Tag Archives: Cerf theory

Reconstructing 4-manifolds from Morse 2-functions – arXiv:1202.3487

A Morse function is a generic map to the real line. A Morse 2-function is generic map to the plane, or other 2-manifold. These functions featured prominently in my dissertation where I used them to give a classification of 2D topological field theories in terms of generators and relations. Now David Gay and Robion Kirby are using them to study 4-manifolds.
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The Classification of Two-Dimensional Extended Topological Field Theories

Ph.D. Dissertation (UC Berkeley) 2009


This work proves the cobordism hypothesis in dimension two. Specifically, this work classifies 2-dimensional extended topological field theories in terms of generators and relations. In this context, a topological field theory is a functor from a bordism category to some target category and an extended field theory is a higher categorical version of this. Why use higher categories? They allow you to mathematically encode the locality of the theory, something which is predicted from the physical point of view and which is very helpful mathematically.

The methods combined algebraic results on symmetric monoidal bicategories with a generalization of Cerf theory. This provides and alternate approach to the cobordism hypothesis to the one given by Hopkins and Lurie, and to date (Feb 2012) remains the only complete account.
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