Monoidal controlled Poincaré duality

Poincaré Duality Algebras Mod Two

Duality and Traces for Indexed Monoidal Categories

By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a tr...

Poincaré duality in dimension 3 CT

The paper gives a review of progress towards extending the Thurston programme to the Poincaré duality case. In the first section, we fix notation and terminology for Poincaré complexes X (with fundamental group G) and pairs, and discuss finiteness conditions. For the case where there is no boundary, π2 is non-zero if and only if G has at least 2 ends: here one would expect X to split as a conne...

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 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.