Ω ∞ ( E + ∧ A ( ∗ )) . Here, E + is the disjoint union of E and a basep oin t ∗ , and Ω ∞ is the inﬁn ite lo op space constru ction. W eiss and Williams constru ct a n atural assem bly map (6.2) α : A % ( E ) − → A ( E ) in [67]. W e will also need the sp ectrum (6.3) Q ( E + ) = Ω ∞ Σ ∞ ( E + ) = lim k Ω k Σ k ( E + ) , TORSION INV ARIANTS FOR F AMILIES 29 where Σ denotes the redu ced susp ension. F or a ﬁbration p : E → B , one h as relativ e functors A B ( E ) → B , A % B ( E ) → B and Q B ( E B ) → B , whic h b ehav e almost as ﬁbrations o v er B wh ere th e functors ab o v e ha ve b een app lied ﬁb rewise to p : E → B . The homotop y Euler char acteristic (6.4) χ h ( E /B ) : B − → A B ( E ) is a section of A B ( E ) → B . It is d eﬁned as the class of E × S 0 o ver E in A B ( E ) if the ﬁb res of p are homoto py ﬁnitely dominate d , th at is homotop y equiv alent to retracts of ﬁn ite CW complexes. If B is a p oin t, then χ h ( E ) enco d es pr ecisely the Eu ler num b er and the W all ﬁniteness obstruction of the ﬁ bre. A ﬂ at ve ctor bund le F → E , or more generally , a bun dle of ﬁn itely generated pro jectiv e R - mo dules for some ring R , induces a map λ F : A ( E ) → K ( R ) indu ced b y taking homology relativ e to E with co eﬃcien ts in F . F or the pro of of Theorem 1.4, one u ses that th e comp osition of maps (6.5) B χ h ( E /B ) − − − − − − → A B ( E ) − − − − → A ( E ) λ F − − − − → K ( R ) classiﬁes the ﬁ brewise cohomolo gy H ( E /B ; F ) → B as a virtual bundle and th us give s the left hand side of (1.18). If p : E → B is a bund le of top ological manifolds, there exists a vertical tangen t microbundle T M → E . It has an Euler class e ( T M ) with co eﬃcien ts in A % B ( E ). Let ℘ d enote the generalise d ﬁbrewise Poinca r´ e d ualit y [31]. Then one can d eﬁne a top olo gic al Euler char acteristic χ t of p with the pr op ert y that (6.6) χ t ( E /B ) = ℘ e ( T M ) : B − → A % B ( E ) . The ﬁbrewise assem bly of (6.2 ) maps it to A B ( E ). One has a P oincar´ e-H opf t yp e index theorem. 6.1. Theorem (Dwy er, W eiss and Williams [31]) . F or a bund le p : E → B of c omp act top olo gic al manifolds, the se ctions χ h ( E /B ) and α ◦ χ t ( E /B ) of A B ( E ) → B ar e homoto pic by a pr eferr e d p ath of se c tions. Conversely, if χ h ( E /B ) lifts to A % B ( E ) , then p is ﬁbr e homot opy e quivalent to a bu nd le of c omp act top olo gic al manifolds. If the v ertical tangent b undle T M → E is a top ologica l disc bu ndle, then p : E → B is calle d a r e gular manifold bund le , wh ic h includ es the im- p ortant special case of a pr op er submersion. In this ca se, one can d eﬁne the Bec ke r Euler cl ass b ( T M ) with coeﬃcien ts in the spher e spectrum . Its ﬁb rewise P oincar ´ e du al giv es the Bec ker-Go ttlieb transfer, regarded as a section (6.7) χ d ( E /B ) = t r E /B = ℘ b ( T M ) : B − → Q B ( E B ) . Ev en though Bec k er-Gottlie b transfer is already deﬁned for ﬁbr ations with ho- motop y ﬁn itely dominated ﬁ bres, we can regard it as a third generalised Euler c h aracteristic χ d for regular manifold bund les by (6.7). Th ere is a natural unit map η : Q B ( E B ) → A % B ( E ), and w e hav e another P oincar´ e-H opf t yp e index theorem. 30 S EBASTIAN GOETTE 6.2. Theorem (Dwy er, W eiss and Williams [31]) . F or a bund le p : E → B of close d r e gular top olo gic al manifolds, the se ctions χ t ( E /B ) and η ◦ tr E /B of A % B ( E ) → B ar e ho motopic by a pr eferr e d p ath of se ctions. Pr o of of The or em 1.4. W e regard the homoto py cla ss of maps E → K ( R ) in- duced by th e ﬁnitely generated pro jectiv e R -mo dule bun dle F → E . As in (6 .5), this map can b e written as a comp osition (6.8) E − − − − → Q ( E ) α ◦ η − − − − → A ( E ) λ F − − − − → K ( R ) . Th us, the righ t h and side of (1.18) in Th eorem 1.4 is classiﬁed by th e comp o- sition (6.9) B tr E /B − − − − → Q B ( E ) α ◦ η − − − − → A B ( E ) − − − − → A ( E ) λ F − − − − → K ( R ) . By Theorems 6.1 and 6.2, this map is homotopic to (6.5) , w hic h classiﬁes the left hand side of (1.18). This completes the pro of.  One notes that b oth sides of (1.1 8 ) in Theorem 1.4 are deﬁ ned for a ﬁ- bration p : E → B with homotop y ﬁnitely dominated ﬁbr es. Ho wev er, for Theorem 6.2 one n eeds the r egular structure c oming from the smooth bundle structure. It is somewhat surp rising that the existence of a smo oth ﬁb re bundle structure is necessary to compare th e v arious Euler charact eristics ab o ve. 6.3. Theorem (Dwy er, W eiss and Williams [31]) . L et p : E → B b e a ﬁbr ation with homot opy ﬁnitely domina te d ﬁbr e s. If χ h ( E /B ) lifts to Q B ( E B ) → B , then p is ﬁbr e homo topy e qu i valent to a bund le of smo oth manifolds. 6.2. T opological higher Reidemeister t orsion. Supp ose that F → E is a bund le of ﬁn itely generated p ro jectiv e R -mo dules that is ﬁb rewise acyclic. Then the three Euler c haracteristics χ h ( E /B ), χ t ( E /B ) and tr E /B of the previous subsection can b e lifted to higher Reidemeister torsions. Assume ﬁ rst that p : E → B is a ﬁbration with h omotop y ﬁn itely dominated ﬁbres. I f F is ﬁbrewise acyclic, then the comp osition in (6.5) is canonica lly homotopic to the trivial map B → K ( R ). F or a single space M , this give s an elemen t τ h ( M ; F ) in the h omotop y ﬁbr e (6.10) Φ h ( M ; F ) = h oﬁb( λ F ) of λ F : A ( M ) → K ( R ) o ver χ h ( M ) ∈ A ( M ). F or the ﬁb ration p , w e get a lift (6.11) τ h ( E /B ; F ) : B − → Φ h ( E /B ; F ) = h oﬁb B ( λ F ) of χ h ( E /B ), where the ﬁbres of Φ h ( E /B ; F ) → B are the homotop y ﬁb res of λ F . If p : E → B is a bundle of topological manifolds , we sim ilarly get a lift of χ t ( E /B ) : B → A % B ( E ) to (6.12) τ t ( E /B ; F ) : B − → Φ t ( E /B ; F ) = h oﬁb B ( λ F ◦ α ) . If p : E → B is a b undle of smo oth or regular manifolds, one gets a lift of χ d ( E /B ) = tr E /B : B → Q B ( E B ) to (6.13) τ d ( E /B ; F ) : B − → Φ d ( E /B ; F ) = h oﬁb B ( λ F ◦ α ◦ η ) . TORSION INV ARIANTS FOR F AMILIES 31 6.4. Deﬁnition. If F → E is ﬁbrewise acyclic , th en τ h ( E /B ; F ), τ t ( E /B ; F ) and τ d ( E /B ; F ) are called th e homotop y, top olo gic al and smo oth Dwyer-Weiss- Wil liams torsion , resp ectiv ely , whenever they are d eﬁned. The natural maps α and η induce maps (6.14) α : Φ t ( E /B ; F ) − → Φ h ( E /B ; F ) and η : Φ d ( E /B ; F ) − → Φ t ( E /B ; F ) . By Theorems 6.1 and 6.2, the Dwy er-W eiss-Williams torsions are related up to a pr eferred ﬁbrewise homotopy by (6.15) τ h ( E /B ; F ) ∼ ατ t ( E /B ; F ) and τ t ( E /B ; F ) ∼ η τ d ( E /B ; F ) if they are d eﬁned. W e will see in the next sect ion 7 that Bism ut-Lott to rsion and Igusa-Klein torsion can detect d iﬀeren t smo oth bundle structures on a giv en top ologica l manifold bun dle p : E → B . Thus, T ( E /B ; F ) and τ ( E /B ; F ) cannot b e re- co vered from τ h ( E /B ; F ) or τ t ( E /B ; F ). On the other h and, we do not know an y example yet w here th e diﬀerence τ ( E /B ; F 1 ) − τ ( E /B ; F 0 ) dep ends on the smo oth ﬁb re bun dle structure if F 0 , F 1 → E are tw o ﬂat v ector b undles of the same rank with unip oten t ﬁ brewise cohomology bundles. It is th us natural to ask if one can reco v er τ ( E /B ; F 1 ) − τ ( E /B ; F 0 ) or T ( E /B ; F 1 ) − T ( E /B ; F 0 ) from τ t ( E /B ; F 1 ) − τ t ( E /B ; F 0 ) or eve n from τ h ( E /B ; F 1 ) − τ h ( E /B ; F 0 ). Let us note at this p oin t that additivit y of the top ological Dwyer-W eiss-Williams torsion τ t ( E /B ; F ) and of the underlying Eu ler c haracteristic χ t ( E /B ) of (6.6) has b een established in [1]. In [3 ], a cohomological v er sion of τ d ( E /B ; F ) is constru cted. It is still d eﬁned if H • ( E /B ; F ) → B is a unip oten t bundle. The follo win g r esult has recen tly b een p ro v ed using Igusa’s axioms. 6.5. Theorem (Badzio c h , D orabia la, Klein, Williams [2 ]) . F or any k > 0 , the c ohomolo gic al smo oth Dwyer-Weiss- Wil liams torsion of [3] is pr op ortional to the Igusa-Klein to rsion in the same de gr e e. In add ition, it w ould b e nice to hav e a natural map f rom Φ d ( E /B ; F ) to I gusa’s Whitehead space W h h ( R, G ) that send s τ d ( E /B ; F ) to the map ξ ( E /B ; F ) of (4.4). 7. Exotic sm ooth bundles Consider tw o smo oth prop er sub mersions p i : E i → B for i = 0, 1. It is p ossible that the ﬁbr es of p 0 and p 1 are diﬀeomo rph ic, and that there exists a h omeomorphism ϕ : E 0 → E 1 suc h that p 0 = p 1 ◦ ϕ , but no su c h diﬀeomor- phism. If th is is the case, then p 0 and p 1 are isomorp hic as top ological, but n ot as smo oth ﬁ bre bu ndles ov er B . In this case, w e w ill say that p 1 giv es an exotic smo oth bund le structur e on the bundle p 0 . Of course, in man y cases there is no distinguished standard smo oth bundle str ucture, so the term “exotic ” ma y b e misleading. Higher torsion inv ariants detect some exotic smo oth bund le structures, as w e will explain in this section. W e also recall Heitsc h-Lazaro v torsion, whic h might b e useful to d etect exotic smo oth structures on foliati ons. 32 S EBASTIAN GOETTE 7.1. Hatc her’s example. It is w ell kn o wn that the higher stable homotop y groups of spheres are ﬁ nite, whereas some higher homotop y groups of the or- thogonal group are not. More pr ecisely , if m is suﬃcien tly large with resp ect to k , then th e k ernel of the J -homomorp hism (7.1) J 4 k − 1 : π 4 k − 1 ( O ( m )) − → π n +4 k − 1 ( S m ) con tains an inﬁn ite cyclic su bgroup. An el emen t γ ∈ ker J 4 k − 1 can b e used to construct a family of em b eddings ˜ γ q : S m × D n − 1 → S m × D n − 1 for q ∈ D 4 k , if n is suﬃcien tly large , whic h are given b y a pair of linear maps S m → S m and D n − 1 → D n − 1 for q ∈ S 4 k − 1 = ∂ D 4 k . Glueing D m +1 × D n − 1 to S m × D m along S m × D n − 1 ⊂ ∂ ( D m +1 × D n − 1 ) for all q ∈ D 4 k , one obtains an ( m + n )- disc b undle ov er D 4 k together with a canonical trivialisation o v er S 4 k − 1 . Thus, this bu ndle can b e extended to a smo oth disc bundle (7.2) p γ : E γ − → S 4 k = D 4 k ∪ S 4 k − 1 D 4 k , as d escrib ed in [43] and [37 ]. This d isc bu ndle w as ﬁrst constructed b y Hatc her. B¨ okstedt pro ved that for γ 6 = 0, the bundle p γ is homeomorphic, bu t not diﬀeomorphic to a trivial disc bund le i n the sen se ab o ve [20]. Note that p γ carries a ﬁbrewise Morse func- tion h : E γ → R w ith t wo critical p oints of index 0 and m in the part S m × D n of the ﬁbre, and another one of in dex m + 1 on D m +1 × D n − 1 . The corresp onding family of Thom-Smale complexes is trivial, but h is n ot framed. If W γ → S 4 k denotes th e R n -bundle with clutc hing function γ | S 4 k − 1 , Igusa’s f raming pr inciple giv es (7.3) τ ( E γ /S 4 k ; C ) = 2( − 1) m 0 J ( W γ ) 6 = 0 ∈ H 4 k ( S 4 k , R ) , see Th eorem 4.8 and [43]. T o construct a smo oth prop er sub mersion, w e tak e the ﬁbrewise dou- ble D E γ → S 4 k . Its Igusa-Klein torsion of Deﬁn ition 4.4 is giv en by (7.4) τ ( D E γ /S 4 k ; C ) = 2  ( − 1) m − ( − 1) n  0 J ( W γ ) , whic h v anishes p recisely if the ﬁbr es are even-dimensional. By Th eorem 5.5, this agrees with th e Bism ut-Lott torsion T ( D E γ /S 4 k ; C ) of Deﬁnition 2.8 , see [37]. If p : E → B is a s mo oth prop er submersion with dim B = 4 k and dim M od d and suﬃciently large, then one c an ta k e o ut r copies of D 4 k × D dim M from E and glue in r copies of E γ | D 4 k instead. This gi v es an e xotic smo oth bundle p r : E r → B . If either Bismut-Lo tt torsion or Igusa-Klein torsion are deﬁned for some ﬂ at bund le F → E , then this torsion will change by ± 2 r 0 J ( W γ ) rk F ∈ H 4 k ( B ; R ) if B is orien ted. Igusa also constru cts a diﬀer enc e torsion satisfying (7.5) τ ( E r /B , E /B ; F ) = ± 2 r 0 J ( W γ ) rk F ev en if H • ( E /B ; F ) ∼ = H • ( E r /B ; F ) → B is not a un ip oten t bun dle. W e still assume that B is orien ted and that dim M is odd and suﬃcient ly large. The gluing co nstru ction abov e can b e generalised to construct a discrete family of exotic smo oth bun dles p ν : E ν → B such that the v alues of their TORSION INV ARIANTS FOR F AMILIES 33 diﬀerence torsions τ ( E r /B , E /B ; F ) form a lattice in the space (7.6) ∞ M k =1 ℘ im  p ∗ : H dim B − 4 k ( E ) − → H dim B − 4 k ( B )  ⊂ ∞ M k =1 H 4 k ( B ) of classes that are P oincar´ e dual to classes pushed do wn from E . This is an ongoing pro j ect with Igu sa. 7.2. The space of stable exotic smo oth structures. There are tw o natural questions: can higher torsion detect all exotic smooth bun dle structures, and can all these structur es b e constru cted? T o answ er these questions, o ne wan ts to understand the space of all suc h exotic smo oth b undle structures. As Wil liams p oint ed out, a certain stable version of this space can b e analysed usin g the metho ds of the p ap er [31]. W e start with a bun dle p : E → B of compact top ological n -manifolds, equipp ed with a v ector bundle V → E of r ank n . A smo oth manifold bun- dle p ′ : E ′ → B is called a ﬁbr ewise tangential smo othing of ( E /B , V ) if there exists a homeomorph ism ϕ : E ′ → E with p ′ = p ◦ ϕ and a ve ctor bun dle iso- morphism k er( dp ′ ) → V o v er ϕ . Let S B ( E , V ) denote the space of all ﬁ brewise tangen tial smo othings. By co nsidering total sp aces of closed, ev en-dimensional linear disk bundles π : D ( ξ ) ⊂ ξ → E after roundin g oﬀ the corners, w e con- struct the space of stable ﬁb r e wise tangential smo othings (7.7) S s B ( E , V ) = lim − → S B ( D ( ξ ) , π ∗ ( V ⊕ ξ )) , where the limit is tak en o v er all v ector b undles. Let H ( ∗ ) b e the stable h -cob ordism sp ace, and constru ct a ﬁb ration H % B ( E ) with ﬁb res Ω ∞ ( M + ∧ H ( ∗ )) as in (6.1 ). 7.1. Theorem (Dwye r, W ei ss and Williams [31]) . If ( E /B , V ) admits stable ﬁbr ewise tangential smo othings, then S s B ( E , V ) is homotopy e quivalent to the sp ac e of se ctions of H % B ( E ) → B . In other w ords, the group π 0 Γ B H % B ( E ) o f homoto py classes o f sect ions acts simply transitiv ely on the isomorphism classes of stable ﬁbrewise tangen tial smo othings. 7.2. Theorem (Igusa and G.) . If the ﬁbr es and b ase of p : E → B ar e close d oriente d manifolds, then (7.8) π 0  S s B ( E , V )  ⊗ Z Q ∼ = ∞ M k =1 H dim B − 4 k ( E ; Q ) . In sp ecial case s, this was already known, see [33]. Th us, if p i : D ( ξ i ) → B are stable ﬁbr ewise tangentia l smo othings for i = 0, 1, we can deﬁne th e r elative D wyer-Weiss-Wil liams torsio n τ d/t 2 k ( p 0 , p 1 ) ∈ H 4 k ( B ; Q ) for k ≥ 1 as the Poinca r´ e d ual of the image of th e corresp ond ing diﬀerence class in H dim B − 4 k ( B ; Q ). 7.3. Theorem (Igusa and G.) . In the situation ab ove, the Igusa-Klein diﬀer enc e torsion is a sc alar multiple of the r elative Dwyer-Weiss-Wil liams torsion. 34 S EBASTIAN GOETTE Details will app ear elsewhere. 7.4. R emark. In general, the space in (7.8) has higher rank than the space in (7.6). Th is implies that higher torsion ca nnot detect all r ational stable ﬁ - brewise tangen tial smo othings. It do es n ot help to c hose diﬀeren t ﬂat vec tor bund les F → E either. O ne reason is that E could b e s imply connected. An- other reason is the fac t that in (7.5) and its analo gue in t he more general setting of (7. 6), the ﬂ at v ector b undle F only con tributes b y its rank. 7.5. R emark. Th us the diﬀerence of the Igusa-Klein or Bismut-Lo tt torsions of E → B with t w o d iﬀeren t ﬂat v ector b undles of the same rank seems to b e indep endent of th e sm o oth stru cture in the examples kno wn so far. This observ ation leads to the question if this diﬀerence can b e computed already from the top ologica l or the homotop y Dwy er-W eiss-Willia ms torsion. Th eo- rem 3.6 shows th at under sp ecial assu mptions, it can ev en b e computed using the Bec k er-Gottlieb transfer only . 7.6. R emark. In th e sp ecial case of asph erical ﬁbres M , Lott deﬁnes a noncom- m utativ e higher analytic to rsion form with co eﬃcient s in a certain s ubalgebra of C ∗ r π 1 ( M ) in [52]. Lott asks if this inv ariant detects all rational exotic s truc- tures. T o the author’s kno wledge, this questio n is still op en . More generally , one w ould lik e to h a ve a similar inv arian t for arb itrary ﬁbres that can d etect all rational stable exotic smo oth str uctures. 7.3. Heitsc h-Lazaro v torsion for foliat ions. Let E b e a sm o oth closed manifold with a smo oth foliation F . Since in general, the space of lea ves E / F is ill-behav ed, w e consider a foliation group oid whose ele ment s are classes of paths on the lea ve s of F . W e will assum e that this group oid G lies b etw een the homotop y and th e holonom y group oid, and that it is Hausdorﬀ and thus giv en by a smooth manifold and t wo su bmersions r , s : G → E . W e will assume that the stron g No viko v-Shubin in v ariants of the lea fwise Hodge-Laplacians are p ositiv e. Heitsc h and Lazaro v giv e a generali sation of Bism ut-Lott torsion in a set- ting that essentia lly av oids noncommutati v e metho d s [41]. Th us, let Ω • c ( E / F ) denote the H¨ aﬂiger F orms, that is, th e coin v arian ts under F in the space of compactly supp orted de Rham forms on a complete transversal to F . Th e cohomology H • c ( E / F ) of (Ω • c ( E / F ) , d ) resembles the compactly s upp orted de Rham cohomology of a manifold. Let F → E b e a ﬂat vecto r bu ndle with metric g F . If one ﬁxes a com- plemen t T H E to T F ⊂ T E and a leafwise metric g T F , there exists a natural connection ∇ T F on T F → E . Using integrati on along the lea v es, one deﬁnes (7.9) Z F e  T F , ∇ T F  c h o  F , g F  ∈ Ω • c ( E / F ) . Let P : Ω • ( F ; F ) → H • = H • ( F ; F ) denote the p ro jection of the lea fwise forms with v alues in F onto the harmonic forms. Using P , one deﬁnes (7.10) c h o  H , g H L 2  ∈ Ω • c ( E / F ) in analogy with (1.5). As in Deﬁnition 2.4, Heitsc h and Lazaro v then constr uct a higher analytic torsion form T ( T H E , g T F , g F ) ∈ Ω • c ( E / F ). TORSION INV ARIANTS FOR F AMILIES 35 7.7. Theorem (Heitsc h and Lazaro v [41 ]) . In th e situatio n ab ove, (7.11) d T  T H E , g T F , g F  = Z F e  T F , ∇ T F  c h o  F , g F  − ch o  H , g H L 2  ∈ Ω • c ( E / F ) . Heitsc h and Lazaro v need a large p ositiv e lo we r b ound for the strong leaf- wise No vik o v-Shubin in v ariants. Thus th ey only r egard examples with compact lea ves. It seems ho wev er, that un iform p ositivit y of the No vik o v-S h ubin inv ari- an ts is suﬃcien t to prov e Theorem 7.7. This is an ongoing joint pro ject with Azzali. Giv en F as ab ov e with dim F o d d and dim E − d im F = 4 k , one can re- mo ve r disjoint foliated regions D 4 k × D dim F and glue in r co pies of the disc bund le E γ | D 4 k of section 7.1. It would b e in teresting to kno w if the Heitsc h- Lazaro v torsion T ( T H E , g T F , g F ) then c hanges by ± 2 r 0 J ( W γ ) rk F as in (7.5) . In this case, w e w ould ha v e a new foliation F r on E that is homeomorphic, but not diﬀeo morph ic to the original foliation F , and th us “exotic” . Note that F and F r ha v e the same dyn amics, since on a complete transv ersal that do es not meet the mo diﬁed regions, nothing c h anges. More generally , one would lik e to classify these exot ic smo oth stru ctures, construct as man y as p ossible explicitly , and see whic h of them can b e distinguished b y Heitsc h-Lazaro v torsion, or a noncomm utativ e generalisation of it. 8. The hypoelliptic Lapla cian and Bismut-Lebeau t orsion In [8] and [15 ], Bism u t an d Leb eau consid er an analytic torsion form that is deﬁ ned using a hyp o elliptic op erator A 2 b, ± on d iﬀeren tial forms on the total space T ∗ M of the v ertical co tangen t b undle of the f amily p : E → B . While th e ﬁbrewise Ho d ge Laplacian generates a Bro wnian motio n on the ﬁbres M of p , the op er ator A 2 b, ± generates a stoc hastic v ersion of the geo desic ﬂo w on T ∗ M , where the v elo cities are pertur b ed by a Bro wnian motion for b ∈ (0 , ∞ ). As b → 0, this pro cess con v erges in an appropriate s ense to the classical Bro wnian motion on M . O n the other hand, as b → ∞ , one r eco ve rs the unp erturb ed geod esic ﬂ o w. One m otiv ation to study the family of op erators ( A b ) is F ried’s conjecture, w hic h relates the torsion of a sin gle manifold M to the closed orb its of a certain class of ﬂows on M , s ee [35] for an o v erview. 8.1. The hypo ellipt ic Laplacian on t he cotangen t bundle. Let p : E → B b e a smooth pr op er s ubmersion, and let F → E b e a ﬂat vec tor bundle as b efore. Let π : T ∗ M → E denote t he v ertical cotangen t bun dle. If one ﬁxes T H E ⊂ T E and g T M as b efore, one obtains a sp litting (8.1) T T ∗ M ∼ = π ∗ ( T H E ⊕ T M ⊕ T ∗ M ) and a corresp onding splitting of th e b undle Ω • ( T ∗ M /B ; π ∗ F ). W e regard the bund le ˜ E = E × (0 , ∞ ) 2 → ˜ B = B × (0 , ∞ ) 2 , and let ( b, t ) d enote the co ordinates of (0 , ∞ ) 2 . On the v ertical part T M ⊕ T ∗ M of T T ∗ M , one deﬁnes a metric g by (8.2) g =  1 t g T M id T ∗ M id T M 2 t g T ∗ M  : T M ⊕ T ∗ M − →  T M ⊕ T ∗ M  ∗ . 36 S EBASTIAN GOETTE T ogether with a metric g F on F → E and the symplectic volume form on T M ⊕ T ∗ M , one obtains an L 2 -metric g on the bu ndle Ω • 0 ( T ∗ M /B ; π ∗ F ) → B of compactly sup p orted forms. On th is bun dle, there exists a g -isometric inv o- lution u with (8.3) ( uα ) ( q , v ) =  id T M 2 t g T ∗ M 0 − id T ∗ M  α ( q , − v ) for all q ∈ E , v ∈ T ∗ q M , and th u s, one ca n deﬁne a nondegenerate He rmitian form h of signature ( ∞ , ∞ ) by (8.4) h ( α, β ) = g ( uα, β ) . As b efore, let A ′ = d E denote the to tal exterior deriv ativ e on Ω • ( T ∗ M ; π ∗ F ), regarded as a sup erconnection on the bundle Ω • 0 ( T ∗ M /B ; π ∗ F ). Deﬁne ¯ A ′ as the h -adjoint of A ′ . Again, A ′ and ¯ A ′ are ﬂat sup erconnections. One has the canonical one-form ϑ ∈ Ω 1 ( T ∗ M ), with (8.5) dϑ = ω H + ω V ∈ Γ  π ∗  Λ 2 ( T H E ) ∗ ⊕ T ∗ M ⊗ T M  ⊂ Ω 2 ( T ∗ M ) , where ω V is the standard symplectic form on the cotangen t b undle of eac h ﬁbre of p . Consider the Hamiltonians (8.6) H ± ( q , v ) = ± t 2 2 b 2 k v k 2 T ∗ M . Then for b = t = 1, the ω V -gradien t o f H + is th e generator (8.7) sgrad H + | ( q , v,b, t ) = g T ∗ M ( v ) ∈ T q M of the geod esic ﬂ o w on T ∗ M o v er the ﬁb res M of p . Regard the ﬂat sup erconnections (8.8) A ′ ± = e − ( H ± − ω H ) A ′ e H ± − ω H and ¯ A ′ ± = e H ± − ω H ¯ A ′ e − ( H ± − ω H ) . Then there exists an h -selfadjoin t sup erconnection A ± and an h -sk ew adj oin t endomorphism X with (8.9) A ± = 1 2 ( A ′ ± + ¯ A ′ ± ) and X ± = ¯ A ′ ± − A ′ ± . One ﬁn ally deﬁnes (8.10) A b,t, ± = A ± | B ×{ ( b,t ) } and X b,t, ± = X ± | B ×{ ( b,t ) } . The op erator A 2 b,t, ± = − X 2 b,t, ± is the su m of a harmonic oscillator along th e ﬁ bres of π : T ∗ M → E , the Lie deriv ativ e by sgrad H ± , and some terms of lo we r ord er or smaller gro w th at inﬁnit y . In particular, ∂ ∂ u − A 2 b,t, ± is hyp o elliptic in the sense of H¨ ormander, for an extra v ariable u ∈ R . By [15], the restriction A [0] , 2 b,t, ± of the op er ator A 2 b,t, ± to the ﬁ bres of ( p ◦ π ) : T ∗ M → B has discrete sp ectrum and compact resolv ent. Recal l that n = dim M . 8.1. Theorem (Bism ut and Leb eau [15]) . The op er ator A ′ [0] b,t, ± acts on the gen- er alise d 0-eigensp ac e ker( A [0] , 2 N b,t, ± ) o f the op er ator A [0] , 2 b,t, ± , wher e N ≫ 0 , and for TORSION INV ARIANTS FOR F AMILIES 37 al l k ∈ Z a nd al l b , t > 0 , (8.11) H k (k er ( A [0] , 2 N b, + ) , A ′ [0] b, + ) ∼ = H k + ( E /B ; F ) = H k ( E /B ; F ) , H k + n (k er ( A [0] , 2 N b, − ) , A ′ [0] b, − ) ∼ = H k + n − ( E /B ; F ) = H k ( E /B ; F ⊗ o ( T M )) . Note that H n + k − ( E /B ; F ∗ ) ∼ = H n − k + ( E /B ; F ) b y ﬁ brewise Po incar ´ e du alit y . 8.2. Bism ut - Leb eau torsion. W e can no w exp lain the higher torsion T b, ± of the co tangen t bund le deﬁned b y Bismut and Leb eau in [15]. W e will see ho w it ﬁts in to Igusa’s axiomatic framework of [45], see section 5.3 ab o v e. A t the momen t, Bismut-Le b eau to rsion is only deﬁned for small p ositiv e v alues of b . A deﬁnition for all b > 0 would b e nicer b ecause as the hyp o elliptic Laplacian con verges to the generator of the geodesic ﬂ o w as b → ∞ , one hop es to reco ver some inf ormation ab out the ﬁbr ewise geod esic ﬂow from the higher torsion. The Hermitian form h of (8.4) restricts to a nond egenerate Her- mitian form h H ± b on H • ± ( E /B ; F ), so one s till has c haracteristic forms ch o ( H • ± ( E /B ; F ) , h H ± b ) ∈ Ω odd ( B ). Bism ut and Leb eau also sh o w that the heat operator e − A 2 b,t, ± is a sm o oth- ing operator and of trace class. Analytic t orsion forms T b, ± ( T H E , g T M , g F ) ∈ Ω ev en B can thus b e d eﬁned as in section 2.2 . Th ey satisfy the foll o wing analog ue of Th eorem 1.5. 8.2. Theorem (Bism ut and Leb eau [15]) . F or b > 0 suﬃciently smal l, (8.12) d T b, ±  T H E , g T M , g F  = Z E /B e  T M , ∇ T M  c h o  F , g F  − ch o  H • ± ( E /B ; F ) , h H ± b  . Note that c h o ( H • − ( E /B ; F )) = ( − 1) n c h o ( H • + ( E /B ; F )) b y (8.11). This g iv es no contradict ion in (8.1 2) b ecause for o dd n , the ﬁrst term on the right hand side v anishes. It is no w n atural to compare T b, ± with the Bismut-Lo tt torsion T of sec- tion 2.2. Recall that w e h a ve deﬁned a m etric g H L 2 on H ( E /B ; F ) in sectio n 1.3. Let g H ± L 2 denote the indu ced metric on H ± ( E /B ; F ). 8.3. Theorem (Bismut and Leb eau [15]) . F or b > 0 suﬃciently smal l, the Hermitian form ( ± 1) n h H ± b is p ositive deﬁnite, and mo dulo exact forms on B one has (8.13) T b, ±  T H E , g T M , g F  = ( ± 1) n T  T H E , g T M , g F  − e c h o  H ± ( E /B ; F ) , g H L 2 , h H ± b  ± tr ∗ E /B 0 J ( T M ) rk F . 8.4. R emark. If one applies the exterior d eriv ativ e d on B to (8 .13), the r esult is compati ble with Theorems 1.5 and 8.2, wh ic h explains the ﬁ rst t wo terms on the r igh t hand s ide of (8.13). The la st term is a Mi ller-Morita-Mumford class in Igusa’s sense. It ca nnot b e guessed from Theorems 1.5 and 8.2. But if w e b eliev e that T b, ±  T H E , g T M , g F  38 S EBASTIAN GOETTE is a h igher torsion inv ariant in the sense of Deﬁnition 5.6, then it is not su r- prising that su c h a class app ears here. O n the other hand, it is surprisin g that T b, ±  T H E , g T M , g F  is give n b y the same linear co m bination of the classes in Theorem 5.7 as Igusa-Klein torsion in the follo w ing sp ecial case. I f F is acyclic and E → B admits a ﬁbr ewise Morse fun ction, then (8.14) T b, −  T H E , g T M , g F  = ( − 1) n τ ( E /B ; F ) b y comparison with Theorem 5.5. If h has trivial stable tangen t bun - dle T s M in an analogo us sense to Deﬁnition 4.1, the class T b, +  T H E , g T M , g F  equals ξ h ( E /B ; F ) ∗ τ up to sign. Conjecturally , th ese equations hold ev en if there is no ﬁbr ewise Morse fun ction. 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