Manifolds and Bordism

Abstract

In this chapter we consider another collection of homology theories, the bordism theories, which arise from manipulations with manifolds, although in a different sense from that in which the K-theories have their origins in manipulations with bundles. We begin with the definition of a manifold, but we shall not prove all the theorems about manifolds which we shall employ, because to do so would lead to a chapter out of all proportion with the rest of the book (it is recommended that the reader unfamiliar with the theory of manifolds see [69] or [52]). Instead we go on to define the Thorn complex of a vector bundle and the various Thorn spectra MG. Then we sketch the proof of Thorn that the homology theories associated with these spectra can be described in terms of singular manifolds.