In this chapter we extend the treatment of coefficients in Chapter III to cover sheaves of abelian groups. We work always with pl cobordism but everything that we say can be extended to an arbitrary theory under the conditions of IV 6. 4. §§4 and 5 in fact extend unconditionally. The general definition of sheaves of coefficients does not have all the best properties one would hope for and we will explain where the difficulties lie at the start of §4.

In §1 we recall the basic properties of stacks and sheaves and in §2 we define the theory of mock bundles with coefficients in a stack. The definition is functorial on the category of all stacks of abelian groups. The main theorem asserts that, if the stack is ‘nice’, then there is a spectral sequence expressing the relation between simplicial cohomology and cobordism with coefficients in the stack. In §3 cobordism with coefficients in a sheaf is defined by means of a simplicial analogue of the Čech procedure. In §4 we discuss an extension of the methods used in the previous sections and give an example of ‘Poincaré duality’ between bordism and cobordism with coefficients in the sheaf of local homology of a Zn-manifold. Finally in §5 we extend the methods further and give examples which suggest the existence of a bordism version of the Zeeman duality spectral sequence [1].

The main purpose of this chapter is to axiomatise the passage from functors defined on pl cell complexes to homotopy functors defined on polyhedra. Principal examples are simplicial homology and mock bundles (see Chapter II).

Our main result, 3. 2, states that the homotopy category is isomorphic to the category of fractions of pl cell complexes defined by formally inverting expansions. Thus to define a homotopy functor, it is only necessary to check that its value on an expansion is an isomorphism. Analogous results for categories of simplicial complexes have been proved by Siebenmann, [3].

In §4, similar results are proved for ∆-sets. This gives an alternative approach to the homotopy theory of ∆-sets (compare [2]).

In §6 and §7, we axiomatise the construction of homotopy functors and cohomology theories. Here we are motivated by the coming application to mock bundles in Chapter II, where the point of studying cell complexes, rather than simplicial complexes, becomes plain as the Thorn isomorphism and duality theorems fall out.

The idea of using categories of fractions comes (to us) from [1] where results, similar to those contained in §4 here, are proved.

Throughout the notes we use basic pl concepts; for definitions and elementary results see [4, 6 or 8].

DEFINITIONS

Ball complexes

Let K be a finite collection of pl balls in some Rn, and write ∣K∣ = υ{σ:σ ∈ K}, Then K is a ball complex if

(1) ∣K∣ is the disjoint union of the interiors of the σ ∈ K, and

(2) σ ∈ K implies the boundary is a union of balls of K.

The purpose of these notes is to give a geometrical treatment of generalised homology and cohomology theories. The central idea is that of a ‘mock bundle’, which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalised bordism theory. Thus every theory has both cycles and cocycles; the cycles are manifolds, with a pattern of singularities depending on the theory, and the cocycles are mock bundles with the same ‘manifolds’ as fibres.

The geometric treatment, which we give in detail for the case of pl bordism and cobordism, has many good features. Mock bundles are easy to set up and to see as a cohomology theory. Duality theorems are transparent (the Poincaré duality map is the identity on representatives). Thorn isomorphism and the cohomology transfer are obvious geometrically while cup product is just ‘Whitney sum’ on the bundle level and cap product is the induced bundle glued up. Transversality is built into the theory – the geometric interpretations of cup and cap products are extensions of those familiar in classical homology. Coefficients have a beautiful geometrical interpretation and the universal coefficient sequence is absorbed into the more general ‘killing’ exact sequence. Equivariant cohomology is easy to set up and operations are defined in a general setting. Finally there is the new concept of a generalised cohomology with a sheaf of coefficients (which unfortunately does not have all the nicest properties). The material is organised as follows.

We describe here the theory of mock bundles. This is a bundle theory giving rise to a cohomology theory which in the simplest case is pl cobordism. In this interpretation, all the usual products have simple definitions, and the Thorn isomorphism and duality theorems have short transparent proofs. Another feature is that mock bundles can be composed yielding a cohomology transfer. The theory also provides a short proof of the pl transversality theorem [12; 1. 2], In a final section, classifying ∆-sets are constructed. The construction is similar to Quinn's [7; §1].

MOCK BUNDLES AS A COHOMOLOGY THEORY

Let K be a ball complex. A q-mock bundle ξq/K with base K and total space consists of a pl projection pξ: Eξ → ∣K∣ such that, for each is a compact pl manifold of dimension q + dim σ, with boundary We denote by ξ(σ), and call it the block over σ.

The empty set is regarded as a manifold of any dimension; thus ξ(σ) may be empty for some cells σ ∈ K. Therefore, q could be negative, and then ξ(σ) = ø if dim σ< -q. The empty bundle ø/K has the empty set for total space, and is a q-mock bundle for all q ∈ Z.

Figure 2 shows a 1-mock bundle over the union of two 1-simplexes.

Mock bundles ξ, η/K are isomorphic, written ξ ≅ η if there is a pl homeomorphism h: Eξ → Eη which respects blocks; h(ξ(σ)) = η(σ) for each σ ∈ K.

In §1 of this chapter we describe a further extension of mock bundles, to the equivariant case – the theory dual to equivariant bordism – and in §2 give a general construction which includes power operations and characteristic classes. The remainder of the paper is concerned with the case of Z2-operations on pl cobordism. In §3 we expound the ‘expanded squares’ (‘expanded’ rather than the familiar ‘reduced’ because of our indexing convention for cohomology) and in §4 we give the relation with torn Dieck's operations [5], §5 describes the characteristic classes associated to Z2-block bundles and in §6 we give a result inspired by Quillen [3] which relates the total square of a mock bundle with the transfer of the euler class of its twisted normal bundle. This leads, in some cases, to the familiar connection between characteristic classes and squares. Finally in §7 we give an alternative definition of squares, based on trans versatility. This is like the ‘internal’ definition of the cup product (see II end of §4).

EQUIVARIANT MOCK BUNDLES

Let G be a finite group and X a polyhedron. By a G-action on X we mean a (pl) map G × X → X satisfying

1. (i) for all g1 g2 ∈ G and x ∈ X, g1(g2x) = (g1g2)x.

2. (ii) if e ∈ G is the identity then ex = x for all x ∈ X.

3. If X, Y are G polyhedra then a map f: X → Y is a G-map if f commutes with the G-action.

In this chapter we give a geometric treatment of coefficients in oriented pl bordism theory. The definition (although not all the theorems) extend to other geometric homology theories and this extension will be covered in Chapter IV, as will the extension to cobordism (mock bundle) theories. Application to general homology theories and connection with other definitions of coefficients will be covered in Chapter VII.

There are two good definitions of coefficients:

1. For a short resolution ρ of an abelian group G we define coefficients in ρ by labelling with generators and introducing one stratum of singularities of codimension 1 corresponding to the relations (see §1).

2. We allow labelling by any group elements, and singularities corresponding to any relation and then, in the bordisms, allow singularities of codimension 2 corresponding to ‘relations between relations’ (see §3).

Definition 1 is very simple geometrically while definition 2 is functorial in G. To prove equivalence of the two definitions involves a further definition, for longer resolutions (in §2). The basic geometrical trick is resolution of singularities and appears in the proof of the universal coefficient sequence in §2. The universal coefficient sequence itself can be seen as the measure of the obstruction to resolution of the final singularity. In §3 it is seen that the universal coefficient sequence is natural for G; consequently by [3] it splits for a large class of abelian groups, including all groups of finite type.

In the summer semester of 1972 I gave a course of lectures on the local theory of differentiable maps at the University of Freiburg. These lectures have formed the basis for the first thirteen chapters of the book, the next three chapters having been written for a summer school organised by the Studienstiftung des deutschen Volkes. My students were responsible for removing many mistakes from the original manuscript which has now been translated into English by L. Lander. He has also made a number of improvements and corrections and provided the last chapter together with its pictures and list of publications. The later chapters discuss a subject which has been the real motivation for writing the book: classical catastrophe theory.

We have both profited greatly from a lecture course on catastrophe theory by K. Jänich, given in Regensburg during the winter semester 1971/72, which contained most of the information and pictures presented in chapter 17.

A small number of copies of the German text of the present book were printed for our students under the title: Der Regensburger Trichter, Band 3, Differenzierbare Abbildungen.

On the pages that follow, the reader will not find any new results or methods. Our purpose is to make it easier for those students, who have properly understood the basic lecture courses on analysis and possess a basic knowledge of algebra, to learn about recent work on differentiable maps, in particular, the mysteries of catastrophe theory.