Poincaré duality pairs of dimension three

Poincaré Duality Pairs of Dimension Three

We extend Hendriks’ classification theorem and Turaev’s realisation and splitting theorems for PD–complexes to the relative case of PD– pairs. The results for PD–complexes are recovered by restricting the results to the case of PD–pairs with empty boundary. Up to oriented homotopy equivalence, PD–pairs are classified by their fundamental triple consisting of the fundamental group system, the or...

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 COMPLEXES IN DIMENSION FOUR 3 Theorem 5

We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of PD–complexes. Generalizing Turaev’s fundamental triples of PD–complexes we introduce fundamental triples for PDn–complexes and show that two PDn–complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjec...

Poincaré Duality Algebras Mod Two

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 ...