Generalized Rochlin Invariants Of Fixed Point Sets (original) (raw)

Noncoincidence index, free group actions, and the fixed point property for manifolds

1989

Let M be a compact oriented connected topological manifold. We show that if the Euler characteristic χ{M) Φ 0 and M admits no degree zero self-maps without fixed points, then there is a finite number r such that any set of r or more fixed-point-free self-maps of M has a coincidence (i.e. for two of the maps / and g there exists x e M so that f(x)-g(x)). We call r the noncoincidence index of M. More generally, for any manifold M with χ(M) Φ 0 there is a finite number r (called the restricted noncoincidence index of M) so that any set of r or more fixed-point-free nonzero degree self-maps of M has a coincidence. We investigate how these indices change as one passes from a space to its orbit space under a free action. We compute the restricted noncoincidence index for certain products and for the homogeneous spaces SU n /K, K a closed connected subgroup of maximal rank; in some cases these computations also give the noncoincidence index of the space.

[114] Integral Invariants of 3-Manifolds, II

Raoul Bott: Collected Papers, 2017

This note is a. sequel to our earlier paper of the same title (4] and describes invariants of rational homology 3-spheres associated to acyclic orthogo nal local systems. Our work is in t he spirit of the Axelrod-Singer papers [1], gene1•a.lizes some of t hei r results, and furnishes a new setting for the purely topological implications of their work.

Computing Topological Invariants Using Fixed Points

Proceedings of the 6th International Congress of Chinese Mathematicians, 2017

When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah-Bott and Berline-Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful tool for computing integrals on a manifold. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This article highlights two applications of the equivariant localization formula. We show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle. Mathematics Subject Classification: Primary: 55R10, 55N25, 14C17; Secondary: 14M17. Many invariants in geometry and topology can be represented as integrals. For example, according to the Gauss-Bonnet theorem, the Euler characteristic of a compact oriented surface in R 3 is 1/2π times the integral of its Gaussian curvature: χ(M) = 1 2π M Kvol. The Euler characteristic can be generalized to other characteristic numbers. For example, if E is a complex vector bundle of rank r over a complex manifold M of complex dimension n, and c 1 ,. .. , c r are the Chern classes of E, then the integrals M c i1 1 · · · c ir r , where r k=1 k · i k = n, This article is based on a talk given at the Sixth International Congress of Chinese Mathematicians , Taipei, in 2013.

A note on the Theta\ThetaTheta-invariant of 3-manifolds

2021

In this note, we revisit the Θ-invariant as defined by R. Bott and the first author in [4]. The Θ-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The Θ-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the Θ-invariant that we can define even if the cohomology group is not vanishing.

A note on the relation between joint and differential invariants

Lobachevskii Journal of Mathematics, 2016

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

On the hauptvermutung for manifolds

1967

The "Hauptvermutung" is the conjecture that homeomorphic (finite) simplicial complexes have isomorphic subdivisions, i.e. homeomorphic implies piecewise linearly homeomorphic. It was formulated in the first decade of this century and seems to have been inspired by the question of the topological invariance of the Betti and torsion numbers of a finite simplicial complex.

On invariants of Hirzebruch and Cheeger–Gromov

Geometry & Topology, 2003

We prove that, if M is a compact oriented manifold of dimension 4k + 3, where k > 0, such that π 1 (M ) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M , we construct a secondary invariant τ (2) : S(M ) → R that coincides with the ρ-invariant of Cheeger-Gromov. In particular, our result shows that the ρ-invariant is not a homotopy invariant for the manifolds in question.

The topological spherical space form problem—II existence of free actions

Topology, 1976

RECENT advances in calculation of projective class groups and of surgery obstruction groups lead us to hope that it will shortly be possible to give a fairly complete account of the classification of free actions of finite groups on spheres. In the present paper, we determine which groups can so act, thus solving a problem of several years' standing. Further, we show that these actions can be taken to be smooth actions on smooth homotopy spheres. Previously known results can be summarised as follows, where we say the finite group 7~ satisfies the "pq-condition" (p, q primes not necessarily distinct) if all subgroups if v of order pq are cyclic. 0.1. (Cartan and Eilenberg[3]). If rr acts freely on S"-', it has periodic cohomology with minimum period dividing n. Moreover, P has periodic cohomology if and only if it satisfies all p2-conditions. And the p* condition is equivalent to the Sylow p-subgroup zrr, of r being cyclic or perhaps (if p = 2) generalised quaternionic. 0.2. (Wolf [19]). If 7~ acts freely and orthogonally on a sphere, it satisfies all pq-conditions. Conversely, if r is soluble and satisfies all pq-conditions, free orthogonal actions exist. However, for rr non-soluble, the only non-cyclic composition factor allowed is the simple group of order 60. 0.3. (Milnor [9], see also Lee [8]). If 7~ acts freely on any sphere, it satisfies all 2p-conditions. 0.4. (Petrie [I 11). Any extension of a cyclic group of odd order m by a cyclic group of odd prime order q prime to m can act freely on S*"-'. Petrie's result shows that pq-conditions are not all necessary for free topological actions. it is therefore not so surprising that THEOREM 0.5. A finite group 7 can act freely on a sphere if and only if it satisfies all 2p-and p '-conditions. We shall elaborate the group theory in the next section: the most interesting groups 7r included are perhaps the groups S&(p) (p prime). As to smooth actions, it will follow from a general result below that THEOREM 0.6. For each free action of T on S"-' constructed in the proof of (0.5). S"-' has a differential structure (z such that rr acts freely and smoothly on S,"-'. Clearly, in many cases one can deduce existence of free smooth actions on S"-', but in this paper we will confine ourselves to general arguments. In principle, the proof of these theorems follows the pattern laid down in [ 151 and elaborated for this problem in a previous paper[l4]. We construct first, a finite simplicial complex X; second, a normal invariant, and hence a normal cobordism class of normal maps M+X; and thirdly we show that the corresponding surgery obstruction vanishes. This yields a manifold homotopy equivalent to X whose universal cover is homotopy equivalent, hence homeomorphic (but not necessarily diffeomorphic, when smooth) to a sphere. In practice, the key idea of the proof is a careful choice of X, and of the normal invariant, so as to allow a simple proof of vanishing of the suergery obstruction. The paper is set out in four sections. In the first, we summarise the group theory, and introduce notations for the groups involved. In the second, we choose the homotopy type of X: this involves circumventing the finiteness obstruction of Swan[ 131, and prepares the way for the surgery. In the third, we discuss normal invariants. General existence of normal invariants follows from the powerful techniques of modern homotopy theory. Topological normal invariants can then be studied using Sullivan's[lO] analysis of the homotopy type of G/Top.