The possibilities for new kinds of locally linear periodic maps of non-prime order on closed, simply connected 4-manifolds with positive definite intersection pairings are explored. On the one hand, certain permutation representations on homology are ruled out under appropriate hypotheses. On the other hand, an interesting homologically nontrivial, pseudofree, action of the cyclic group of order 25 on a connected sum of ten copies of the complex projective plane is constructed. 1

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental

Periodic maps of composite order on positive definite 4–manifolds

We show that any 3–dimensional homotopy lens space M 3 that is simplehomotopy equivalent to a lens space L(p,q) is topologically s-cobordant to the lens space. It follows that M has the same multi-signature as L(p,q) and the action of π1(M) on the universal cover of M embeds in a free orthogonal action on S 7.

Abstract. Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order ≥ 7 is proved and nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order ≥ 5 is explicitly determined, provided that the action is homologically nontrivial. 1.

Abstract. Let G be a cyclic group of order 3, 5 or 7, and X = E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of [18]. 1.

Abstract. We show that every closed, simply connected, spin topological 4-manifold except S 4 and S 2 × S 2 admits a homologically trivial, pseudofree, locally linear action of Zp for any sufficiently large prime number p which is nonsmoothable for any possible smooth structure. 1.