241536 Homotopy Theory and Topological Complexity (V) (SoSe 2020)

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Requirements for participation, required level

This course is aimed primarily at students who have some experience in
topology and alegbraic topology.

The course will begin by introducing its participants to the rudiments
of homotopy theory. We will develop the basics of fibre and cofibre
sequences, and subsequently extend these fundamental ideas by defining
homotopy pushouts and pullbacks, and other basic homotopy (co)limits.
A strong emphasis will be placed throughout on the role of duality in
the unstable setting, and the primary goal of the course is to
introduce and prove Mather's Cube Theorems. These two theorems provide
a fascinating and unexpected link relating two fundamentally different
approaches to unstable homotopical problems.

A theme present throughout the course that we will look to for a
source of examples will be that of Topological Complexity. This is a
topological invariant which was originally introduced to study robot
motion planning problems. As it turned out, it presents a
homotopy-theoretic problem with deep theoretical implications, and has
been the topic of much recent research. Although only a small part of
the course, we hope to give some applications for the abstract theory
which we will develop, and spend some time discussing the more
exciting topics which have been introduced in the last few years.