Dariusz M. Wilczyński

Group actions on the complex projective plane

0. Introduction. In this paper we consider homologically trivial, locally smooth actions of finite and compact Lie groups on topological 4-manifolds having the same homology as CP2, the complex projective plane. Our main result states that all such actions are lowally complex linear and that the acting groups can also act linearly on CP2. In particular, we obtain the classification of groups acting on both manifolds homotopy equivalent to CP2: G acts on the complex projective plane (resp. on the Chern manifold) if and only if G is isomorphic to a subgroup (resp. a pseudofree subgroup) of PU(3). The paper is organized as follows. In §1 a precise statement of the main result is given. §2 examines tangent representations of some cyclic groups; the main tool being the G-signature theorem. The relevant number theoretic computations are carried out in Appendix 1. §§3 and 4 discuss abelian and nonabelian groups in terms of the numbers of fixed points (Euler characteristic of the fixed point set). §5 proves that all tangent representations are complex. §§6-9 discuss several classes of groups acting without fixed points. The terminology that we use to distinguish among different classes of fixed point free groups is adopted from the linear case. §10 contains the proof of our main theorem and, finally, §11 is devoted to group actions on the Chern manifold. For completeness, a list of fixed point free linear groups is included in Appendix 2. ACKNOWLEDGMENT. I would like to thank Professor John Ewing for suggesting this problem to me and also for his support and encouragement. I have been informed that I. Hambleton and R. Lee have independently obtained results similar to those presented here.

Periodic maps on simply connected four-manifolds

IN THIS paper we describe certain invariants of periodic maps on closed, simply connected four-manifolds with the purpose of classifying such maps up to homeomorphism. The manifolds themselves are classified by Freedman [9] in terms of the intersection pairing on second homology and the Kirby--Sicbcnmann invariant. Gcncralizing Freedman’s result to some nonsimply connected four-manifolds Hambleton and Kreck [IO] have recently shown that in the case of a cyclic fundamental group of odd order the manifold is determined up to homeomorphism by the same two invariants togcthcr with the fundamental group (with the intcrscction pairing being understood now as dcfincd on the torsion free part of the second homology group). A similar statement for cvcn order groups fails to bc true. although its failure is less dramatic for algebraic surfaces as follows from the work by the same authors [I I]. From the point of view of the fundamental group acting on the universal cover these results concern the topological classification of fret periodic maps on simply conncctcd four-manifolds. In gcncral, howcvcr. a periodic map is allowed to have fixed points or periodic points of period smaller than the map period. When this happens, the quotient orbifold has singularities. We shall show that in the cast of isolated singularities one can still classify periodic maps by means of some invariants associated with this orbifold, although in the singular case one must also take into account the torsion part of its second homology. In the more rigid situation of a free periodic map no torsion information is needed. This can bc partially accounted for by the fact that many four-manifolds do not support any such maps. On the other hand, by a result of Edmonds [5]. every closed, simply connected fourmanifold supports infinitely many periodic maps with isolated fixed points. The paper is organized in four sections. In Section 2 we discuss the homotopy classification of locally linear, pseudofree actions of finite groups on simply connected fourmanifolds. There we introduce some terminology and prepare the obstruction theory needed for the results of the next two sections. Our main result is the topological classification of semifree periodic maps of odd period. This is the content of Theorem 3.1. A related but somewhat weaker result for periodic maps of even period is contained in Theorem 3.2. As an application of our techniques, we prove in Section 4 that any locally linear, pscudofree action of a finite group on the complex projective plane is topologically conjugate to a linear action. This last statcmcnt about actions on CP’ has also been asserted in [I93 in the context of finite group actions on the Chern manifold. There we concentrated our efforts on actions without fixed points referring the reader for the proof in the semifrce case to the combined results of [7] and [ 121. This point requires now further elaboration for maps of composite period as the authors of [ 121 have meanwhile added to their result a technical hypothesis of

Representing homology classes by locally flat surfaces of minimum genus

A necessary and sufficient condition is given for a nontrivial homology class of a simply connected 4-manifold to be represented by a simple, topologically locally flat embedding of a compact Riemann surface. This result extends the embedding theorem of the 2-sphere, proved by the authors in K-Theory7 (1993), to the case of a surface of any genus. As a consequence, the computation of the least genus of a locally flat surface is obtained for any homology class of even divisibility. The computation of the least genus is also settled for certain (and perhaps all) classes of odd divisibility. In the case of a minimum genus surface it is shown that the constructed embedding is often nonsmoothable. For example, all but nine homology classes in the complex projective plane contain such a nonsmoothable surface. A similar result holds for any class of nonnegative self intersection in a K3 surface.

Genus inequalities and four-dimensional surgery

Abstract We obtain a new genus inequality for a topologically locally flat surface in a 4-dimensional manifold. This inequality provides new information about the fundamental group of the complement of such a surface and in many cases gives the minimum genus among surfaces within the same homology class. The general problem of finding an embedded surface of a small genus allowed by the inequality remains undecided and is directly related to the surgery conjecture of 4-dimensional topology.