Introduction to Algebraic Topology (K-Theory)

Abstract

It is the goal of this part to develop a larger portion of algebraic topology by means of a theorem of Raoul Bott concerning the topology of the general linear group GL(N,¢) on the basis of linear algebra, rather than the theory of “simplicial complexes” and their “homology” and “cohomology”. There are several reasons for doing so. First of all, it is of course a matter of taste and familiarity as to which approach to “codifying qualitative information in algebraic form” (Atiyah) one prefers. In addition, there are objective criteria such as simplicity, accessibility and transparence, which speak for this path to algebraic topology. Finally, it turns out that this part of topology is most relevant for the investigation of the index problem.

“In perhaps most cases when we fail to find the answer to a question, the failure is caused by unsolved or insufficiently solved simpler and easier problems. Thus all depends on finding the easier problem and solving it with tools that are as perfect as possible and with notions that are capable of generalization.” (D. Hilbert, 1900)