nLab Brown-Comenetz duality

Idea

Brown-Comenetz 76 introduced the spectrum $I_{\mathbb{Q}/\mathbb{Z}}$ which represents the cohomology theory that assigns to a spectrum $X$ the algebraic Pontrjagin duals $Hom_{\mathbb{Z}}(\pi_{- \ast} X, \mathbb{Q}/\mathbb{Z})$ of its homotopy groups with respect to Q/Z.

If $X$ is a spectrum whose stable homotopy groups are finite groups, then the natural “double-duality” map $X \to I_{\mathbb{Q}/\mathbb{Z}} I_{\mathbb{Q}/\mathbb{Z}} X$ (the unit of the corresponding continuation monad) is an equivalence (see at dualizing object in a closed category).

The Brown-Comenetz spectrum $I_{\mathbb{Q}/\mathbb{Z}}$ has an “integral” refinement which is the Anderson spectrum $I_{\mathbb{Z}}$ inducing Anderson duality.

Properties

• Eilenberg-Mac Lane spectra are Brown-Comenetz self-dual: $I_{\mathbb{Q}/\mathbb{Z}} H \mathbb{F}_p \cong H \mathbb{F}_p$.

Related concepts

• Anderson duality

• Spanier-Whitehead duality

References

• E. H. Brown, Jr. and M. Comenetz, Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math. 98 (1976), no. 1, 1–27.

• Vesna Stojanoska, Duality for Topological Modular Forms (arXiv:1105.3968)