We consider the flow dual to the 1form via some metric. Nicolaescu this is a stateoftheart introduction to the work of franz reidemeister, meng taubes, turaev, and the author on the concept of torsion. Higher torsion invariants in differential topology and algebraic k. A third formula for the zeta function, in terms of xed points of return maps, is given in equation. Riemannian geometry, topology and dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. Rigidity and gluing for morse and novikov complexes.
Reidemeister torsion for groups 1 is used to study the structure of basic. Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions. Nielsen theory and reidemeister torsion banach center publications, volume 49 institute of mathematics polish academy of sciences warszawa 1999 dynamical zeta functions, congruences in nielsen theory and reidemeister torsion alexander felshtyn fachbereich mathematik, universit. Reidemeister torsion in generalized morse theory in two previous papers with yijen lee, we dened and computed a notion of reidemeister torsion for the morse theory of closed 1forms on a nite dimensional manifold. Symmetry of reidemeister torsion on su2representation spaces. There are many connections between reidemeister torsion and knots, particularly the alexander polynomials of knots see milnor 193, 194, 195 and turaev 293, as well as chap. The use of the term geometric topology to describe. The intersection r torsion for finite cone xianzhe dai xiaoling huangy 1 introduction torsion invariants were originally introduced in the 3dimensional setting by k. Reidemeister torsion in floernovikov theory and counting pseudoholomorphic tori, i lee, yijen, journal of symplectic geometry, 2005. The torsion of a cellular simplicial complex was introduced in the 30s by w. Quantum reidemeister torsion, open gromovwitten invariants. Those handles are index 1 handles in the sense of morse theory.
The mathematics genealogy project is in need of funds to help pay for student help and other associated costs. Theorem of the day reidemeisters theorem two knots are topologically equivalent if and only if their projections may be deformed into each other by a sequence of the three moves shown below. Circlevalued morse theory and reidemeister torsion arxiv. M hutchings, reidemeister torsion in generalized morse theory, eprint math. Reidemeister torsion and morsesmale flows article pdf available in ergodic theory and dynamical systems 1602.
Reidemeister torsion and theorem f 20 references 26 1. We now introduce a notion of topological reidemeister torsion following turaev 28, and an analogous notion of morsetheoretic torsion. Reidemeister 23 in 1935 who used them to give a homeomorphism classi cation of 3dimensional lens spaces. R is a morse function if it has nitely many critical points all of. We show in theorem 3 that the reidemeister torsion of a level surface of a lyapunov function and of the attraction domain of an attractor. If you have additional information or corrections regarding this mathematician, please use the update form. Given a circlevalued morse function of a closed oriented manifold, we prove that reidemeister torsion over a noncommutative formal laurent polynomial ring equals the product of a certain noncommutative lefschetztype zeta function and the algebraic torsion of the novikov complex over the ring. Mathematics genealogy project department of mathematics north dakota state university p. Here is a more detailed presentation of the contents.
Torsion, as a function on the space of representations. We start with the main result of 14 which we reformulate to say that the set of possible singular sets of berwise generalized morse functions on m produce a spanning set in the homology of m in the correct degrees. Reidemeister torsion in generalized morse theory, forum math. We show in theorem 3 that the reidemeister torsion of a. Reidemeister torsion in generalized morse theory reidemeister torsion in generalized morse theory hutchings, michael 20020129 00. In particular, we consider them on a 1dimensional smooth part of the space, which is canonically oriented and metrized via a reidemeister torsion volume form. The examples come from the theory of generalized morse functions. The present paper gives an a priori proof that this morse theory invariant is a topological invariant.
Combina torial no vik o vmorse theor y r obin f orman intr oduction in classical morse theory one b egins with a smo oth manifold m and a smo oth function on m the. Morse theory and higher torsion invariants ii request pdf. Notes on the reidemeister torsion university of notre dame. Jul 04, 2007 project euclid mathematics and statistics online. To submit students of this mathematician, please use the new data form, noting this mathematicians mgp id of 43253 for the advisor id. By counting gradient flow lines between critical points, one can recover the homology of the manifold. We can compare the morsetheoretic and topological reidemeister torsion using. In two previous papers with yijen lee, we dened and computed a notion of reidemeister torsion for the morse theory of closed 1forms on a nite dimensional manifold. Complex valued raysinger torsion ii dan burghelea and stefan haller abstract. Is the torsion function of 3manifolds a finite invariant in a reasonable sense.
Read reidemeister torsion, spectral sequences, and brieskorn spheres. We adapt classical reidemeister torsion to monotone lagrangian submanifolds using the pearl complex of biran and cornea. Through morse theory we are able to get valuable insights into the topology of a manifold by studying differentiable functions on that space. Higher reidemeister torsion and parametrized morse theory, rend. Michael hutchings the mathematics genealogy project. Reidemeister torsion in generalized morse theory, forum.
Jan 29, 2002 reidemeister torsion in generalized morse theory reidemeister torsion in generalized morse theory hutchings, michael 20020129 00. Where morse theory is used to analyse manifolds, discrete morse theory is used to analyse cwcomplexes. It is not usually at all obvious how to determine whether one knot an embedding in three dimensions of a closed loop is equivalent to another. G, and the fourier transform of tmdis a generalized function. Wallace, and others, including a proof of the generalized poincare hypothesis in high dimensions. Pdf circlevalued morse theory, reidemeister torsion.
We define an invariant i which counts closed orbits of the gradient of f, together with flow lines between the critical points. There are three independent constructions of this invariant using morse theory igusaklein torsion, homotopy theory dwyerweisswilliams torsion and analysis bismutlott analytic torsion. Novikov 25 generalized this to multiplevalued functions, i. A special case of novikovs theory is circlevalued morse theory, which michael hutchings and yijen lee have connected to reidemeister torsion and seibergwitten theory. Pdf circlevalued morse theory and reidemeister torsion. Let x be a closed manifold with zero euler characteristic, and let f.
Reidemeister torsion, the alexander polynomial and u 1,1. Euler structures, nonsingular vector fields, and torsions of reidemeister type v g turaev 1990 mathematics of the ussrizvestiya 34 627. Numerous and frequentlyupdated resource results are available from this search. Reidemeister torsion in generalized morse theory core. Another facet of morse s theory is the morse complex a di eren tial complex whic h w e denote b y m whic h is constructed from the critical. The main theorem of discrete morse the ory gives a condition on the minimal number of cells required to build a cwcomplex of a certain homotopy type. Morse homology is a special case for the oneform df. A note on exceptional groups and reidemeister torsion. Morse cobordisms are used to compare various morse type complexes without the need of bifurcation theory. In the generalized setting, novikov found the appropriate topological. Torsion reidemeister torsion in generalized morse theory lee yijen lee, reidemeister torsion in floernovikov theory and counting pseudoholomorphic tori i, ii gluing with cliff taubes, gluing pseudoholomorphic curves along branched covered cylinders ii in this post, inspired by a question of jiayong li and katrin wehrheim and other people.
Morse theory has provided the inspiration for exciting developments in differential topology by s. According to our current online database, michael hutchings has 10 students and 10 descendants. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The classical franzreidemeister torsion and its cousins, the whitehead tor sion and raysinger. Reidemeister torsion for the morse theory of closed 1forms on a. In morse theory, one starts with a generic realvalued smooth function on a closed smooth manifold. By continuing to use our website, you are agreeing to our use of cookies. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. In this talk we will consider a discrete version of morse theory by first introducing a discrete morse function and gradient vector field.
Hutchings, reidemeister torsion in generalized morse theory. A generalized morse function gmf on a smooth manifold m is a. Sergei novikov generalized this construction to a homology theory associated to a closed oneform on a manifold. Journal of journal of geometry and physics 199410523 geometryarid northholland physics reidemeister torsion, the alexander polynomial and u 1, 1 chernimons theory l. We did the rst calculation of higher torsion on circle bundles in 12 using a dilogarithm formula for pictures which i developed 8. Citeseerx reidemeister torsion in generalized morse theory. These singular sets are examples of \strati ed subsets of m. Under some mild assumptions on, we prove a formula relating 1. Some applications, including relations between the reidemeister torsion and other. In two previous papers with yijen lee, we defined and computed a notion of reidemeister torsion for the morse theory of closed 1forms on a finite dimensional manifold. In the three dimensional case, combining this result with the conjecture in 1, we obtain a formula for the full seibergwitten invariant, which was conjectured by turaev. Noncommutative reidemeister torsion and morsenovikov theory.
We discuss mengtaubes theorem and the improvements due to turaev. As an application we show that the reidemeister torsion function on the 1dimensional subspace has symmetry about the metrization. Contents introduction reidemeister zeta function and torsion. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. First we give a description of the general results for evaluating surgery obstructions. This paper proves a refinement of the main theorem of 1, using a different method. Bibliography for kiyoshi igusa brandeis university. Introduction consider ma smooth c1 closed and oriented manifold which is of dimension n 1. Jul 31, 2003 the gluing result is a type of mayervietoris formula for the morse complex. If you would like to contribute, please donate online using credit card or bank transfer or mail your taxdeductible contribution to.
The definition involves gene we use cookies to enhance your experience on our website. Another facet of morses theory is the morse complex, a differential complex, which. It is hoped that this will provide a model for possible generalizations to floer theory. Noncontractible periodic orbits, gromov invariants, and floertheoretic torsions.
The purpose of this book is to develop the theory of higher franz reidemeister torsion fr torsion from its foundations up to the current results. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Noncontractible periodic orbits, gromov invariants, and. Franz reidemeister torsion kiyoshi igusa r tecwmische nformat. Chapter 4 discusses more analytic descriptions of the reidemeister torsion. Discrete morse theory from a simplehomotopy point of view. It is used to express algebraically the novikov complex up to isomorphism in terms of the morse complex of a fundamental domain. We give several descriptions of the higher fr torsion, show that they are equivalent, use them to define the higher fr torsion in a very general setting and also. Nicolaescu this is a stateoftheart introduction to the work of franz reidemeister, meng taubes, turaev, and the author on the concept of torsion and its generalizations. In particular, when v is a nonsingular morse smale ow, we show that the reidemeister torsion can be recovered from the resonances lying on the imaginary axis. Kappeler witten deformation of the analytic torsion and reidemeister torsion amer. It is easily checked that any two handlebodies are homeomorphic if, and only if, they have the same genus. We also outline our recent proof 83 of the extension of the mengtaubesturaev theorem to rational homology spheres. We announce a proof of the conjecture of ray and singer that for a compact riemannian manifold the analytic torsion and reidemeister torsion are equal.
Circlevalued morse theory and reidemeister torsion. Reidemeister torsion in generalized morse theory math berkeley. Consider the generalized quaternion group q4p of order 4p. It proves that for such representations the notion of reidemeister torsion is welldefined. Dan burghelea papers and preprints available to download. Ams proceedings of the american mathematical society. Reidemeister torsion, spectral sequences, and brieskorn. In the current chapter the treatment of reidemeister torsion in 194 will be generalized to define a relative k theory invariant for chain complexes. In two papers with yijen lee hl1, hl2, we defined a notion of reidemeister torsion for the morse theory of closed 1forms on a finite dimensional manifold.