Poincare duality and Periodicity, II. James Periodicity

TL;DR

This paper explores conditions under which iterated suspensions of a complex can be modified to satisfy Poincare duality, revealing connections between James periodicity and four-fold periodicity in L-theory.

Contribution

It introduces quadratic self duality and links James periodicity to Poincare duality and four-fold periodicity in surgery obstruction groups.

Findings

Quadratically self dual complexes lead to spines at powers of two.

James periodicity influences the ability to achieve Poincare duality.

New interpretation of four-fold periodicity in L-theory via bordism.

Abstract

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then S^j K is a spine whenever j is a suitable power of two. The powers of two come from the James periodicity theorem. We briefly explain how our main results, considered up to bordism, give a new interpretation of the four-fold periodicity of the surgery obstruction groups. We therefore obtain a relationship between James periodicity and the four-fold periodicity in L-theory.