Poincare duality and Periodicity

TL;DR

This paper constructs new periodic families of Poincare complexes, explores their embedding properties, and describes four-fold periodicity in knot cobordism using homotopy theory, addressing longstanding questions in topology.

Contribution

It introduces novel constructions of Poincare complexes with specific periodicity and embedding properties, and provides a homotopy theoretic explanation for four-fold periodicity in knot cobordism.

Findings

Constructed periodic families of Poincare complexes.

Identified complexes whose top cell falls off after suspension.

Described four-fold periodicity in knot cobordism.

Abstract

We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.