# Jeffrey Rolland's Homepage

Teaching:

Marquette University
Department of Mathematics, Statistics, and Computer Sciences
Office: Room 321, Cudahy Hall
Office Phone: (414) 288-3790
Email: jeffrey.rolland@marquette.edu

Math 1451 Calculus II Section 101
MWRF 10:00 AM - 10:50 AM

Math 2450 Calculus III Section 103
MWRF 11:00 AM - 11:50 AM

Research:
I have a Ph.D. from the Geometric Topology group at 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

Publications:
"Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds", doctoral dissertation http://arxiv.org/abs/1502.04338
"A Geometric Reverse to the Plus Construction and Some Examples of Pseudo-Collars on High-Dimensional Manifold",
submitted http://arxiv.org/abs/1508.03670
"Some More Examples of Pseudo-Collar Structures on High-Dimensional Manifolds" (joint with Marston Conder), in preparation

Presentations:
Workshop in Geometric Topology 2009 (UW-Milwaukee)
Workshop in Geometric Topology 2014 (UW-Milwaukee)
Dissertation Defense (01/2015) (UW-Milwaukee)

Jeffrey Rolland.
jeffrey.rolland@marquette.edu