University
of Wisconsin -
Milwaukee Calendar: TBD Teaching: |

Research:

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 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:Workshop in Geometric Topology 2009 (UW-Milwaukee)

Workshop in Geometric Topology 2014 (UW-Milwaukee)

Miscellaneous:

My
PGP Public Key (NB: It's a PGP 2.6 legacy key, so you need
the IDEA cypher to use it.)

Jeffrey Rolland. rollandj@uwm.edu