数学研究所
学术报告
拓扑研讨班
Speaker:Prof. Stephen Theriault （University of Southampton）
Inviter: 黄瑞芝 副研究员
Language: English
Title: Properties of Poincare Duality complexes - from rational to integral
Time&Venue: 2025年4月2日（星期三） 14:30-15:30 & 南楼N818
Abstract:A provocative theorem of Halperin and Lemaire states that if M is a simply-connected Poincare Duality complex of dimension n and the rational cohomology of M is not generated as an algebra by a single element, then the based loops on M retracts off the based loops on its (n-1)-skeleton. This implies that the attaching map for the n-cell of M kills off homotopy groups but does not introduce any new ones. It is rare that a cell attachment has this property and it is surprising that it holds in such generality for Poincare Duality complexes.In this talk we show that this theorem secretly originates in integral homotopy theory. We give conditions for when the based loops on M integrally retracts off the based loops on its (n-1)-skeleton, and use these to recover the rational result.
------------------------------------------------------------------------------
Title: The homotopy theory of connected sums with projective spaces
Time&Venue: 2025年4月9日（星期三） 14:30-15:30 & 南楼N818
Abstract:Let M be a simply-connected 2n-dimensional Poincare Duality complex. We describe a homotopy decomposition of the based loops on the connected sum of M and a 2n-dimensional complex projective space that holds integrally if n is even and localized away from 2 if n is odd. This generalizes results of Duan that relied on geometric methods; our proof is purely homotopy theoretic. This is joint work with Ruizhi Huang.
附件：