Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, March 6, 2017 - 12:00 to 12:45

Location:

BEXL 207

Local Speaker:

Abstract:

Consider a relative two-complex (L,K) where K is an aspherical CW complex (of arbitrary dimension). The pair (L,K) is said to be â€œaspherical" if the second relative homotopy group of the pair (L,K) is trivial, which is equivalent to saying that the inclusion of K in L induces a monomorphism of fundamental groups and that the complex L is aspherical in the traditional sense. In this case one can deduce a lot of group theoretic information about the fundamental group of L in terms of that of K. In this talk, I will consider the alternatives to asphericity, examining in detail the case where the quotient complex L/K is "dunce cap" modeled on the presentation (x:xxx^-1). Asphericity was classified in this setting by M. Edjvet in 1993. I will survey the non-aspherical cases and conclude that except for a small handful of unresolved cases, if (L,K) is not aspherical, then the fundamental group of L either contains a "forbidden" finite subgroup or else splits as an amalgamated free product over a virtual three-manifold group. Some interesting and well-known finite groups and three-manifolds occur in this survey.