Poincare duality and Periodicity, II. James Periodicity

Details Poincare duality and Periodicity, II. James Periodicity Poincare duality and Periodicity, II. James Periodicity

2008-12-29

arxiv.org/abs/0812.4988v1

Mathematics

Algebraic Topology

Scientific paper

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.

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