University
of Wisconsin -
Milwaukee
Department of Mathematical Sciences
Office: EMS W464
Phone: (414) 229-4343
Email: rollandj@uwm.edu
Calendar: Calendar.pdf
Work
School
Teaching:
UWM D2L
UW-Parkside D2L
Research:
I am a dissertator in the Geometric
Topology group at UWM.
My thesis advisor is Craig
Guilbault.
My thesis topic is "An Extension of Semi-s-cobordisms
to Almost Perfect Kernels".
In 1978, Jean-Claude Hausmann and Pierre Vogel developed a
theory for taking a
given closed manifold M and and a given (super-)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 involves creating some examples of semi-s-cobordisms. (The literature has very few actual examples of Hausmann-Vogel's theory.) In particular, the first 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 second part of my thesis involves creating semi-s-cobordisms when the kernel group is not perfect, but instead is almost perfect, so that any element of P is a commutator of the form gpg-1p-1, where g is an element of G (the fundamental group of M–) and p is an element of the almost perfect group P.
Miscellaneous:
My
PGP Public Key (NB: It's a PGP 2.6 legacy key, so you need the IDEA cypher to use it.)