MW 5:00 PM - 6:15 PM
Math 2450 Calculus III Section 101
I have a Ph.D. in Geometric
Topology from UWM.
My dissertation advisor was Craig Guilbault.
My dissertation topic was "Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds".
In 1978, Jean-Claude Hausmann and Pierre Vogel developed a theory for taking a given closed manifold M and and a given (locally) perfect group P and creating a cobordant manifold M_ whose fundamental group is a group extension with quotient group the fundamental group of M and kernel group P. The homology groups of M and M_ are isomorphic. The cobordism between M and M_ is called a 1-sided s-cobordism or a semi-s-cobordism because the inclusion of M into the cobordism is a simple homotopy equivalence (as in an s-cobordism), but the inclusion of M_ into the cobordism is not a homotopy equivalence at all.
Note that if (W, M, M_) is a semi-s-cobordism, then (W, M_, M) is a plus cobordism. (This justifies the use of M_ for the right-hand boundary of a semi-s-cobordism, a play on the traditional use of M+ for the right-hand boundary of a plus cobordism.) Thus, a semi-s-cobordism may be thought of as an inverse to Quillen's plus construction in the manifold category.The first part of my dissertation involves creating an explicit procedure for creating semi-s-cobordisms, when the fundamental group of M_is a semi-direct product of the fundamental group of M and a prescribed super-perfect group S.
The second part of my thesis involves creating some examples of semi-s-cobordisms. (The literature has very few actual examples of semi-s-cobordisms.) In particular, the second part of my thesis involves "stacking" semi-s-cobordims, so that the right-hand boundary of cobordism i ("M_") becomes the left-hand boundary of cobordim i+1 ("M"). This "stacking" of semi-s-cobordisms out to infinity is called a pseudo-collar in several papers by my thesis advisor and his co-author, Fred Tinsley.
The third part of my dissertation involves creating
pseudo-collar-like manifolds which do not have perfectly semi-stable
fundamental group at infinity, extending a result of my dissertation
advisor and his co-author, Fred Tinsley. Hypo-Abelian groups, groups
whose perfect cores are trivial, play a large role in creating these absolutely inward tame but not pseudo-collarable 1-ended open manifolds
My PGP Public Key (NB: It's a PGP 2.6 legacy key, so you need the IDEA cypher to use it.)