Michael Hutchings
In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological invariant. It is hoped that this will provide a model for possible generalizations to Floer theory.
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In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.
In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps.
In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for closed hyperbolic 3-manifolds.
Let X be a compact oriented Riemannian manifold and let $ϕ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $ϕ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $ϕ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $ϕ$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $ϕ$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs)…
We use higher parallel transport -- more precisely, the integration A_{infty}-functor constructed by Block-Smith and Arias Abad-Schaetz -- to define Reidemeister torsion for flat superconnections. We hope that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu, thus permitting for a generalization of the Cheeger-Mueller Theorem.
On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.
Analytic and Reidemeister torsion for representations in finite type Hilbert modules
对于闭黎曼流形，我们将解析挠和Reidemeister挠的定义推广到有限vonNeumann代数上有限型Hilbert模上基本群的正交表示.如果表示是行列式类，我们证明，推广Cheeger-Müller定理，分析和Reidemeister扭转是平等的。特别地，这证明了猜想，对于具有正的Novikov-Shubin不变量的闭黎曼流形，L2解析挠率和Reidemeister挠率是相等的。
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 SL_2(C) to the corresponding Reidemeister torsion. 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.
We study the Reidemeister torsion and the analytic torsion of the $m$ dimensional disc in the Euclidean $m$ dimensional space, using the base for the homology defined by Ray and Singer in \cite{RS}. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma \cite{BM}, that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary \c…
In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We continue to study analytical properties of the Nielsen zeta function. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced ma…
We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL(2;R). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsio…
This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I_F. (See math.DG/0111313 for part I). Having constructed I_F and outlined a proof of its invariance based on bifurcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These gluing theorems are different from (and much harder than) the more conventional versions in that they deal with broken trajectories or broken orbits connected at degenerate rest points. The issues of orientation and signs are also se…
Let X be a closed manifold with zero Euler characteristic, and let f: X --> S^1 be a circle-valued Morse function. We define an invariant I which counts closed orbits of the gradient of f, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679-695]. We proved a similar result in our previous paper [Topology, 38 (1999) 861-888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered…
In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler structures, introduced earlier by the second author. We show that the sign of our PR-scalar product is determined by the Stiefel-Whitney classes and the semi-characteristic of the m…
Adjoint twisted Reidemeister torsion and Gram matrices
本文计算了闭定向及带边双曲3-流形的伴随扭曲Reidemeister挠，基于Dehn填充和理想三角剖分给出了新公式，并在量子不变量的渐近展开研究中发挥重要作用。
The paper consists of four parts. Part I presents a brief survey of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with congruences for the Reidemeister and Nielsen numbers. Part IV deals with the Reidemeister torsion . In Chapter 2 we prove that the Reidemeister zeta function of a group endomorphism is a rational function with functional equation in the following cases: the group is finitely generated and an endomorphism is eventually commutative; the group is finite ; the group is a direct sum of a …
This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II). The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance--the Reidemeister torsion (τ_F) of the Floer-Novikov complex of (possibly non-hamiltonian) symplectomorphisms. (τ_F) turns out NOT to be invariant under hamiltonian isotopies, but this failure may be fixed by introducing certain ``correction term'': We define…
Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted product of zeta-regularized determinants of Laplace operators on differential forms. The celebrated Cheeger-Mueller Theorem, established independently by J. Cheeger and W. Mueller, proved equality between the analytic …
Torsions for manifolds with boundary and glueing formulas
本文将解析挠和Reidemeister挠的定义从闭紧致黎曼流形推广到具有边界$(M, \partial M)$的紧致黎曼流形，给出了有限型$\Cal A$模的平坦丛$\Cal F$和边界$\partial M =\partial_- M \cup \partial_+ M$分解为不相交分支.特别地，我们将$L-2$解析挠率和Reidemeister挠率推广到带边界的紧致流形上.如果系统$(M,\partial_-M, \partial_+M, \Cal F)$是行列式类，我们计算了解析挠率和Reidemeister挠率的商，并证明了它们的胶合公式。特别地，我们肯定地回答了[LL]中的猜想7.6。
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes.
In this paper we prove the Cheeger-Müller theorem for $L^2$-analytic torsion form under the assumption that there exists a fiberwise Morse function and the Novikov-Shubin invariant is positive.
The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion
该文研究紧致双曲轨道曲面上扭曲 Ruelle zeta 函数在零点取值与 Reidemeister-Turaev 挠率的关系，推广 Fried 猜想至非酉表示并解决相位符号问题，同时计算消失阶与首项系数。
It is shown that for any piecewise-linear closed orientable manifold of odd dimension there exists an invariantly defined metric on the determinant line of cohomology with coefficients in an arbitrary flat bundle E over the manifold (E is not required to be unimodular). The construction of this metric (called Poincare - Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister type construction with Poincare duality. The main result of the paper states that the Poincare-Reidemeister metric computes combinatorially the Ray-Singer metric. It is shown also that the Ray-S…
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is of determinant class, is always defined. When the pair is of determinant class the relative torsionis…
For any compact Lie group G we discuss the relation of the equivariant Reidemeister and analytic torsion of G-manifolds with their G-CW structures.
We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.
Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the difference of τ(M/B;F) and the higher analytic torsion \Cal T(M/B;F). We also generalise the examples given in math.DG/0111222 .
We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa's higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_i: M -> B with diffeomorphic fibres that are homeomorphic but not …
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.
Witten deformation for non-Morse functions and gluing formula for analytic torsions
本文针对非 Morse 函数族提出 Witten 变形的新解析方法，证明解析挠率的粘合公式，并揭示其与 Bismut-Zhang 定理及移动边界问题的联系，为高阶推广奠定基础。
Generalized Morse Functions, Excision and Higher Torsions
本文提出通过广义莫尔斯函数消除奇异临界点，推广Bismut-Goette定理，无需假设纤维方向横截条件。
The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics
综述紧致黎曼流形上莫尔斯 - 斯梅尔流的广义弗里德猜想，通过构建框架、刻画向量场等建立相关不变量，最终证明射线 - 辛格度量与米尔诺度量一致，解析挠率有特定等式关系。
We prove that the anomaly term appearing in the ratio between Reidemeister torsion and analytic torsion for a space with a conical singularity vanishes in the smooth case. Moreover, we show that such anomaly term is non trivial for the cone over a torus.
Witten deformation of the analytic torsion and the spectral sequence of a filtration
设F是紧致黎曼流形M上的平坦向量丛，f是Morse函数.设g是F上的光滑欧氏度量，g_t=e^{-2tf}g，ρ（t）是F的Ray-Singer解析挠率.假设向量场grad（f）满足Morse-Smale横截性条件，我们给出了\log（ρ（t））对于t \to + \infty的形式为a_0+ a_1 t + bPDMATHABMM \left（\frac tπ \right）+o（1）的渐近展开式，其中系数b是只依赖于F的Betti数的半整数.在f的所有临界值都是有理数的情况下，我们根据与Morse函数相关联的滤波的谱序列来明确地计算系数a_0和a_1。这些结果是Bismut和Zhang的一个定理的应用。
Witten deformation of Ray-Singer analytic torsion
考虑紧致黎曼流形M上的平坦向量丛F，设f是M上的自指标Morse函数。设g是F上的光滑欧氏度量。设g_t=exp（-2tf）g，ρ（t）为F的Ray-Singer解析挠率.假设向量场$grad f$满足Morse-Smale横截性条件，我们给出了t个\to \infty的log（ρ（t））的渐近展开式a_0+ a_1 t +b log（t）+o（1）.我们给出了系数a_0，a_1和b的显式公式。特别是，我们证明了b是一个半整数。
The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements.
We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle an…
In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced $L^2$ homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dime…
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