C. T. C. Wall

Poincare Complexes: I

Recent developments in differential and PL-topology have succeeded in reducing a large number of problems (classification and embedding, for example) to problems in homotopy theory. The classical methods of homotopy theory are available for these problems, but are often not strong enough to give the results needed. In this paper we attempt to develop a branch of homotopy theory applicable to the classification problem for compact manifolds. A Poincare complex is (approximately) a finite cw-complex which satisfies the Poincare duality theorem. A precise definition is given in ? 1, together with a discussion of chain complexes. In Chapter 2, we give a cutting and gluing theorem, define connected sum, and give a theorem on product decompositions. Chapter 3 is devoted to an account of the tangential properties first introduced by M. Spivak (Princeton thesis, 1964). We then start our classification theorems; in Chapter 4, for dimensions up to 3, where the dominant invariant is the fundamental group; and in Chapter 5, for dimension 4, where we obtain a classification theorem when the fundamental group has prime order. It is complicated to use, but allows us to construct two interesting examples. In the second part of this paper, we intend to classify highly connected Poincare complexes; to show how to perform surgery, and give some applications; by constructing handle decompositions and computing some cobordism groups. This paper was originally planned when the only known fact about topological manifolds (of dimension >3) was that they were Poincare complexes. Novikovs proof [301 of topological invariance of rational Pontrjagin classes and subsequent work in the same direction has changed this, but we can still easily summarize the basis of the relation of Poincare complexes to smooth and PL-manifolds. The first point of difference lies in the structure group of the normal bundle; G, 0, or PL in the three cases. If this group be appropriately reduced (from G to 0 or PL), surgery can be performed as in [13] to try to construct a manifold; certain algebraic obstructions arise in the middle dimension (see [13] for the simply-connected and [26] for the general case). These algebraic structures provide the second point of difference; one in which we are particularly interested. Chapter 5 was originally written for the purpose of constructing examples to illustrate these (5.4.1 and 5.4.2). A fuller discussion of these points is planned to appear as a sequel to [26].

Quadratic forms on finite groups, and related topics

THE RESULTS of this paper form part of a series of investigations of classification problems in differential topology. Since, however, this part is entirely algebraic (or perhaps, since specific rather than general systems are under consideration, I should say number-theoretic), it has seemed desirable to separate this paper from our other publication [5]. This also permits us to consider the algebra in more detail than we shall actually need for the application.

Finiteness Conditions for CW Complexes. II

A CW complex is a topological space which is built up in an inductive way by a process of attaching cells. Spaces homotopy equivalent to CW complexes play a fundamental role in topology. In the previous paper with the same title we gave criteria (in terms of more-or-less standard invariants of the space) for a CW complex to be homotopy equivalent to one of finite dimension, or to one with a finite number of cells in each dimension, or to a finite complex. This paper contains some simplification of these results. In addition, algebraic machinery is developed which provides a rough classification of CW complexes homotopy equivalent to a given one (the existence clause of the classification is the interesting one). The results would take a particularly simple form if a certain (rather implausible) conjecture could be established.

Diffeomorphisms of 4-Manifolds

The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3manifold M3. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of M3 of a special type with respect to the considered diffeomorphism. Mathematics Subject Classification: 37B25, 37D15, 57M30.

Geometric structures on compact complex analytic surfaces

ALTHOUGH the techniques of high-dimensional manifold topology have been successfully extended by Freedman [ 143 to the topology of 4-manifolds, the results of Donaldson [ 1 l] show that one must seek quite a different pattern in studying smooth 4-manifolds, for which low-dimensional techniques may be more appropriate. Since our most coherent account of three-dimensional topology is given by Thurston’s geometrization theorem [47], this motivates the study of geometrical structures (in the sense of Thurston) in dimension 4. By “geometry in the sense of Thurston” I understand a pair (X, G,) with X a l-connected manifold, G, a Lie group acting transitively on X, such that:

Resolutions for extensions of groups

Let G be an extension of the normal subgroup K by its quotient group H . Suppose we are given free resolutions for H , K (see below for definition). We shall show how to construct from them, by a ‘twisted tensor product’, a free resolution for G , which may also be used to give the spectral sequence of the extension.

On the Classification of Hermitian Forms

We suppose throughout that R is a ring (associative, with unit), and ! a 2-sided ideal in R such that R is conplete in the /-adic topology. Write/~ for the quotient ring R/I; similarly for an R-module P, write P=P/PI=P| etc. In this section we obtain the basic relations between the algebraic K- and L-theory of R and /~. We first discuss objects (where the results are well known), then morphisms.

The topological spherical space form problem—II existence of free actions

RECENT advances in calculation of projective class groups and of surgery obstruction groups lead us to hope that it will shortly be possible to give a fairly complete account of the classification of free actions of finite groups on spheres. In the present paper, we determine which groups can so act, thus solving a problem of several years’ standing. Further, we show that these actions can be taken to be smooth actions on smooth homotopy spheres. Previously known results can be summarised as follows, where we say the finite group 7~ satisfies the “pq-condition” (p, q primes not necessarily distinct) if all subgroups if v of order pq are cyclic. 0.1. (Cartan and Eilenberg[3]). If rr acts freely on S”-‘, it has periodic cohomology with minimum period dividing n. Moreover, P has periodic cohomology if and only if it satisfies all p2-conditions. And the p* condition is equivalent to the Sylow p-subgroup zrr, of r being cyclic or perhaps (if p = 2) generalised quaternionic. 0.2. (Wolf [19]). If 7~ acts freely and orthogonally on a sphere, it satisfies all pq-conditions. Conversely, if r is soluble and satisfies all pq-conditions, free orthogonal actions exist. However, for rr non-soluble, the only non-cyclic composition factor allowed is the simple group of order 60. 0.3. (Milnor [9], see also Lee [8]). If 7~ acts freely on any sphere, it satisfies all 2p-conditions. 0.4. (Petrie [I 11). Any extension of a cyclic group of odd order m by a cyclic group of odd prime order q prime to m can act freely on S*“-‘. Petrie’s result shows that pq-conditions are not all necessary for free topological actions. it is therefore not so surprising that

Nets of Quadrics, and Theta-Characteristics of Singular Curves

The major part of this paper is devoted to enumerating all the many types of nets of quadrics in C4. An introductory section puts the invariant theory in context, and gives a framework for the classification. A quadric xT (λA0+μA1 + νA2)x of the net has dual equation XT adj (λA0+μA1 + νA2) X = 0. The adjugate system is the system of curves in the (λ,μ,ν) plane given by these equations. The set B of base points of this system on the curve 0 =Δ ≡ det (λA0+μA1+ νA2), together with the curve Δ, give system to the enumeration. In a final section of the paper the calculations are used to provide evidence for conjectures of the following type (generalizing results known when Δ = 0 is non-singular): each net determines and is determined by a square root of the canonical bundle on the curve r obtained from Δ by blowing B up; the set of square roots is an affine space over F2, and those arising are the zeros of a certain quadratic map.

Geometry of quartic curves

In recent work [ 5 ] which involved enumeration of singularity types of highly singular quintic curves, it was necessary to use rather detailed information on the geometry of quartic curves (for the case when the quintic consists of the quartic and a line). The present paper was written to supply this background. The cases of primary interest for this purpose are the rational quartics, and we concentrate on these.