Planar Sets are aspherical
by
Andreas Zastrow
Ruhr-Universitaet Bochum (Dept. Maths.)

In order to prove the fact as stated in the title, firstly an embedding theorem is proven: Any compact and totally and locally pathwise connected subset of the plane can be homotopy-equivalently embedded into a ``Modified Sierpinski Carpet", i.e. a space which can be obtained by sometimes omitting to take out a square in the scheme of constructing the classical Sierpinski carpet. Such a space offers the opportunity of defining infinite combinatorial objects that surject on the variety of all paths. The corresponding objects are defined and the equivalence relations that (a) corresponds to the kernel of this surjection and (b) describe when two paths are relatively homotopic are described from the combinatorial viewpoint. On a combinatorial basis minimal elements in these equivalence classes are be selected, and by using these minimal representatives a system of paths can be selected the connects each point of the plane in each relative homotopy class to the base point. Such a system of paths provides a natural source for how to contract a continuous image of a higher dimensional sphere. Thanks to the combinatorial control of our system of paths and to the properties of minimal representatives it can be shown that the above construction actually gives a continuous contraction process. This proves nullhomotopy in an arbitrary situation, and so nothing is left that could generate higher homotopy groups. Hence asphericity.

Date received: January 20, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-13.