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Parametrized Morse Theory on Foams and Network TQFT
by
Charles Frohman
The University of Iowa
Coauthors: Dennis Roseman
We analyze generic codimension zero and codimension one singularities of smooth functions on Foams.
A foam is a singular suface whose singularities are modeled on a cylinder over the letter Y. The facets of a foam are oriented and they induce a consistent orientations on the seams and the orientation on the boundary makes the boundary into an oriented trivalent graph whose vertices are sources or sinks. Finally, there is a choice of cyclic ordering of the facets coming up to any seam, which leads to a cyclic ordering of the edges coming up to a boundary web. The facets of a foam are colored.
A network TQFT assigns a vector space to any colored web, and a linear map to each foam.
We use the Morse theory of smooth functions on foams to derive basic data, and the consistency conditions on that data needed for a network TQFT to exist.
This research is inspired by the paper "Network TQFT" by Serge Natanzon. Our Foams are both more restrictive and less restrictive than his. We do not have vertices, yet our facets can have genus and his cannot.
Date received: March 25, 2008
Copyright © 2008 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cawt-12.