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Minimal flows and global cross sections which are disks
by
William Basener
Boston University / Colby Sawyer College
We cover some recent results concerning global cross sections which are disks. This is a generalization of the usual global cross section which is a compact manifold either without boundary or whose boundary is invariant under the flow. Every fixed point free flow on a compact manifold of dimension n > 2 has a global cross section which is an n-1 dimensional disk. We show how the first return map to the disk captures many topological properties of the flow. As an application, we give several maps of a disk which are first return maps for a global cross section to a flow on a 3 or 4 dimensional manifold, including one for S3. The example for S3 is interesting because if the first return map can be shown to be minimal it would prove the Gottschalk Conjecture (which asks whether there exists a minimal flow on S3).
Date received: January 17, 2001
Copyright © 2001 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 # cagh-08.