Reinhard Schultz

Vanishing of whitehead torsion in dimension four

IN TOPOLOGY and geometry it is often useful or important to recognize n-dimensional manifolds that are isomorphic to products X x [0, l] of a compact manifold X with the unit interval. The s-cobordism theorem of Barden, Mazur and Stallings (see [13, 171) states that a compact (n + 1)-manifold W of dimension n + 1 2 6 with boundary 8 W = M,, u Ml is a product M0 x [0, l] if and only if W is an h-cobordism (i.e., W is homotopically a product) and a certain algebraic invariant z( W; M,,) called the Whitehead torsion is trivial (for n = 4 the results of [8,9] yield a topological version of the s-cobordism theorem for a large class of the fundamental groups). The Whitehead torsion invariant takes values in an abelian group called the Whitehead group of M that depends only on the fundamental group and is denoted by Wh(n,( W)). The vanishing condition on Z( W; M,-,) is essential because the Whitehead group is nonzero in many cases and every element can be realized as the torsion of some h-cobordism ( W”+l; M”,, My) for n 2 4; a proof in the case n 2 5 appears in [13], and the case n = 4 is treated in [2]. In fact, it is possible to choose the h-cobordisms so that M,-, ?z Ml (compare [2, Prop. 3.3 and the first sentence in the paragraph on p. 515 before Prop. 3.21). On the other hand, our understanding of the case n = 3 is still quite limited. For example, the following realization question from [16, Problem 4.91 is still open.

Equivariant surgery theories and their periodicity properties

Summary: Background material and basic results.- to equivariant surgery.- Relations between equivariant surgery theories.- Periodicity theorems in equivariant surgery.- Twisted product formulas for surgery with coefficients.- Products and periodicity for surgery up to pseudoequivalence.

Nonlinear analogs of linear group actions on spheres

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

DESUSPENSION OF GROUP ACTIONS AND THE RIBBON THEOREM

action of a group G on a sphere, it is natural to ask if the action can in some sense be realized as a single or multiple suspension of an action on a lower-dimensional sphere. In many cases the answer to such questions have extremely far-reaching implications for problems of independent interest. For example, the entire classification theory of free involutions on spheres is built around the desuspendability question (compare Lopez de Medrano [25], Browder-Livesay

Homological properties of periodic homeomorphisms of

Given a mapping f from a space X into itself, it is often possible to obtain significant information about f from the algebraic endomorphisms induced by f on the homology and cohomology of X. For example, ifX is a compact polyhedron or topological manifold, then the Lefschetz fixed-point theorem relates the existence of fixed points for f to a function of the eigenvalues of the rational homology or cohomology self-maps defined by f (i.e., the Lefschetz number; compare [G-H]). Frequently, some natural assumptions on f and X allow one to retrieve much more information about f than in the general case. In particular, if X is a compact differentiable manifold and f is a diffeomorphism such that fN lx for some X (in other words, if f is periodic), then the Lefschetz number of f equals the Euler characteristic z(F) or the set of points F left fixed by f (compare [Kob]). Furthermore, if f 4:1 but fP for some prime p, then the action off on the homology groups Hk(X; Z) makes the latter into Z[Zp]-modules, and results of R. Swan [Sw1] imply strong restrictions on these modules. For example, if X is an (n 1)connected 2n-manifold (n > 2) and e C is a primitive pth root of 1, then the Z[]-module

4

A WELL-KNOWN theorem of Cartan and Hadamard states that a complete Riemannian n-manifold with everywhere nonpositive sectional curvature has a universal covering that is diffeomorphic to Euclidean n-space ([23, p. 184). This is one of many motivations for interest in spaces with contractible universal coverings; such spaces are called aspherical. Over the past dozen years considerable information has been obtained regarding group actions on such manifolds (compare[9-12]), and recently analogous questions for manifolds admitting suitably nontrivial maps into certain aspherical manifolds have been considered. In particular, the second named author has considered group actions on closed n-manifolds that map to the n-torus by a degree one map[22,23] and Schoen and Yau have considered smooth or real analytic actions on n-manifolds that- map to closed Riemannian manifolds of nonpositive curvature [21]. (Compare also [29]). In this paper, we shall prove results that overlap or contain the results of Borel-Conner-Raymond on aspherical manifolds, the results of [22], and the theorems of Schoen and Yau on group actions. One class,of results deals with manifolds that map onto a closed aspherical manifold by a map of nonzero degree; this class of manifolds contains all closed aspherical manifolds and in fact all connected sums of closed aspherical manifolds with other manifolds. The results of Borel, Conner and Raymond generalize completely; the only connected compact Lie groups that can act are tori, and all such actions are injective in the sense of [9]. In fact, if f : A4 + N is the map from M to the closed aspherical manifold N and i : T + M is an orbit map, then the composite is injective. Furthermore, if the map f has degree one, then (i) a finite group G acting on M with fixed points induces an injection from G to Aut(n,(M, WI,,)), where m. is a fixed point, (ii) if rr,(M, mo) is centerless, then there is an injection of G into the outer automorphism group of r,(M, mo) even if G has no fixed points. The results of [21] also indicate that one has strong restrictions on the differential symmetry of M, even if dim N > dim M, so long as f maps the fundamental class of M sufficiently nontrivially and N has nonpositive curvature. Our results show that similar restrictions also hold on the topological symmetry of M. Technically speaking, the central ideas of the proofs are (i) fairly explicit descriptions of the fundamental groups of certain orbit spaces (ii) some general facts about the fundamental groups of aspherical manifolds in general and nonpositively curved manifolds in particular. This paper consists of three sections. The first provides the necessary information on orbit spaces and their fundamental groups, while the second discusses group actions on manifolds with nonzero degree maps into aspherical manifolds and the third discusses group actions on manifolds with suitable maps into higher dimensional manifolds of nonpositive curvature.

-manifolds

Smooth circle actions are constructed on certain homotopy spheres not previously known to admit such actions. In this paper we shall prove the following two results: PROPOSITION A. Let ?8 be any homotopy 8-sphere. Then there is a smooth semifree circle action on ?8 with S4 as itsfixed point set. PROPOSITION B. Let 210 be any homotopy 10-sphere bounding a spin nanifold. Then there is a smooth semifree circle action on ?210 with S4 as its fixed point set. Combining these with other results, we know that any smooth manifold ?lt which is piecewise-differentiably homeomorphic to Sn, bounds a spin manifold, and satisfies n _ 13 has a smooth circle action. The above propositions imply the cases n= 8, 10, while the cases n=7, 11, 12 follow because Ir? 0=OP1 in these cases and every homotopy sphere in OP,+1 (n>5) has a semifree circle action with a homotopy (n-4)-sphere as its fixed point set (e.g., see [3]). Finally, the cases n=9, 13 follow from the above remark on OP,+1 and results of Bredon [1]. Undoubtedly, the central difficulty in obtaining connected Lie group actions on homotopy spheres is the lack of a manageable construction for an arbitrary such manifold. The value of such a realization is obvious in the construction of large orthogonal group actions on homotopy spheres bounding nr-manifolds. Bredons construction of smooth S1 and S3 actions on homotopy spheres in the image of the Milnor-MunkresNovikov pairing [1] is another illustration of the usefulness of an explicit construction for a given homotopy sphere. In this paper we shall show that certain homotopy spheres in the image of the Milnor plumbing pairing 0: (SO) x 7Tp(SOqF),+,+l (see [4] or [5]) also have smooth circle Received by the editors July 26, 1971 and, in revised form, March 24, 1972. AMS 1970 subject classf/lcations. Primary 57D60, 57E15, 57E25; Secondary 57D50, 57D55.

COMPACT GROUP ACTIONS AND MAPS INTO ASPHERICAL MANIFOLDS

In topology it is often profitable to characterize the algebraic invariants of spaces or morphisms by axioms. Of course, the most basic examples along these lines are the Eilenberg-Steenrod axioms for homology and cohomology [EiS], and in many cases there are axiomatic characterizations for more refined algebraic structures that exist in homology and cohomology. For example, axioms for the Steenrod cohomology operations appear in [EpS], and axioms for various sorts of transfers known in the nineteen sixties were described in unpublished work of J. M. Boardman [Bo]. In this paper we shall describe axioms for the bundle transfers considered by the first named author and D. H. Gottlieb in [BG2–3] and from a different perspective by A. Dold, M. Clapp and D. Puppe in [Cl, D1–3, DP]. Previous work of F. Roush [Rou] and L. G. Lewis [L] describes axioms for the special case of finite coverings and bundles whose fibers are compact Lie groups that act smoothly on the fibers; another axiomatic characterization of finite covering transfers appears in [BeS, Sect. 13]. The characterization in this paper is valid for topological fiber bundles for which the fibers and the bases are finite-dimensional compact ANR’s and finite CW complexes respectively. We shall also describe axioms for the homomorphisms in generalized homology and cohomology theories that are determined by bundle transfers.

Circle actions on homotopy spheres bounding plumbing manifolds

If a smooth manifold has a Riemannian metric with positive scalar curvature, it follows immediately that the universal covering also has such a metric. The paper establishes a converse if the manifold in question is closed of dimension at least 5 and the fundamental group is an elementary abelian p-group of rank 2, where p is an odd prime. A conjecture of J. Rosenberg [Rs3] states that a closed smooth manifold with odd order fundamental group and dimension ≥ 5 admits a Riemannian metric with positive scalar curvature if and only if its universal covering admits such a metric. Standard considerations involving transfers (cf. [KS1]) reduce the conjecture to the special case of p-groups (where p is an odd prime). Results of Rosenberg [Rs1]– [Rs3] and S. Kwasik and the author [KS1] prove the conjecture if the fundamental group G is a cyclic p-group. In this work we study the conjecture when G is a finite abelian p-group. The following interim conclusion reflects many of the basic ideas and disposes of the first examples not covered by [KS1], [Rs1]–[Rs3]. Theorem. Let p be an odd prime, and let M be a closed smooth manifold of dimension n ≥ 5 with fundamental group Zp × Zp. Then M admits a Riemannian metric with positive scalar curvature if and only if its universal covering M does. In [RsS] J. Rosenberg and S. Stolz prove a stable result that yields a weaker conclusion for G ∼= Zp × Zp but applies to all finite groups. In particular, if G is a finite group of odd order and M is a closed smooth manifold with fundamental group G, their result states that the universal covering M has a Riemannian metric with positive scalar curvature if and only if some product M ×X × · · · ×X does, where X is an 8-dimensional Bott manifold; i.e., it is a spin manifold whose Â-genus is equal to 1. Although the methods should yield quantitative information on the number of factors of X that are needed, it is not clear how precise such estimates would be. Our result implies that if G ≈ Zp×Zp, then no copies of X are needed if n ≥ 5 and at most one copy of X is needed if n < 5. A more geometric proof of this result in dimensions ≤ 2p− 2 was obtained independently by Stolz (unpublished). It seems likely that the methods of this paper can be combined with the multiple Kunneth formula decomposition of [Hn] and finite induction to prove at least a semistable version of Rosenberg’s conjecture for all elementary abelian p-groups Received by the editors February 13, 1995 and, in revised form, September 13, 1995. 1991 Mathematics Subject Classification. Primary 53C21, 55N15, 57R75; Secondary 53C20, 57R85. c ©1997 American Mathematical Society

Axioms for bundle transfers and traces

Unpublished results of S Straus and W Browder state that two notions of homotopy equivalence for manifolds with smooth group actions—isovariant and equivariant— often coincide under a condition called the Gap Hypothesis; the proofs use deep results in geometric topology. This paper analyzes the difference between the two types of maps from a homotopy theoretic viewpoint more generally for degree one maps if the manifolds satisfy the Gap Hypothesis, and it gives a more homotopy theoretic proof of the Straus‐Browder result. 55P91, 57S17; 55R91, 55S15, 55S91