Ted Petrie

Homotopy representations of finite groups

© Publications mathématiques de l’I.H.É.S., 1982, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Smith equivalence of representations

An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set Σ G consisting of two points p and q , are the tangential representations T p Σ and T q Σ of G at p and q equal? Put another way: Describe the representations ( V, W ) of G which occur as ( T p Σ T q Σ) for Σ a sphere with smooth action of G and Σ G = p ∪ q . Under these conditions we say V and W are Smith equivalent (21) and write V ~ W . A stronger equivalence relation is also interesting. We say representations V and W are s -Smith equivalent if ( V, W ) = ( T p Σ, T q Σ) and Σ is a semi-linear G sphere (23), i.e. Σ K is a homotopy sphere for all K and Σ G = p ∪ q . In this case we write V ≈ W .

Spherical isotropy representations

© Publications mathématiques de l’I.H.É.S., 1985, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

SMITH EQUIVALENCE OF REPRESENTATIONS FOR ODD ORDER CYCLIC GROUPS

THEOREM A. There exist odd order cyclic groups which have Smith equivalent but nonisomorphic representations. After a short background discussion, we shall give a list of conditions (1.1) which are important in the development of this paper and state more precise versions of our results. Theorem B provides nonisomorphic Smith equivalent representations if one has nonisomorphic representations of G which satisfy (1.1). Corollary C shows the existence of infinitely many groups which have nonisomorphic representations which satisfy (1.1). The orders of some cyclic groups of this type are described by Lemma 2.5 and Corollary 2.6. Taken together, Theorem B and Corollary C prove Theorem A. The concept of Smith equivalence is based upon a question posed by P. A. Smith in a 1960 survey article [32, p. 4061. In our terminology the question is whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [3] and Milnor [17] established an affirmative answer for cyclic groups of odd prime power order and for representations with semifree action. By definition a group acts semifreely if each point is left fixed by only the trivial group element or by each group element. An extension of this work by Sanchez [28] implied the answer also to be affirmative if G is cyclic of order pq where p and q are odd primes. Bredon [6] showed for 2-groups that Smith equivalent representations are isomorphic if their dimension is large in comparison to the order of the group. The first negative answers to Smith’s question were established by the second author for odd order abelian groups with at least four noncyclic Sylow subgroups. See [23], [24]. Subsequently a number of papers have established a negative answer for even order groups, but the case of odd order cyclic groups has remained open. For cyclic groups of even order see the papers [7], [25], [ 111, [3 13, For noncyclic abelian groups and nonabelian 2-groups, see [34] and [S]. There are many interesting results and ideas in these papers. Results and techniques used in answering Smith’s question varj markedly according to the parity of the order of G. In Remark 1.2(ii) we explain the difference of the geometric and number theoretic treatment of even and odd order groups as it arises from the Atiyah-Bott extension of the Lefschetz Fixed Point theorem. One interesting question, untouched here, is to relate the differentiable structure of a homotopy sphere Z and the isotropy representations of an action of G on Z with exactly two fixed points. Some general remarks on this appear in the problem section of the forthcoming proceedings of the 1983 conference on group actions at Boulder [393. It can be shown that the homotopy spheres with group actions that are constructed by our method are all standard spheres. The methods of Schultz (e.g. in [29] and other papers) are relevant to realizing these actions on exotic spheres.

Stably trivial equivariant algebraic vector bundles

Let G be a reductive algebraic group over C, let F be a G-module, and let B be an affine G-variety, i.e., an affine variety with an algebraic action of G. Then B x F is in a natural way a G-vector bundle over B, which we denote by F. (All vector bundles here are algebraic.) A G-vector bundle over B is called trivial if it is isomorphic to F for some G-module F. From the endomorphism ring R of the G-vector bundle S, we construct G-vector bundles over B. The bundles constructed this way have the property that when added to S they are isomorphic to F e S for a fixed G-module F. They are called stably trivial. The set of isomorphism classes of G-vector bundles over B which satisfy this condition is denoted by VEC(B, F; S). For such a bundle E we define an invariant p(E) which lies in a quotient of R. This invariant allows us to distinguish non-isomorphic G-vector bundles. When B is a Gmodule, a G-vector bundle over B defines an action of G on affine space. We give criteria which in certain cases allow us to distinguish the underlying actions. The construction and invariants are applied to the following two problems:

One fixed point actions on spheres, II

The object of this paper is to treat the following old question of Montgomery and Samelson [M-S] and some of its consequences: Which groups act smoothly on a closed homotopy sphere with exactly one fixed point and what are the isotropy representations of G which occur on the tangent space at the fixed point? The first and only previously existing example of such an action was given by E. Stein for the group SL(2, Z,) [St]. A related question was solved by Oliver [O,]: Which groups act smoothly on a disk without fixed points? A group which acts on a sphere with one fixed point acts on a disk without fixed points. In [P,] the author announced:

Homology Planes an Announcement and Survey

We review recent advances dealing with homology planes, i.e., non singular affine acyclic surfaces over the complex numbers C. Here acyclic means vanishing integral reduced homology. From a topological point of view these are the simplest affine surfaces.

Algebraic automorphisms of smooth affine surfaces

Smooth acyclic affine surfaces will in this paper be referred to as homology planes. In a remarkable paper [R] Ramanujam exhibited an example of a smooth contractible affine surface R not homeomorphic to the complex plane. Quite recently Gurjar-Miyanishi have produced an infinite number of distinct smooth contractible affine surfaces, one of Kodaira dimension 2 and the rest of Kodaira dimension 1. These are referred to as GM and G(u) respectively. Here u ranges over a certain discrete set. (See section 6). See [GM] also. One of the main results of this paper is this theorem:

Smooth conjugacy of algebraic actions on affine varieties

The motivation for this result is the linearity conjecture which asserts that any algebraic action of a reductive algebraic group on affine n-space is conjugate to a linear action. This topic has been popularized by H. Bass and H. Kraft. See for example [4], [2] and [S] for the current status of the conjecture. In addition to being interesting, the conjecture is tough. It is not even known whether the set of algebraic conjugacy classes of algebraic actions of a reductive group of affine n-space is countable, as implied by the linearity conjecture. Theorem 1 is a version of this countability question. We shall see in the course of the proof of Theorem 1 that there are an uncountable number of conjugacy classes of smooth actions of a nontrivial compact Lie group on real n-space. Theorem 1 shows that only countably many of these are conjugate to algebraic actions on real n-space. The main tools in the proof are the compactification theorem of Corollary 4 and the Palais Rigidity Theorem [lo]. Whether this rigidity theorem has an algebraic analog is an interesting question whose consequences we briefly explore at the end. Before giving the proof, we introduce some notation. In this discussion varieties are affine. The ground field is the reals [w or the complex numbers @. An algebraic map between affine spaces (over [w or C) is a map whose coordinate functions are polynomials. An algebraic map between affine varieties is a map which extends to an algebraic map of affine spaces containing the varieties as subvarieties. Let W be a nonsingular variety resp. a smooth manifold and Aut( W, c) for c=a or s denote the group of algebraic resp. smooth automorphisms of W with the C” topology. Let Hom(G, Aut( W, c)) denote the space of c actions of G on W. A point d of this space is a homomorphism d:BG+Aut( W, c) such that the induced map from G x W to W is a c-map. Topologize Hom(G, Aut( W, c)) as the subspace of the space of all smooth maps of G x W to Win the C” topology. The group Aut( W,c) acts by conjugation on the space of c-actions and the orbit space is denoted by Hom(G, Aut( W, c))/Aut( W, c). In the case W is a nonsingular variety V, weakening structure defines a continuous function from the space of a-actions to the space of s-actions and induces a map of corresponding orbit spaces. We are interested in the cardinality of the image. The idea is to factor this map through a countable set. In the case V is simply connected at infinity (i.e. given any compact set C in Vthere is a compact set D containing C