nLab Poincaré duality space

Definition

Definition

A topological space for which there is $d \in \mathbb{N}$ and a class $[X] \in H_d(X)$ in its ordinary homology such that the cap product induces isomorphisms

$$(-) \cap [X] \;\colon\; H^\bullet(X) \overset{\simeq}{\longrightarrow} H_{d-\bullet}(X)$$

between ordinary cohomology and ordinary homology groups as indicated, is called a Poincaré duality space.

If $X$ is moreover a CW-complex then this it is sometimes called a Poincaré complex or even a Poincaré manifold.

See at Poincaré duality for more.

References

Textbooks:

• James Munkres, Duality in Manifolds, Chapter 8 in: Elements of Algebraic Topology, Addison-Wesley (1984) [pdf]

Exposition:

• John R. Klein: Poincaré duality spaces [pdf, pdf]

In the general concext of spectral geometry (spectral triples):

• Alain Connes, page 10 of Noncommutative geometry and reality, J. Math. Phys. 36 (11), 1995 (pdf)

Concerning string topology over (the homology of the free loop space of) Poincaré duality spaces:

• Alastair Hamilton, Andrey Lazarev: Symplectic $A_\infty$-algebras and string topology operations, Amer. Math. Soc. Transl. 224 2 (2008) 147–157 [arXiv:0707.4003]