Geometry and Topology seminar
Speaker: Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Exotic Open Contractible 3-Manifolds and Asymptotic Limits
Time, day and date: 2:30:00 PM - 3:30:00 PM, Thursday, October 15
Venue: Ramanujan Hall
Abstract: In 1934, Whitehead proposed a proof of the Poincaré conjecture. It rested on the claim that every open contractible 3-manifold is homeomorphic to Euclidean 3-space. The following year he found the claim to be false. He then published a counterexample, now known as the Whitehead manifold. McMillan later produced uncountably many such manifolds, no two of them homeomorphic.
Telling Euclidean space apart from these exotic manifolds is delicate: they are contractible, so no ordinary homotopy invariant distinguishes them, and the whole of the difference sits at infinity, where they are wild. The invariant that does see it is the Brown-Grossman proper fundamental group, extracted from an inverse sequence of ordinary fundamental groups by a tool developed by Brown. We call this tool the asymptotic limit functor.
In this talk, we will explain the construction, and show that the asymptotic limit behaves considerably better than the inverse limit and its first derived functor. We will then use it to quantify the wildness algebraically: every open contractible 3-manifold not homeomorphic to Euclidean 3-space has exactly one end, and its Brown-Grossman proper fundamental group is uncountable and perfect. This contrasts sharply with the ordinary fundamental group of a manifold, which is always countable.
Joint work with Rekha Santhanam. Based on arXiv:2609.14063.