I am a dissertator in the Geometric
Topology group at UWM.
My thesis advisor is Craig Guilbault.
My thesis topic is "A Geometric Reverse to Quillen's Plus Constructiion".
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 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 thesis recall's Quillen's procedure for producing a plus cobordism for a high-dimensional manifold whose fundamental group contains a perfect normal subgroup, and develops some of the properties of the plus construction. I also recall the procedure for solving the group extension problem, that is, given groups a quotient group Q and a kernel group K, finding all groups G which satisfy the short exact sequence relationship 1 -> K -> G -> Q -> 1.
The second part of my thesis 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 third 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 third 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 last part of my thesis involves outlining the relation between Hausmann-Vogel's theory of semi-s-cobordisms and my procedure for producing semi-s-cobordisms.Presentations:
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