Vorlesung "`Algebraic Topology III, WS 13/14"'

Contents of the course:

The last semester had two foci: the first was the general development of homology and cohomology, with cup product, universal coefficient theorems,\ldots The second focus then was the specific homological properties of manifolds, in particular Poincare duality.

In this third part, we go back to general features of (co)homology.

Our first theme will be the relation between homotopy groups and homology groups. We start with the \textbf{Hurewicz theorem}, which states that the first homology group (with integer coefficients) is the abelianization of the fundamental group, and then generalize this to the \textbf{Whitehead theorem}, which lets us in good situations compute the homotopy groups from the homology groups and allows us to detect from the effect in homology when a map is a homotopy equivalence.

A second focus will be the idea of \emph{classifying spaces}. We will show that there are spaces $K(A,n)$ for each (abelian group) $A$ and each $n\in\integers$ such that there is a canonical isomorphism $H^n(X;A)\iso [X,K(A,n)]$ and will derive information about these spaces. This is the content of the \textbf{Brown representability theory}. It paves the way for a systematic study of (co)homology theory using classifying spaces which leads to a very powerful tool and intensely studied branch of topology: \textbf{spectra} (of topologists) and their homotopy theory.

A third major focus will then be the introduction of a powerful homological tool, namely \emph{spectral sequences}. We will derive the underlying homological algebra and then construct in particular the \emph{Leray-Serre spectral sequence} for a fibration. This gives information about the (co)homology of the total space of a fibration in terms of the (co)homology of the base and the (co)homology of the fiber. This theorem is very powerful (although the inner workings of spectral sequences are a bit complicated) and we will give numerous applications.

We will also derive a spectral sequence (generalizing the universal coefficient theorem) which relates an arbitrary generalized cohomology theory to ordinary cohomology, the \textbf{Atiyah-Hirzebruch spectral sequence}.

Literatur (among others):