Notes on the Reidemeister Torsion Liviu I. Nicolaescu University of Notre Dame Notre Dame, IN 46556 http://www.nd.edu/˜lnicolae/ January 2002 To the memory of my father, Ion I. Nicolaescu, who shared with me his passion for books. Introduction The torsion of a cellular (simplicial) complex was introduced in the 30s by W. Franz [29] and K. Reidemeister [90] in their study of lens spaces. The lens spaces L(p, q) (p fixed) have the same fundamental groups and thus the same homology groups. However, they are not all homeomorphic. They are not even homotopically equivalent. This can be observed by detecting some below the radar interactions between the fundamental group and the simplicial structure. The torsion captures some of these interactions. In particular, it is able to distinguish lens spaces which are homotopically equivalent but not homeomorphic, and moreover completely classify these spaces up to a homeomorphism. This suggests that this invariant is reaching deep inside the topological structure. What is then this torsion? What does is compute? These are the kind of ques- tions we try to address in these notes, through many examples and various equivalent descriptions of this invariant. From an algebraic point of view, the torsion is a generalization of the notion of determinant. The most natural and general context to define the torsion would involve the Whitehead group and algebraic K-theory as in the very elegant and influential Milnor survey [72], but we did not adopt this more general point of view. Instead we look at what Milnor dubbed R-torsion. This invariant can be viewed as a higher Euler characteristic type invariant. Much like the Euler characteristic, the torsion satisfies an inclusion-exclusion (a.k.a. Mayer– Vietoris) principle which can be roughly stated as Tors(A ∪ B) = Tors(A) + Tors(B) − Tors(A ∩ B) which suggests that the torsion could be interpreted as counting something. The classical Poincaré–Hopf theorem states that the Euler–Poincaré characteristic of a smooth manifold counts the zeros of a generic vector field. If the Euler-Poincaré characteristic is zero then most vector fields have no zeroes but may have periodic orbits. The torsion counts these closed orbit, at least for some families of vector fields. As D. Fried put it in [34], “the Euler characteristic counts points while the torsion counts circles”. One of the oldest results in algebraic topology equates the Euler-Poincaré char- acteristic of simplicial complex, defined as the alternating sums of the numbers of simplices, with a manifestly combinatorial invariant, the alternating sum of the Betti numbers. Similarly, the R-torsion can be given a description in terms of chain com- plexes or, a plainly invariant description, in homological terms. Just like the Euler characteristic, the R-torsion of a smooth manifold can be given a Hodge theoretic description, albeit much more complicated. More recently, this invariant turned up in 3-dimensional Seiberg–Witten theory, in the work of Meng–Taubes ([68]). This result gave us the original impetus to understand Introduction vii the meaning of torsion. This is a semi-informal, computationally oriented little book which grew out of our efforts to understand the intricacies of the Meng–Taubes–Turaev results, [68, 115]. For this reason a lot of emphasis is placed on the Reidemeister torsion of 3-manifolds. These notes tried to address the author’s own struggle with the overwhelming amount of data involved and the conspicuously scanty supply of computational examples in the traditional literature on the subject. We considered that at an initial stage a good intuitive argument or example explaining why a certain result could be true is more helpful than a complete technical proof. The classical Milnor survey [72] and the recent introductory book [117] by V. Turaev are excellent sources to fill in many of our deliberate foundational omissions. When thinking of topological issues it is very important not to get distracted by the ugly looking but elementary formalism behind the torsion. For this reason we devoted the entire first chapter to the algebraic foundations of the concept of torsion. We give several equivalent definitions of the torsion of an acyclic complex and in particular, we spend a good amount of time constructing a setup which coherently deals with the torturous sign problem. We achieved this using a variation of some of the ideas in Deligne’s survey [18]. The general algebraic constructions are presented in the first half of this chapter, while in the second half we discuss Turaev’s construction of several arithmetically defined subrings of the field of fractions of the rational group algebra of an Abelian group. These subrings provide the optimal algebraic framework to discuss the torsion of a manifold. We conclude this chapter by presenting a dual picture of this Turaev subrings via Fourier transform. These results seem to be new and simplify substantially many gluing formulæ for the torsion, to the point that they become quasi-tautological. The Reidemeister torsion of an arbitrary simplicial (or CW) complex is defined in the second chapter. This is simply the torsion of a simplicial complex with Abelian local coefficients, or equivalently the torsion of the simplicial complex of the maximal Abelian cover. We present the basic properties of this invariant: the Mayer–Vietoris principle, duality, arithmetic properties and an Euler–Poincaré type result. We com- pute the torsion of many mostly low dimensional manifolds and in particular we explain how to compute the torsion of any 3-manifold with b1 > 0 using the Mayer–Vietoris principle, the Fourier transform, and the knowledge of the Alexander polynomials of links in S3. Since the literature on Dehn surgery can be quite inconsistent on the various sign conventions, we have devoted quite a substantial appendix to this subject where we kept an watchful eye on these often troublesome sings. The approach based on Alexander polynomials has one major drawback, namely it requires a huge volume of computations. We spend the whole section §2.6 explaining how to simplify these computation for a special yet very large class of 3-manifolds, namely the graph manifolds. The links of isolated singularities of complex surfaces are included in this class and the recent work [75, 76] proves that the Reidemeister torsion captures rather subtle geometrical information about such manifolds. We conclude this chapter with some of the traditional applications of the torsion in topology. viii Introduction Chapter 3 focuses on Turaev’s ingenious idea of Euler structure and how it can be used to refine the concept of torsion by removing the ambiguities in choosing the bases needed for computing the torsion. Turaev later observed that for a 3-manifold a choice of an Euler structure is equivalent to a choice of spinc-structure. After we review a few fundamental properties of this refined torsion for 3-manifolds we then go on to present a result of Turaev which in essence says that the refined torsion of a 3-manifold with positive b1 is uniquely determined by the Alexander polynomials of links in S3 and the Mayer–Vietoris principle. This uniqueness result does not include rational homology spheres, and thus offers no indication on how to approach this class of manifolds. We spend the last part of this chapter analyzing this class of 3-manifolds. In §3.8 we describe a very powerful method for computing the torsion of such 3-manifolds, based on the complex Fourier transform results in Chapter 1, and an extremely versatile holomorphic regularization technique. These lead to explicit for- mulæ for the Fourier transform of the torsion of a rational homology sphere in terms of surgery data. These formulæ still have the two expected ambiguities: a sign ambiguity and a spinc ambiguity. In §3.9 we describe a very simple algorithm for removing the spinc ambiguity. This requires a quite long topological detour in the world of quadratic functions on finite Abelian groups, and surgery descriptions of spin and spinc struc- tures, but the payoff is worth the trouble. The sign ambiguity is finally removed in §3.10 in the case of plumbed rational homology spheres, relying on an idea in [75], based on the Fourier transform, and a relationship between the torsion and the linking form discovered by Turaev. Chapter 4 discusses more analytic descriptions of the Reidemeister torsion: in terms of gauge theory, in terms of Morse theory, and in terms of Hodge theory. We discuss Meng–Taubes theorem and the improvements due to Turaev. We also out- line our recent proof [83] of the extension of the Meng–Taubes–Turaev theorem to rational homology spheres. As an immediate consequence of this result, we give a new description of the Brumfiel–Morgan [7] correspondence for rational homology 3-spheres which associates to each spinc structure a refinement of the linking form. On the Morse theoretic side we describe Hutchinson–Pajitnov results which give a Morse theoretic interpretation of the Reidemeister torsion. We barely scratch the Hodge theoretic approach to torsion. We only provide some motivation for the ζ- function description of the analytic torsion and the Cheeger–Müller theorem which identifies this spectral quantity with the Reidemeister torsion. ∗∗∗ Acknowledgements1 I spent almost two years thinking about these issues and I was helped along the way by many people. I am greatly indebted to Frank Connolly who introduced me to this subject. His patience and expertise in answering my often half 1This work was partially supported by NSF grant DMS-0071820. Introduction ix baked questions have constantly kept me on the right track. I also want to thank my colleagues Bill Dwyer, Stephan Stolz, Larry Taylor and Bruce Williams for many illuminating discussions which made this scientific journey so much more enjoyable.