In this final chapter we draw together all the ideas of the previous chapters by showing that an arbitrary cohomology theory is a geometric theory in an essentially unique way. Thus the geometric definitions of coefficients, operations etc. all apply to an arbitrary theory. This is achieved by examining the geometry of CW complexes. We will define a new concept, that of a transverse CW complex, which has all the geometric properties of ordinary cell complexes. In particular, it has a dual complex and transversality constructions can be applied. The transversality theorem (in §2) is a version for a CW complex of the theorem in part II §4. However the proof uses even less and is elementary!

If X is a based transverse CW complex and X* its dual complex, then the subcomplex χ(X) ⊂ X* consisting of dual objects other than the object dual to the basepoint, behaves with X exactly like the base of a Thorn complex behaves with the whole complex. A map f : M → X can be made transverse to χ(X) (in fact the transversality theorem does exactly that) and the transverse map is determined by its values near f−1χ(X). In this way, an arbitrary spectrum is seen to be a ‘Thom’ spectrum for a suitable theory (with singularities), see §§4 and 5.

Another consequence of this chapter is that CW complexes, already useful as homotopy objects, now have a beautiful intrinsic geometric structure.