Slawomir Kwasik

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.

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 study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

-manifolds

Abstract We present almost complete computations of the surgery obstruction for 3-manifolds, as well as consequences for a suitable version of the structure set in dimension three. Applications leading to new results and conjectures concerning the Borel conjecture in dimension ⩾5 and UNil-groups are also discussed.

Symmetric symplectic homotopy K3 surfaces

The Atiyah-Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic K-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraicK-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new innite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.

Three-dimensional surgery theory, UNil-groups and the Borel conjecture

By definition, two homeomorphisms f0 ,fi : X + Y are isotopic if there is a homotopy H : X x [0, l] + Y such that each map H, = H I X x {t} is a homeomorphism and Hi =f;: for i = 0,l; if X is a locally compact Hausdorff space then the map H, :X x [0, l] + Y x [0, 1) defined by H, (x, t) = (H(x, t), t) is a homeomorphism. Two homeomorphismsfO,fi as above are said to be pseudo-isotopic if there is a homeomorphism H, :X x [0, l] + Y x [0, l] such that H, (x, i) = (h(x), i) for i = 0,l; thus isotopy implies pseudo-isotopy for reasonable spaces. The concept of isotopy formalizes some basic intuitive concepts of topology (cf. [12, p. 1771). On the other hand, the weaker concept of pseudo-isotopy turns out to be considerably more tractable (e.g. see Cl]), and it is a reasonable substitute for isotopy in many situations (in particular, [12, Theorem 8.2.3, p. 1851 remains valid if one replaces “isotopic” by “pseudo-isotopic”; cf. [ 19, paragraph 5.4, p. 540, and the discussion on p. 5231). It is also known that pseudo-isotopy implies isotopy for homeomorphisms of simply connected manifolds in all dimensions greater than three (see c7, 23, 241). The purpose of this paper is to illustrate a previously undetected phenomenon in the topology of 3-dimensional manifolds. Namely, we show that on closed (reducible) 3manifolds there exist homeomorphisms that are pseudo-isotopic, but not isotopic, to the identity. Detailed statements are given at the end of the Introduction. In higher dimensions there are systematic families of pseudo-isotopic but not isotopic homeomorphisms. For example, such homeomorphisms exist on the n-dimensional torus T” for all n 2 4 (see [7, 13, 161). The construction of these examples depends on the complicated and powerful algebraic machinery developed by Hatcher and Wagoner [9]; (also see [14]). In fact, the known constructions of pseudo-isotopic but no isotopic homeomorphisms generally involve some substantial algebraic input (e.g. see [ 13, 27, lo]). The adaptability of this input to the 3-dimensional case is at best problematic, and hence our approach to the relation beween pseudo-isotopy and isotopy problem is different from those employed in the papers cited. For large classes of closed irreducible 3-manifolds it is already known that homotopic homeomorphisms are isotopic (references for many cases are collected in [17], Section 1) and well-known geometrization conjectures in the theory of 3-manifolds imply this is true for arbitrary irreducible 3-manifolds. Thus it is not surprising that our examples involve

Toral and exponential stabilization for homotopy spherical spaceforms

A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to complete classication of such manifolds with thickness 1, 3 or 2k, for k 1. On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.

PSEUDO-ISOTOPIES OF 3-MANIFOLDS

We study the homeomorphism types of manifolds h-cobordant to a fixed one. Our investigation is partly motivated by the notion of special manifolds introduced by Milnor in his study of lens spaces. In particular we revisit and clarify some of the claims concerning h-cobordisms of these manifolds.

Tangential thickness of manifolds

In topology and geometry it is often instructive to consider objects with isolated singularities. Frequently such objects turn out to be relatively tractable and to provide useful insights into more general situations. For actions of finite cyclic groups on manifolds the standard notion of singularity is a point that is left fixed by some non-trivial element of the group (but not necessarily by the whole group). If the singular set is isolated the action is said to be pseudofree . Special cases of pseudofree actions have been studied in several independent contexts; in particular, previous papers of Cappell and Shaneson [4] and the authors [15] considered classes of ‘nice’ pseudofree actions on spheres. References to other works are given at the beginning of [15] (and in the meantime there has also been considerable further activity).

How different can h-cobordant manifolds be?

One of the best known three-dimensional manifolds is the Poincare homology three-sphere 1(2, 3,5); this closed 3-manifold has the same homology groups as the standard sphere S3 but is not simply connected (see [Pnc, p. 106] or [Br2, Section 1.8]). Numerous investigations during the past century have shown that this manifold has many remarkable properties; several equivalent descriptions of 1(2,3,5) are summarized in [KiS]. From the viewpoint of transformation groups, one noteworthy property is that 1(2, 3, 5) is the only nonsimply connected homology sphere admitting a transitive action of a compact Lie group [Brl]. The manifold 1(2, 3,5) also figures importantly in regularity questions for group actions on spherelike manifolds. Linear (or orthogonal) actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions, and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions. One distinguishing feature of linear actions is that their fixed point sets (if nonempty) are also spheres. The well-known results of P. A. Smith (cf. [Br2]) show that continuous p-group actions on homology spheres all resemble linear actions in this respect; i.e., the fixed point sets are always Zp-homology (cohomology) spheres. On the other hand, outside the family of p-groups, there are many exotic group actions. Perhaps the most elementary example of this type involves 1(2, 3,5). If A is the subgroup of the rotation group SO3 given by the symmetries of a regular icosahedron (so that A is isomorphic to the alternating group A5), then the quotient space S03/A is the Poincare homology 3-sphere and the induced action of A5 ? A has exactly one fixed point. This example suggests that general actions of finite groups on homology spheres can be quite different from linear actions, and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action.