In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing

1. (1) singularities

2. (2) labellings

3. (3) restrictions on normal bundles.

The final notion of a ‘geometric theory’ is in fact sufficiently general to include all theories (this being the main result of Chapter VII).

A further extension, to equivariant theories, will be covered in Chapter V.

In the present chapter, we also deal with coefficients in an arbitrary geometric theory. A geometric theory with coefficients is itself an example of a geometric theory and it is thus possible to introduce coefficients repeatedly!

The chapter is organised as follows. In §1 we extend the treatment of coefficients in the last chapter to cover oriented mock bundles and in §§2 and 3 we deal with singularities and restrictions on the normal bundle. In §§4 and 5 we give interesting examples of geometric theories, including Sullivan's description of K-theory [11] and some theories which represent (ordinary) Zp-homology. Finally §6 deals with coefficients in the general theory.

COBORDISM WITH COEFFICIENTS

We now combine Chapters II and III to give a geometric description of cobordism with coefficients. It is first necessary to introduce oriented mock bundles (the theory dual to oriented bordism). We give here the simplest definition of orientation, an alternative definition will be given in §2.