Kai-Uwe Bux (Bielefeld): Combinatorial Morse Theory: how and why

Thursday, 25.11.2010 16:30 im Raum M5

Mathematik und Informatik

The idea of Morse theory is that the topology of a manifold M can be understood with the aid of a sufficiently nice smooth function h : M --> R. We think of h as a height on M and consider sublevel sets M_r := { x in M | f(x) <= r }. As the level r increases, the sublevel sets M_r grow; and the goal is to control how the homotopy type of M_r changes during this process. The goal of combinatorial Morse theory is to understand the homotopy types of a simplicial or CW-complex X by means of a function h : X --> R. Instead of sublevel sets, one considers sublevel complexes, i.e., subcomplexes X_r spanned by all vertices up to height r. Under favorable conditions, one can again control the change how the homotopy type of X_r varies as r increases. I shall explain and demonstrate the method, giving plenty of examples and applications. The complexes arising will be mostly Eilenberg-MacLane spaces for groups. Hence, understanding their homotopy type yields information about the homology of the underlying group.


Angelegt am Tuesday, 26.10.2010 10:02 von gstei_02
Geändert am Thursday, 18.11.2010 10:23 von gstei_02
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