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1.
We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology. 相似文献
2.
For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti. 相似文献
3.
Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the
parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that
these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and
the Reidemeister torsion of CW-complexes. 相似文献
4.
We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over S2. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information. 相似文献
5.
In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4],
we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding
Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely
in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector
fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We
then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these
zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire
complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that
the torsion is equal to the value at of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work
on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere.
Received 2 August 1995; in final form 25 September 1996 相似文献
6.
We generalize Turaev's definition of torsion invariants of pairs ( M,&\xi;), where M is a 3-dimensional manifold and &\xi; is an Euler structure on M (a non-singular vector field up to homotopy relative to ∂ M and modifications supported in a ball contained in Int( M)). Namely, we allow M to have arbitrary boundary and &\xi; to have simple (convex and/or concave) tangency circles to the boundary. We prove that
Turaev's H
1( M)-equivariance formula holds also in our generalized context. Using branched standard spines to encode vector fields we show
how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation
of torsions a more effective one. Euler structures of the sort we consider naturally arise in the study of pseudo-Legendrian
knots ( i.e.~knots transversal to a given vector field), and hence of Legendrian knots in contact 3-manifolds. We show that torsion, as
an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely
analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic.
Received: 3 October 2000 / Revised version: 20 April 2001 相似文献
7.
For a closed Riemannian manifold ( M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of 1 ( M) on a finite dimensional vector space to a representation on a A-Hilbert module W of finite type where A is a finite von Neumann algebra. If ( M, W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L
2-analytic and L
2-Reidemeister torsions are equal.The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done. 相似文献
9.
We propose a refinement of the Ray–Singer torsion, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. Given a closed, oriented manifold of odd dimension with fundamental group Γ, the refined torsion is a complex valued, holomorphic function defined for representations of Γ which are close to the space of unitary representations. When the representation is unitary the absolute value of the refined torsion is equal to the Ray–Singer torsion, while its phase is determined by the η-invariant. As an application we extend and improve a result of Farber about the relationship between the absolute torsion of Farber–Turaev and the η-invariant. To cite this article: M. Braverman, T. Kappeler, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 相似文献
10.
In this paper we study, the Reidemeister zeta function. We prove rationality and functional equations of the Reidemeister zeta function of an endomorphism of finite group. We also obtain these results for eventually commutative endomorphisms. These results are applied to the theory of Reidemeister and Nielsen numbers of self-maps of topological spaces. Our method is to identify the Reidemeister number of a group endomorphism with the number of fixed points in the unitary dual. As a consequence, we show that the Reidemeister torsion of the mapping torus of the unitary dual is a special value of the Reidemeister zeta function. We also prove certain congruences for Reidemeister numbers which are equivalent to a Euler product formula for the Reidemeister zeta function. The congruences are the same as those found by Dold for Lefschetz numbers. 相似文献
11.
Over Pruüfer domains every module has a torsion free cover with a flat kernel. In this paper we show that the commutative Noetherian rings that have this property are the Gorenstein rings with Krull dimension at most one. Here we use the torsion theory defined by the nonzero-divisors of the commutative ring R. 相似文献
12.
We introduce a topological-type invariant for a cocompact properly discontinuous action of a discrete group on a Riemannian manifold generalizing classical notions of Reidemeister torsion. It takes values in the weak algebraic K-theory of the von Neumann algebra of . We give basic tools for its computation like sum and product formulas and calculate it in several cases. It encompasses, for instance, the Alexander polynomial and is related to analytic torsion. 相似文献
13.
In the spirit of Ray and Singer we define a complex-valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex-valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations. 相似文献
14.
In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same
odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product
as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions
and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis
and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently
vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined
by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions
and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal
invariants are computed in the process. 相似文献
15.
For a general chain complex ( C*, ∂*), one can associate the Reidemeister torsion of it. We prove the relation between Reidemeister torsion and Fubini-Study 2-form ωFS of the complex projective space CPn. 相似文献
16.
In this paper, we extend the Cappell–Miller analytic torsion to manifolds with boundary under the absolute and relative boundary conditions and using the techniques of Brüning-Ma and Su-Zhang, we get the anomaly formula of it for odd dimensional manifolds. Then by the methods of Brüning-Ma, Cappell–Miller and Su-Zhang, we get the Cheeger–Müller theorem for the Cappell–Miller analytic torsion on odd dimensional manifolds with boundary up to a sign. As a consequence of the main theorem, we get the gluing formula for the Cappell–Miller analytic torsion which generalizes a theorem of Huang. 相似文献
17.
All-or-nothing transforms have been defined as bijective mappings on all s-tuples over a specified finite alphabet. These mappings are required to satisfy certain “perfect security” conditions specified using entropies of the probability distribution defined on the input s-tuples. Alternatively, purely combinatorial definitions of AONTs have been given, which involve certain kinds of “unbiased arrays”. However, the combinatorial definition makes no reference to probability definitions. In this paper, we examine the security provided by AONTs that satisfy the combinatorial definition. The security of the AONT can depend on the underlying probability distribution of the s-tuples. We show that perfect security is obtained from an AONT if and only if the input s-tuples are equiprobable. However, in the case where the input s-tuples are not equiprobable, we still achieve a weaker security guarantee. We also consider the use of randomized AONTs to provide perfect security for a smaller number of inputs, even when those inputs are not equiprobable. 相似文献
18.
In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a
union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered
already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we mention [2] and [3]
(see also references therein).
The restriction of the number of pages to three (or any other number ≥3) provides a convenient way to encode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence
of links by analogy with the Reidemeister moves for planar link diagrams.
This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090.
Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 25–37, October–December,
1999.
Translated by I. A. Dynnikov 相似文献
19.
We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Müller theorem on Ray-Singer torsion and Reidemeister torsion, to the case of infinite Galois covering spaces. Our result is stated in the framework of extended cohomology, and generalizes in this case a recent result of Braverman-Carey-Farber-Mathai. It does not use the determinant class condition and thus also (potentially) generalizes several results on L2-torsions due to Burghelea, Friedlander, Kappeler and McDonald. We combine the framework developed by Braverman-Carey-Farber-Mathai on the determinant of extended cohomology with the heat kernel method developed in the original paper of Bismut-Zhang to prove our result. 相似文献
20.
We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy ?2 are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational. 相似文献
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