Duality phenomena

Duality is a recurring phenomenon in algebraic topology. We start in Section 11.1 with its categorical formulation in a symmetric monoidal \(\infty \)-category. We then compare dualizability and compactness in Section 11.2, showing in particular that the dualizable modules over a commutative ring spectrum are precisely the perfect modules. In Section 11.3, we specialize to spectra: Spanier–Whitehead duality relates the cohomology of a dualizable spectrum to the homology of its dual, while Atiyah duality identifies the dual of the suspension spectrum of a closed smooth manifold \(M\) with the Thom spectrum \(\th (-T_M)\) of its negative virtual tangent bundle. The latter result was first proved by Atiyah (1961), with a precursor in [Milnor and Spanier (1960)]. Combining it with the Thom isomorphism, we recover generalized Poincaré duality in Section 11.4.

We refer to Becker and Gottlieb (1999) for a historical overview of duality phenomena in algebraic topology.

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