L. HARTMANN AND M. SPREAFICO

Transcription

1 ON THE CHEEGER-MÜLLER THEOREM FOR AN EVEN DIMENSIONAL CONE L. HARTMANN AND M. SPREAFICO Abstract. We prove the equaity of the L -anaytic torsion and the intersection R torsion of the even dimensiona finite metric cone over an odd dimensiona compact manifod. 1. Introduction The cassica Cheeger-Müer theorem proves equaity between L -anaytic torsion and Reidemeister R torsion for cosed Riemannian manifods [4, 4, 8]. When the manifod has a boundary, a boundary term appears. This boundary term given by Lück in [1] when the metric is a product near the boundary) has been expicity given in the genera case in some recent works of J. Brüning and X. Ma [, 3]. For a compact connected oriented Riemannian manifod M, g M ) with boundary the Cheeger-Müer theorem reads a simiar formua is vaid for the reative case) og T abs M, g M ); ρ) og τ R M, g M ); ρ) rkρ)χ M) og + rkρ)a BM,abs M), where T abs M, g M ); ρ) and τ R M, g M ); ρ) are the anaytic torsion with absoute BC on the boundary, and the Reidemeister R torsion of M, g M ), with respect to an orthogona representation ρ of the fundamenta group of M and with the basis for homoogy fixed as in [8] see Sections. and 3.4 for detais), respectivey, χ is the Euer characteristic, and A BM is the anomay boundary term. In this work we prove the foowing extension of the Cheeger-Müer theorem, where M is the cone CW over a compact connected oriented Riemannian manifod W, g) see Section.4 for detais) and where the usua R torsion is repaced by the intersection torsion Iτ R defined by A. Dar in [7] where the homoogy basis is fixed using the L -harmonic forms via the suitabe De Rham map, see Section 3.4 for detais). The proof foows at once from the formua for the anaytic torsion given in Theorem.1 in Section.4, and the duaity formua for the intersection torsion proved in Proposition 4.1 in the ast section. Since the metric on CW is fixed by the metric of W, and the unique representation of the fundamenta group is the trivia one, that may be assumed of rank one, both these quantities wi be omitted in the notation. Theorem 1.1. Let W, g) be a compact connected oriented Riemannian odd dimensiona manifod without boundary. Let CW denote the cone over W. Then, og T abs CW ) og Iτ R CW ) + A BM,abs W ), og T re CW ) og Iτ R CW, CW ) + A BM,re W ). We concude this introduction with some remarks. 010 Mathematics Subject Cassification: 58J5, 57Q10. 1

2 L. HARTMANN AND M. SPREAFICO If the dimension of W is even, resuts for the anaytic torsion see [18, 5]) exist and some extra terms appear, whose interpretation is sti not cear. For this reason we omit not iuminating formuas and we concentrate here on the odd case. There is work in progress in this direction. The resut of this work wi give the extension of the Cheeger-Müer theorem for a genera space with conica singuarities as define in [5]) once it wi be avaiabe a suitabe guing formua extending the one proved by S.M. Vishik in [35] for compact manifods under the assumption that the metric is product near the guing. This is a ikey resut, for recenty on one side a guing formua for compact manifods without any assumption on the metric near the boundary was given by J. Brüning and X. Ma in [3], and on the other side M. Lesch [0] extended the resut of S.M. Vishik to a pseudomanifod. In particuar, since in genera pseudomanifod are modeed on cones, a generaization of the resut of this work for a cone over a pseudomanifod coud be used to obtain a generaization of the Cheeger-Müer theorem for pseudomanifods. The resut of Theorem 1.1 coud be read in terms of Euer basis [9] and in term of the Reidemeister metric [1, 7] ), and woud affirm that the Euer basis of C W coincides with the quotient of the De Rham basis by the L -anaytic torsion in term of metrics, the intersection Reidemeister metric coincides with the Ray and Singeer metric, namey the quotient of the De Rham metric by the L -anaytic torsion). However, whie the anaytic side is cear, the compete deveopment of a precise theory for the intersection Reidemeister basis metric) is sti under construction see Section 5 of [17]). There is work in progress on this topic.. Background and anaytic torsion This section is essentiay based on [15] and [16], and we refer to those papers for further detais see aso [34] for the anaytic torsion)..1. Geometric setting. Let M, g M ) be a compact connected oriented Riemannian manifod of dimension m with boundary M and Riemannian structure g M. Let ρ : π 1 M) Ok, R) be a representation of the fundamenta group of M, and et E ρ be the associated vector bunde over M with fibre R k and group Ok, R), E ρ R k ρ M. Let ΩM, Eρ ) denote the graded inear space of smooth forms on M with vaues in E ρ. The exterior differentia on M defines the exterior differentia on Ω q M, E ρ ), d : Ω q M, E ρ ) Ω q+1 M, E ρ ). The metric g defines an Hodge operator on M and hence on Ω q M, E ρ ), : Ω q M, E ρ ) Ω m q M, E ρ ), and, using the inner product, in R k, an inner product on Ω q M, E ρ ) is defined by ω, η) ω η. M Near the boundary there is a natura spitting of ΛM as direct sum of vector bundes ΛT M N M ΛT M, where N M is the dua to the norma bunde to the boundary, and the smooth forms on M near the boundary decompose as ω ω tan + ω norm, where ω norm is the orthogona projection on the subspace generated by dx, the one form corresponding to the outward pointing unit norma vector to the boundary, and ω tan is in C M) Λ M). We write ω ω 1 + dx ω, where ω j C M, ΛT M)), and ω dx ω.

3 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 3 Define absoute and reative boundary conditions by B abs ω) ω norm M ω M 0, B re ω) ω tan M ω 1 M 0. Let Bω) Bω) Bd + d )ω)). The adjoint d and the Lapacian d + d ) operators are defined on the space of sections with vaues in E ρ, the Lapacian with boundary conditions Bω) 0 is sef adjoint, and the spaces of the harmonic forms with boundary conditions are H q abs M, E ρ) {ω Ω q M, E ρ ) q) ω 0, B abs ω) 0}, H q re M, E ρ) {ω Ω q M, E ρ ) q) ω 0, B re ω) 0}... De Rham maps. Let K be a ceuar or simpicia decomposition of M and L of M. Let C q M; E ρ ) R k ρ C q M; Zπ 1 M)) be compex of the twisted chains see the paragraph before equation 3.3) for detais on this construction). Then we have the foowing de Rham maps A q that induce isomorphisms in cohomoogy), with A q abs :Hq abs M, E ρ) C q M; E ρ ), A q re :Hq re M, E ρ) C q M, M); E ρ ), A q abs ω)v ρ c) A q re ω)v ρ c) ω, v), where v ρ c beongs to C q M; E ρ ), and c is identified with the q-subcompex simpicia or ceuar) that c represents. Foowing Ray and Singer [8], we introduce the de Rham maps A q : A re q :H q re M, E ρ) C q M, M); E ρ ), A re q A abs q :H q abs M, E ρ) C q M; E ρ ), A abs q both defined by.1) A re q ω) A abs q ω) 1) m 1)q j,i c :ω 1) m 1)q Pq 1 A m q abs ω), :ω 1) m 1)q Pq 1 A m q re ω), ĉ q,j ω, e i ) ) c q,j ρ e i, where the sum runs over a q-simpices c q,j of M M in the first case, but runs over a q-simpices c q,j of M in the second case. Here P q : C q K, L; Z) C m q ˆK ˆL; Z) is the Poincaré map, and ĉ denotes the dua bock ce of c. The extension of the de Rham map for pseudomanifods is based on the works on Cheeger [5], see the end of Section 3.4 beow..3. Zeta function and anaytic torsion. The Lapace operator on forms q), with boundary conditions B abs/re, has a pure point spectrum Sp q) abs/re consisting of rea non negative eigenvaues. The sequence Sp + q) abs/re is a totay reguar sequence of spectra type accordingy to [15] Section 4, and the forms vaued zeta function is the associated zeta function, defined by ζs, q) abs/re ) ζs, Sp + q) abs/re ) λ s, λ Sp + q) abs/re when Res) > m, and by anaytic continuation esewhere. The anaytic torsion T abs/rew, g); ρ) of W, g) with respect to the representation ρ is defined by og T abs/re W, g); ρ) 1 m 1) q qζ 0, q) abs/re ). q1

4 4 L. HARTMANN AND M. SPREAFICO.4. The anaytic torsion of a cone. Let W, g) be an orientabe compact connected Riemannian manifod of dimension m without boundary and with Riemannian structure g. We denote by C W the space [0, ] W )/{0} W ) 0, ] W {p}, where p is the vertex of the cone, i.e. the image of {0} W under the quotient map, with the metric g C dx dx + x g, on 0, ] W, and we ca it the finite metric cone over W see [16] 3.1 for detais). The anaytic torsion of a cone over a sphere i.e. W S m ) was studied in [15]. The resut is based on one side on works of J. Cheeger on the Hodge theory of L forms [5, 6, 6], and on the other on works of M. Spreafico on zeta invariants for doube sequences [9, 30, 31, 3]. In the genera case, extending the approach used for the spheres in [15], we have the foowing resut: Theorem.1. The anaytic torsion of the cone C W on an orientabe compact connected Riemannian manifod W, g) of odd dimension m p 1 is og T abs C W, g C ) 1 1) q rkh q W ; Q) og p q) + 1 og T W, g) + S C W ), where the singuar term S C W ) ony depends on the boundary of the cone: S C W ) 1 j 1 Res 0 Φ k+1,q s) k ) q 1) h Res 1 ζ s, s0 j k h)) q p + 1) j k), j0 k0 h0 sj+ 1 the functions Φ k+1,q s) are some universa functions, expicity known by some recursive reations, and is the Lapace operator on forms on the section of the cone) and coincides with the anomay boundary term of Brüning and Ma, namey S C W ) A BM,abs C W ). The proof of Theorem.1 is based on anaytic toos and is essentiay the same as the proof of simiar resuts for the spheres given in [15]. In the genera case treated here, using the same method and a simiar strategy, we just need to sove severa technica probems, that can be quite hard, and require ong difficut anaysis. A detais are in [16]. See aso [34]. 3. Intersection torsion Intersection torsion for pseudomanifods was introduced in works of A. Dar [7, 8]. In these works the case of pseudomanifods without boundary is considered, and in genera a intersection homoogy theory is deveoped for the boundaryess case. Here we need to consider the boundary case, but a particuar situation where the boundary is in fact a smooth manifod, disjoint from the singuar ocus. In this particuar case it is easy to rework a definitions and the main resuts of the boundaryess case, as expect. This is the purpose of this section Pseudomanifods with smooth boundary. We define pseudomanifods with smooth boundary adapting the definition of pseudomanifods of [10] 1.1, [13] 1, [19] 4.1. If X is a topoogica space, we denote by CX the cone over X. By definition the cone and the open cone over the empty set are a point. A topoogica pseudomanifod of dimension 0 is a countabe set with the discrete topoogy. A topoogica pseudomanifod of dimension n with smooth boundary is an Hausdorff paracompact topoogica space X with a fitration by cosed subspaces X 1 X 0 X 1 X n 3 X n X n 1 Σ X n X,

5 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 5 caed stratification, such that: 1) X Σ is dense in X; ) there exists a cosed subspace B of X, with B Σ, such that M X Σ is an n-manifod with boundary M B, and for each j n, for each point x X j X j 1 there exists a compact topoogica pseudomanifod L of dimension n j 1 with fitration L 1 L 0 L 1 L n j 3 L n j L n j 1 L, and a neighborhood U x of x in X with an homeomorphism ϕ : U x R j CL, which respects the stratifications, namey ϕ maps homeomorphicay U x X j+k+1 onto R j CL k. The cosed subspace Σ X n is caed the singuar ocus of X. It is a consequence of the definition that each subspace X j X j 1 is a manifod of dimension j with boundary, and, by condition ), the boundary of X is disjoint from the singuar ocus. When the singuar ocus has dimension 0, then a stratification of X is X 1 Σ X 0 X 1 X n 1 X, and X is caed a space with isoated singuarities. In this work we are mainy concerned with this type of pseudomanifods. If X is a manifod with boundary, then X is a pseudomanifod with a stratification consisting with ony one stratum X. For our purpose it is sufficient to work in the piecewise inear category, as in [10]. A piecewise inear p) space X is a topoogica space with a cass of ocay finite simpicia trianguations T X): if T T then any inear) subdivision of T beongs to T X), and it T 1, T T X), then they have a common subdivision in T X). A cosed p-subspace of X is a subspace which is a subcompex of a suitabe admissibe trianguation of X. We wi identify a trianguation of a space with the associated simpicia compex. A ppseudomanifod X of dimension n with smooth boundary is a p-space X of dimension n containing two cosed disjoint p-subspaces X and Σ, with Σ of codimension greater or equa to, such that X Σ is an oriented p-manifod of dimension n dense in X and with smooth boundary X. Equivaenty, for an admissibe) trianguation of X, then X is the union of the cosed n- simpices and each n 1)-simpex is face of one or two n-simpices, and X is the subcompex of the n 1)-simpices that are faces of just one n-simpex. By the same proof as in [13] Prop. 1.4, any p-pseudomanifod with smooth boundary admits a p-stratification: a stratification of X is given by setting X k T k), where T is an admissibe) trianguation of X, and this stratification is subordinate to the trianguation, meaning that the strata are subcompexes. From now on we assume pseudomaniofods are finite p-pseudomanifods, that p-pseudomanifods have a fixed stratification the previous one if a trianguation is given), and that a trianguations are admissibe, i.e. compatibe with the p-structure. Our definition of pseudomanifod with smooth boundary is consistent with the definition of pseudomanifod with boundary of [10] 5., taking a manifod for boundary, namey assuming the singuar ocus of the boundary vanishes. 3.. Intersection homoogy and reative intersection homoogy for pseudomanifods with smooth boundary. Let first reca the basic ingredients for the definition of intersection homoogy, as in [10]. A perversity is a finite sequence of integers p {p j } n j such that p 0 and p j+1 p j or p j + 1. The perversity: m {m j [j/] 1} is caed ower midde perversity. The nu perversity is 0 j 0, and the top perversity is t j j. Given a perversity p, the compementary perversity p c is p c j t j p j j p j. Now et X be a pseudomanifod with boundary and with a given stratification. If j is an integer and p a perversity, a p-subspace A of X is said p, j)-aowabe if dima) j, dima X n k ) j k + p k, k.

6 6 L. HARTMANN AND M. SPREAFICO In standard references intersection homoogy is usuay defined for pseudomanifods without boundary, and reative intersection homoogy for pairs X, A) where A is an open subspace of a cosed pseudomanifod. In order to extend the definition to pseudomanifods with smooth boundary we have, at east, two possibe equivaent approaches. The first approach is based on [], and use smoothy encosed subspaces, as foows. Gue the infinite cyinder X [0, ) to X through the boundary, et Z X X X [0, ). Embed Z into the suitabe R kk1+k, in such a way that i X) ix) R k1 {0,..., 0}, iσ) {x R k x j > 0, j > k 1 } R k1 R k + R k1 {0,..., 0}, and i X [0, )) {x R k x j 0, j > k 1 } R k1 R k, where i denotes the embedding. Then a Whitney stratification of R k is given by setting Z 0 R k iz), Z 1 ix Σ), Z k ix k X k 1 ) for k see [] Section 7.1.), and ix) is the cosure of Z 1. The subsets S ± R k1 R k ± are smoothy encosed in R n, as in the definition of Section 1.3. of []. Now identifying the different spaces with their images under i) it is cear that S + Z X. By definition [] Section 1..3, X is smoothy encosed in Z, and X S Z) S + Z) S Z) X is the intersection with another smoothy encosed subset of Z. It foows from [] Section 1..3 that both the intersection chain compex I p CX) of X, and the reative intersection chain compex I p CX, X) of the pair X, X) are defined, the first as in the boundary ess case, the ast one by setting I p C q X, X) I p C q X)/I p C q X). The intersection homoogy groups are the homoogy groups of I p CX), and the reative intersection homoogy group I p H q X, X) of the pair is defined as the q-homoogy group of the chain compex I p CX, X). Moreover, there is the foowing homoogy ong exact sequence associated to the pair X, X) see aso [11] 1.11):... I p H q X) I p H q X) I p H q X, X) I p H q 1 X).... The second approach proceeds as in [1] 1.4, and consists in repacing the pseudomanifod X with boundary X by the pseudomanifod X X. For et X be a pesudomanifod with smooth boundary X. Let Co X) be an open coar neighborhood of x. Then, X X is a pseudomanifod, with open subspace Co X) X, and usua intersection homoogy theory and reative intersection homoogy theory are defined for X X, and X X, Co X) X), [1] 1.3 [19] 4.6. Since the boundary is disjoint from the singuar stratum, there exists a stratum preserving homotopy sef equivaence X X X, and the same for the pair X, X) X X, Co X) X), as defined in [19] 4.8. It foows by [19] 4.8.5, that I p H q X) I p H q X X), and I p H q X, X) I p H q X X, Co X) X). We reca now the definition of the intersection homoogy chain compex and groups, as in [10]. Let T be an admissibe) trianguation of X such X is trianguated by a subcompex L T of T. Let C T X) CT ) denote the chain compex of simpicia chains of X with respect to T. Let CX) denote the direct imit chain compexes under refinement of the C T X) over a trianguations of X compatibe with the p-structure. Since X is p-subspace of X, C T X) CL) is defined, and is a sub compex of C T X), and the reative chain compex is aso defined C T X, X) CT, L) CT )/CL). The construction commutes with direct imit, and hence the CX) and CX, X) are defined. The intersection chain group of perversity p, is the subgroup I p C q X) of C q X) consisting of those chains c such that c is p, q)-aowabe and c is p, q 1)-aowabe. The reative intersection chain group of perversity p, is I p C q X, X) I p C q X)/I p C q X), where I p C q X) C q X) for each p, since X is actuay a manifod. The group I p C q X)/I p C q X) is the subgroup of C q X, X) consisting of those chains c in T that are not in L, such that c is p, q)-aowabe and c is p, q 1)-aowabe. Intersection cohomoogy is defined as the agebraic dua of intersection homoogy see for exampe [19] 4..8). Poincaré duaity is recovered for pseudomanifods using intersection homoogy: with

7 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 7 coefficients in a fied, when X, there is an isomorphism [10] ) IP q : I p H q X) I pc H m q X). For a pseudomanifod with smooth) boundary, the duaity reads [33] 3.) IP q : I p H q X) I pc H m q X, X) Basic sets. In order to define intersection torsion and reative intersection torsion, we introduce some chain compexes of free modues. Let X be a pseudomanifod of dimension n with smooth boundary, and fixed stratification. First, we define the basic R sets as in [10] 3.4. Let T be a trianguation of X compatibe with the fitration. Let Rq p be the subcompex of the first barycentric subdivision T of T consisting of a simpices which are p, q)-aowabe. Then, Rq p is a subcompex of the q-skeeton of T. It is cear that Rq p is a subcompex of R p q+1. Define the compex C p X) by setting Cq p X) H q Rq, p R p q 1 ), and boundary defined by the homoogy ong exact sequence of the pair Rq, p R p q 1 ). This is a free abeian group generated by finitey many chains with contractibe support. So Cq p X) is in one one correspondence with the group of simpicia q-chains c q with c q Rq, p and c q R p q 1. The homoogy of C p X) is canonicay isomorphic to ImH q Rq) p H q R p q+1 )). By [10] 3.4, there is an isomorphism Ψ : ImH q Rq) p H q R p q+1 )) I p H q X). For this is stated, without proof, in [10] 3.4, however, if in the present case we remove the boundary, we obtain the isomorphism for the pseudomanifod X X, and it is cear that the groups at the two sides of the isomorphism are the same for X and X X, since the singuar ocus is disjoint from the boundary. Aso note that the isomorphism is natura, that is to say is induced by the incusion of Rq p into T. This is cear from the construction of the simiar isomorphism caed Ψ for the basic sets Q in [10] 3.. Let Pq p R p q+1 L. Then, P q p is an R set Rq p of X, and dimrq) p q 1. Actuay, Pq p L q) is the q-skeeton of L. For X is a manifod and hence a the simpices of any trianguation of X are aowabe for any perversity. Define the chain compex C p X) as above. Then, the homoogy of C p X) is canonicay isomorphic to ImH q Pq p ) H q P p q+1 )), and there is a natura isomorphism ImH q Pq p ) H q P p q+1 )) I p H q X) H q X), that is the restriction of Ψ. Next, we dea with the reative chain compex. We define the compex C p X, X) by setting C p q X, X) H q R p q L, R p q 1 L ), and boundary defined by the homoogy ong exact sequence of the pair Rq p L, R p q 1 L ). This is a free abeian group generated by finitey many chains with contractibe support, and is in one one correspondence with the group of the simpicia q-chains c q with the interior of c q Rq p Rq p L ), and the interior of c q R p q 1 Rp q 1 L ). It is possibe to show that the homoogy of C m X, X) is canonicay isomorphic to Imi q,q : H q Rq p L ) H q R p q+1 L )), and that Imi q,q : H q Rq p L ) H q R p q+1 L )) is isomorphic to the reative intersection homoogy of the pair X, X) R torsion. In order to define intersection torsion we briefy reca the definition of the torsion of a chain compex. We foows the cassica definition of Minor [3], but with a itte change of notation. Let R be a ring with the invariant dimension property, and M a finitey generated free eft) R-modue. Let U be a subgroup of the group R of units of R, and et K U R) K 1 R)/U denotes the quotient of the Whitehead group of R by the subgroup generated by the casses of the eements of U. Let x {x 1,..., x n } and y {y 1,..., y n } be two bases for M. We denote by y/x)

8 8 L. HARTMANN AND M. SPREAFICO the non singuar n-square matrix over R defined by the change of bases y j k y/x) jkx k ), and we denote by [y/x] the cass of y/x) in the Whitehead group K U R). Let C : C m m C m 1 m 1... C 1 1 C 0, be a bounded chain compex of finite ength m of finite dimensiona) free eft R-modues. Denote by Z q ker q : C q C q 1 ), B q Im q+1 : C q+1 C q ), and H q C) Z q /B q the homoogy groups of C. Assume that a the chain modues C q have preferred bases c q {c q,1,..., c q,mq }, and the homoogy modues H q C) are free with preferred bases h q. Aso assuming that the boundary modues B q are free with preferred bases or using staby free bases, we fix a set of eements b q {b q,1,..., b q,nq } of C q such that q b q ) is a basis for B q 1 for each q in other words we are choosing a ift of a basis of B q 1 ). Then the set { q+1 b q+1 ), h q, b q } is a basis for C q for each q. The Whitehead torsion of C with respect to the basis h {h q } is the cass τ W C; h) m 1) q [ q+1 b q+1 ), h q, b q /c q )], in the Whitehead group K U R). The definition is we posed since it is possibe to show that the torsion does not depend on the bases b q. If K is a connected finite ce compexes of dimension m, with universa covering K, identify the fundamenta group π π 1 K) with the group of the ceuar) covering transformations of K, the action π makes each chain modue C q K; Z) into a free modue over the group ring Zπ, finitey generated by the natura choice of the q-ces of K. Denote the resuting compex of free finitey generated modues over π with preferred basis obtained by the ifts of the ces) by C K; Zπ). If the homoogy modues H q K; Zπ) are free with preferred basis h q, the Whitehead torsion of K with respect to the graded basis h is the cass τ W K; h) wτ W C K; Zπ); h)), of K π Zπ). If ρ : π Aut R M) is a representation of the fundamenta group in the group of the automorphisms of some free right modue M over some ring with unit R, we form the twisted compex CK; M ρ ) of free finitey generated R-modues, by setting 3.3) C q K; M ρ ) M ρ C q K; Zπ). Fixing a basis m for M, bases for these modues and for cyces and boundary submodues) are given by tensoring with m. Assuming that the homoogy modues H q CK; M ρ )) are free with preferred graded bases h {h q }, then, we define the R torsion τ R K; ρ, h) of K with respect to the representation ρ and the graded basis h to be the cass of τ W CK; M ρ ); h) in K 1 ZAut R M))/ρπ). We have m τ R K; ρ, h) 1) q [ρ q+1 b q+1 ), h q, b q /c q )]. By the same procedure we define the reative R torsion of the pair K, L), and we write τ R K, L); ρ, h) τ W CK, L); M ρ ); h). In particuar, if W, g) is a compact connected oriented Riemannian manifod, and ρ is an orthogona representation, we use the de Rham maps of section. in order to fix the basis for the homoogy, foowing Ray and Singer [8]. We define the absoute R torsion of W, g) by τ R W, g); ρ) τ R W ; ρ, A abs a)),

9 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 9 where a is an orthonorma graded basis for the harmonic forms. It is possibe to prove that the definition does not depend on the basis a. Reative R torsion for a manifod with boundary is defined accordingy. Let X be an m-pseudomanifod with smooth boundary, et T be a trianguation of X such that the boundary X of X is a subcompex T of T, and et T be the universa covering compex of T, and T the ift of T. Let R q p be the ifts of the basic sets Rq p to T, and identify the fundamenta group π π 1 X) with the group of the covering transformations of T. Note that covering transformations are simpicia, so if we set Cq p X) H q R q, p R p q 1 ), the action of the group of covering transformations group makes each chain group Cq p X) into a free modue over the group ring Zπ, and each of these modues is finitey generated by fixing ifts of the natura choice of the q-chains that generate Cq p X). We obtain a compex of free finitey generated modues over Zπ that we denote by C p X; Zπ), with preferred basis. The same procedure appies for the reative chain compex Cq p X, X) Hq R q p T, R p q 1 T ), and gives the Zπ-compex C p X, X); Zπ), with preferred basis obtained by ifting the chains whose supports do not intersect the boundary. Assuming that the homoogy modues H q C p X; Zπ)), H q C p X, X); Zπ)) are Zπ-free with preferred graded bases h {h q }, we define the intersection Whitehead torsion of X and the reative intersection Whitehead torsion of the pair X, X) with respect to the graded basis h to be the casses Iτ p W X; h) τ WC p X; Zπ); h), Iτ p W X, X); h) τ WC p X, X); Zπ); h), in the Whitehead group W hπ 1 X)) K π Zπ), respectivey. Proceeding as in the smooth case, given a representation ρ of π 1 X) we define intersection R torsion I p τ R X; ρ, h) τ W C p X; M ρ ); h) of X with respect to the representation ρ and to the graded basis h, and the reative intersection R torsion of the pair X, X), I p τ R X, X); ρ, h) τ W C p X, X); M ρ ); h). If in particuar a Riemannian structure is defined on the non singuar part of X, L forms can be used to extend the construction of Ray and Singer and to define suitabe de Rham maps from L harmonic forms to intersection homoogy, and to fix the basis h [7, Section, pg. 197] and ocay cited [6]). Note in particuar, that the basis h fixed in this way is sef dua, i.e. IP q h q ) is the agebraic dua of h n q. Foowing A. Dar [7, pg. 197], we use the notation I p τ R X, g); ρ) and I p τ R X, X, g); ρ) to denote the torsion I p τ R X, X); ρ, h) when the basis h is fixed as the image of an orthonorma basis of L harmonic forms via de De Rham map, and we define the intersection R torsion of X, and the reative intersection R torsion of X, X) with respect to the representation ρ by Iτ R X, g); ρ) 1 ) I m τ R X, g); ρ) + I mc τ R X, g); ρ), Iτ R X, X, g); ρ) 1 ) I m τ R X, X, g); ρ) + I mc τ R X, X, g); ρ). In a definitions, if X is an oriented manifod stratified with ony one stratum X then we obtain the cassica Whitehead torsion and the cassica R torsion. 4. Duaity theorems for intersection R torsion of pseudomanifods with smooth boundary We give in this section some duaity theorems for the intersection torsion of a pseudomanifod with smooth boundary that extend the duaity theorems of A. Dar [7] for the boundary ess case. First, we need a emma for the L harmonics form on the suspension and on the cone. Due to the independent interest of these resuts, we give them aso for the case of an even dimensiona section.

10 10 L. HARTMANN AND M. SPREAFICO We present an expicit proof for the suspension. The proof for the cone is simiar, and in the odd dimensiona case was given in Lemma 3.5 of [16]. Next, we give some formuas for the torsion, that we use to prove the fina duaity resuts. Let Σ W 0, ) W {p 0, p } be the suspension of W, reaized as the guing of two copies of C W aong the boundaries. Lemma 4.1. If dim W p 1 is odd, then α q q p)) H q W ), 0 q p 1, H q Σ W ) {0}, q p, { x α q 1 dx ϕ q 1), ϕ q 1) H q 1 W ) }, p + 1 q p. If dim W p is even, then H q W ), 0 q p, H q Σ W ) {0}, q p + 1, H q 1 W ), p + q p + 1. Proof. The soutions of the harmonic equation u 0 on Σ W are { f 1 x), 0 x, ux) f x), x <, where f 1 and f are forms of the foowing four types: ψ q) ±,1,n xa±,q,n ϕ q) cc,n, ψ q) q 1) xa±,q 1,n ±,,n dϕ cc,n + a ±,q 1,nx a±,q 1,n 1 dx ϕ q 1) cc,n, ψ q) ±,3,n xa±,q 1,n+ q 1) dϕ cc,n + a,q 1,nx a±,q 1,n+1 dx ϕ q 1) cc,n, ψ q) ±,4,n xa±,q,n+1 dx dϕ q ) cc,n. and a ±,q,n α q ± µ q,n. The harmonics of are obtained requiring that u, du and d are square integrabe, and satisfy the foowing conditions f 1 x) x f x) x, df 1 )x) x df )x) x, d f 1 )x) x d f )x) x, pus the idea boundary condition of Cheeger if dim W p is even [5, 6], described beow. Note that the first condition is aways satisfied, whie the other two conditions coincide respectivey with the conditions df 1 ) norm x) x 0 and d f 1 ) tg x) x 0. An expicit verification of these conditions gives the resut. Let us consider the case of dim W p 1 odd in some detais. The soution of type I ψ q) +,1,n satisfies the square integrabiity condition for a q, whie ψ q),1,n satisfies this condition ony if q p 1. The condition df 1 ) norm x) x 0 is satisfied if and ony if a ±,q,n 0, and this is true for a +,q,n when λ q,n 0 and 0 q p 1, and for a,q,n when λ q,n 0 and p q p 1. Since λ q,n 0, ϕ q) cc,n is an harmonic on W, and thus d f 1 0. The soution of type II ψ q) +,,n satisfies the square integrabiity condition for a q, whie ψq),,n satisfies this condition ony if q p. The condition d f ) norm x) x 0 is satisfied if and ony if

11 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 11 either a ±,q,n 0, or ϕ q 1) cc,n is an harmonic h on W. The first possibiity woud require λ q 1,n 0, and therefore woud give ψ q) ±,,n 0. The second possibiity gives ψq) ±,,n a ±,q 1,nx a±,q 1,n 1 dx h. Appying d f 1 ) tg x) x 0, we obtain p q p 1 for ψ q) +,,n, and 0 q p for ψq),,n, where however, the case q p corresponds to the nu form. Proceeding in a simiar way, we find that there are no new forms of the types III and IV satisfying the conditions above. If dim W p is even, the same anaysis gives the answer in a dimension q p + 1. In dimension q p + 1, we obtain the form ϕ dx h, where h is an harmonic on W, that satisfies a the conditions above and is square integrabe with its exterior derivative, thus we need the idea boundary conditions, that we briefy reca here. Let H p W ; Q) V a V r be a maxima sef annihiating for the cup product) decomposition. Let {e j } h j1 be an orthonorma basis for the corresponding spaces of harmonic p-forms H p W ) coherent with the decomposition. Then, for any h H p W ), p p h h a,j x)e j + h r,j x)e j. We say that a form j1 jp+1 ω α + dx β in H p Σ W ) satisfies the idea boundary condition is in the domain of d) for the decomposition V a V r if for the decomposition in V a V r of the projection of α onto H p W ) the foowing condition hods: f a,j0) f r,j 0) 0. Simiary, a p + 1-form ω is in the domain of d for the decomposition V a V r if ω is in the domain of d for the decomposition V r V a. Consider the p + 1-form ϕ dx h. Its dua is ϕ h H p Σ W ), and ceary does not satisfy the idea BC. Lemma 4.. If dim W p 1 is odd, then α q q p)) { H q abs C H q W ), 0 q p 1, W ) {0}, p q p 1. { H q re C {0}, 0 q p, W ) { x α q 1 dx ϕ q 1), ϕ q 1) H q 1 W ) }, p + 1 q p. If dim W p is even, then { H q abs C H q W ), 0 q p, W ) {0}, p + 1 q p + 1; { H q re C {0}, 0 q p, W ) { x α q 1 1 dx ϕ q 1), ϕ q 1) H q 1 W ) }, p + 1 q p + 1. For the suspension, we have a short exact sequence of chain compexes: 4.1) 0 C p C W ) C p C W ) C p C W ) C p Σ W ) 0.

12 1 L. HARTMANN AND M. SPREAFICO A formua for the torsion of an exact sequence of compexes is given by Minor in [3] Section 3. In the present case, we can fix the chain basis of the midde compex consistenty, using the basis determined by the simpices, and hence we have the foowing formua og I p τ R C W, g C ); ρ 0 ) og τ R W, g); ρ 0 ) + og I p τ R Σ W, g Σ ); ρ 0 ) + og τs p m), where the compex S p m is defined by the exact ong homoogy sequence associated to the exact sequence in equation 4.1), that is the Mayer Vietoris sequence S p m :... I p H q C W ) I p H q C W ) I p H q C W ) I p H q Σ W )..., i.e., S p m,3q Ip H q W ), S p m,3q+1 Ip H q C W ) I p H q C W ) and S p m,3q+ Ip H q Σ W ). Lemma 4.3. Let W be a compact connected oriented manifod of odd dimension m p 1 without boundary. Let ρ 0 : π 1 C W ) O1, R) be the rank one trivia orthogona representation of the fundamenta group. Then, og I m τ R C W, g C ); ρ 0 ) og τ R C W, g); ρ 0 ) + og I m τ R Σ W, g Σ ); ρ 0 ) + og τs m m), where r q rkh q W ), and og τs) m og τs) mc 1) q og ) rq. p q Proof. First, we reca the intersection homoogy for the cone and the suspension with midde perversity. Since both spaces have isoated singuarities, the unique vaue which imports of the perversity is the vaue m p m c p p 1. We have see for exampe [10] 6, or [19] 4.7., 4.7.3) { I m Hq C H q C W ) I mc H q C W ) W ), q < p, 0, q p, and H q C W ), q < p, I m H q Σ W ) I mc H q Σ W ) Im H q W ) H q Σ W )) 0, q p, H q Σ W ), q > p, Note that, beside the homoogy, aso the basic R set with the two midde compementary perversities coincide, by the very definition. This impies that the chain compex used in the definition of the intersection torsion for these two perversities coincide and therefore the torsions coincide. We proceed by taking p m, and this wi aso cover the compementary case. Next, in order to compute the torsion of the compex S m m, we need the chain bases. These are the bases for the homoogy determined by the geometry using the de Rham maps as described in Sections. and 3.4. We study the two cases q < p and q p separatey. When q < p, consider the foowing part of the compex S m... I m H q+1 Σ W ) q+1 I m H q C W ) I m H q C W ) I m H q C W ) I m H q Σ W ) q.... The geometry impies that the homomorphisms q+1 q are nu, and using the previous resuts, a the vector spaces are isomorphic to V H q W ), and the sequence spits as 4.) 0 H q C W ) V I m H q C W ) V I m H q C W ) V I m H q Σ W ) V 0.

13 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 13 In order to fix the homoogy bases, et a q be an orthonorma base for H q W, g). Then the norm of a q,j with the metric g is a q,j g a q,j ga q,j q a q,j g a q,j q a q,j g q. W W So a orthonorma base for H q W, g) is q a q, and appying the de Rham maps we obtain A q, g q a q,j ) q A q, ga q,j ) q P 1 q A q g a q,j ) q q P 1 A q g a q,j ) q A q,g a q,j ). Then the basis for H q C W ) is q A q,g a q ). Next, consider the cone C W, g C ). By Lemma 4., the constant extension of the forms in a q gives a basis for H q abs C W ). The norm of this basis eements is a q,j g C a q,j gc a q,j x q dx a q,j g a q,j p q W p q a q,j g p q p q. C W So an orthonorma base for H q C W ) is ) A abs p q 1 ) p q q,g C a q,j p q p q p q 0 p q p q p q This give the basis for I m H q C W, g C ): q p q p q H q Σ W ) we obtain the basis of I m H q Σ W ): ) 1 a q, and, using duaity 3.), ) 1 A abs p q q,g C a q,j ) p q ) 1 p q p q ) 1 A q,g a q,j ). ) p q 1 p q p q p q ) 1 ) P 1 q A q g a q,j ) A q,g a q ). ) 1 A q,g a q ). IPq 1 A p q re gc a q,j ) Repeating the same process for We can now compute the determinants of the change of basis in the vector spaces of the sequence in equation 4.). At ) rq I m H q C W ) the determinant is 1, at I m H q C W ) H q C W ) is and at I m H q Σ W ) p q is rq. We consider now the case q p. The reevant part of the sequence S m reads 0 I m H q+1 Σ W ) q+1 H q C W ) 0 Since the de Rham maps are sef dua, as observed at the end of Section 3.4, and the intersection homoogy with perversities m and m c coincide, we can use duaity 3.1) to obtain the basis for I mc H q+1 Σ W ) starting with the basis for the same space obtained when q < p. This gives the ) basis q p+ 1 q+1 p A q,g a q ) for I mc H q+1 Σ W ). Using the basis fixed above for the other space, ) rq the determinant of the change of basis at H q C W ) is 1 and at I m H q+1 Σ W ) is. Now q+1 p appying the definition of Reidemeister torsion to the compex S m, we obtain where D denotes

14 14 L. HARTMANN AND M. SPREAFICO the determinant of the matrix of the change of basis) og τs m ) 6p p 1) q og DS m,q) 1) q og DI m H q Σ W )) + 1) q+1 og DI m H q C W ) I m H q C W )) 1) q og DI m H q Σ W )) + + p qp+1 1) q+1 og DI m H q C W ) I m H q C W )) 1) q og rq 1) q og rq 1) q og and this competes the proof. 1) q+1 og + 1) q+1 og + p q ) rq, 1) q og DI m H q Σ W )) p q p q ) rq + p qp+1 ) rq + 1) q og 1) q og q p ) rq p q ) r q 1 Considering the short exact sequence of chain compexes associated to the pair C W, C W ), by Minor [3] 3, we have 0 C p C W ) C p C W ) C p C W, C W ) 0, og I p τ R C W, g C ); ρ 0 ) og τ R C W, g); ρ 0 ) + og I p τ R C W, C W ), g C ); ρ 0 ) + og τt p m). The cacuation of the torsion of T p m :... I p H q C W ) I p H q C W ) I p H q C W, C W )..., where T p m,3q Ip H q W ), T p m,3q+1 Ip H q C W ) and T p m,3q+ Ip H q C W, C W ), wi give the foowing resut. Lemma 4.4. Let W be a compact connected oriented manifod of odd dimension m p 1 without boundary. Let ρ 0 : π 1 C W ) O1, R) be the rank one trivia orthogona representation of the fundamenta group. Then, og I m τ R C W, g C ); ρ 0 ) og τ R C W, g); ρ 0 ) + og I m τ R C W, C W ), g C ); ρ 0 ) + og τt m m ). where r q rkh q W ), and og τt) m og τt) mc 1) q og ) rq. p q

15 ON THE CHEEGER-MÜLLER THEOREM FOR A CONE 15 Proof. The intersection homoogy of C W was given in the proof of Lemma 4.3, and that of C W, C W ) can be computed using the sequence of the pair, we get I m H q C W, C W ) I mc H q C W, C W ) { 0, q p, H q 1 C W ), q > p. Torsion and homoogy for the compementary perversities m and m c coincide, so fix p m. When q < p, consider the foowing part of the compex T m I m H q C W, C W ) 0 I m H q C W ) I m H q C W ) I m H q C W, C W ) 0. Let a q be an orthonorma base for H q W ). Then, as in the proof of Lemma 4.3, a basis for ) H q C W ) is q A q,g a q ), and a basis for I m H q C W, C W ) is p q 1 p q A q,g a q ). The de- terminant of the change of basis is 1 at I m H q C W ), and is q p, the reevant part of the sequence T m is p q ) rq at I m H q C W ). When 4.3) 0 I m H q+1 C W, C W ) H q C W ) 0 By Lemma 4., a basis for harmonic forms with reative boundary conditions H q re C W ) is ω q x αq 1 1 dx a q 1. Their norm is ω q,j g C C W x q dx a q 1,j g a q 1,j So an orthonorma base for H q re C W ) is q p q p 0 x q dx a q 1,j g q p q p. ) 1 ω q, using duaity 3.), ) A re q p 1 ) q p q,g C ω q,j q p q p q p q p q p q p ) 1 A re q p q,g C ω q,j ) q p ) 1 q p q p ) 1 A q 1,g a q 1,j ), ) 1 IPq 1 A p q abs gc ω q,j ) ) P 1 q 1 A q 1) g a q 1,j ) and this gives the basis for I m H q C W, C W ). The determinants of the change of basis in 4.3) are: ) rq 1 at H q C W ), and q p+ at I m H q+1 C W, C W ). Appying the definition of Reidemeister torsion to the compex T, m we obtain

16 16 L. HARTMANN AND M. SPREAFICO og τt m ) 6p p qp+1 p qp+1 1) q og DT m ) 1) q og DI m H q C W, C W )) + 1) q+1 og DI m H q C W )) 1) q og 1) q og and this compete the proof. p q p q ) r q 1 ) rq, + 1) q+1 og q p Proposition 4.1. Let W be a compact connected oriented manifod of odd dimension m p 1 without boundary. Let ρ 0 : π 1 C W ) O1, R) be the rank one trivia orthogona representation of the fundamenta group. Then r q rkh q W )), ) rq og Iτ R C W, g C ); ρ 0 ) og Iτ R C W, C W, g C ); ρ 0 ) 1 og τ R C W, g); ρ 0 ) + 1 ) rq 1) q og. p q Proof. By definition, if m is odd, og Iτ R C W, C W, g C ); ρ 0 ) og I m τ R C W, C W, g C ); ρ 0 ) og I mc τ R C W, C W, g C ); ρ 0 ). Then the statement foows using Lemma 4.3 once we reca that, when m is odd, by [7].8, og I mc τ R Σ W, g Σ ); ρ 0 ) og I m τ R Σ W, g Σ ); ρ 0 ). Theorem 4.1. Let W be a compact connected oriented manifod of odd dimension m p 1 without boundary. Let ρ 0 : π 1 C W ) O1, R) be the trivia orthogona representation of the fundamenta group. Then, og I m τ R C W, g C ); ρ 0 ) 1) m og I mc τ R C W, C W, g C ); ρ 0 ), og Iτ R C W, g C ); ρ 0 ) 1) m og Iτ R C W, C W, g C ); ρ 0 ). Proof. The proof foows by Proposition 4.1 and the two previous emmas. References [1] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müer, Astérisque 005, 199. [] J. Brüning and Xiaonan Ma, An anomay formua for Ray-Singer metrics on manifods with boundary, GAFA ) [3] J. Brüning and Xiaonan Ma, On the guing formua for the anaytic torsion, Math. Z ) [4] J. Cheeger, Anaytic torsion and the heat equation, Ann. Math ) [5] J. Cheeger, Spectra geometry of singuar Riemannian spaces, J. Diff. Geom ) [6] J. Cheeger, On the Hodge theory of Riemannian pseudomanifods, Proc. Sympos. Pure Math )

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