Poincaré duality and periodicity

Poincaré Duality and Periodicity, Ii. James Periodicity

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension ΣK so that the resulting space satisfies Poincaré duality. When this is possible, we say that ΣK is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then ΣK is a spine whenever j is a suitable power of two. The ...

Poincaré duality and periodicity

We construct periodic families of Poincaré spaces. This gives a partial solution to a question posed by Hodgson in the proceedings of the 1982 Northwestern homotopy theory conference. In producing these families, we formulate a recognition principle for Poincaré duality in the case of finite complexes having one top cell that splits of after a single suspension. We also explain how a Z-equivari...

Poincaré Duality Algebras Mod Two

Poincaré Duality and Commutative Differential Graded Algebras

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré du-ality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold.

Twisted K-theory and Poincaré duality

Using methods of KK-theory, we generalize Poincaré K-duality to the framework of twisted K-theory.