nLab Spanier-Whitehead duality

Idea

The stable homotopy category $Ho(Spec)$ is a symmetric monoidal category via the symmetric smash product of spectra. Monoidal duality in $Ho(Spec)$ is called Spanier-Whitehead duality or S-duality .

Examples

• For $X$ a compact smooth manifold. The dual of its suspension spectrum $\Sigma^\infty_+ X$ is given by the Thom spectrum $Th(N X)$ of its stable normal bundle. This is also called Atiyah duality (due to Atiyah).

The explicit interpretation in terms of monoidal duality is (DoldPuppe, theorem 3.1).

Using this one shows that the trace on the identity on $\Sigma^\infty_+ X$ – its categorical dimension – is the Euler characteristic of $X$.

Related entries

• S-theory

• Anderson duality

• Brown-Comenetz duality

• Spanier-Whitehead category

• Costenoble-Waner duality

• stable Cohomotopy

References

The original references are

• Edwin Spanier, ; Henry Whitehead, (1953), A first approximation to homotopy theory, Proc. Nat. Acad. Sci. U.S.A. 39: 655–660, MR0056290

• Edwin Spanier, ; Henry Whitehead, (1955), Duality in homotopy theory , Mathematika 2: 56–80, MR0074823

The interpretation of the duality as ordinary monoidal duality in the stable homotopy category is apparently due to

• Albrecht Dold, Dieter Puppe, Duality, trace and transfer , Proceedings of the Steklov Institute of Mathematics, (1984), issue 4

An English version of that paper is:

• Albrecht Dold, Dieter Puppe, Duality, trace and transfer , Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pp. 81–102, PWN, Warsaw, 1980. pdf

Atiyah duality is due to

• Michael Atiyah, Thom complexes , Proc. London Math. Soc. (3) , no. 11 (1961), 291–310.

Lecture notes include

• Frank Adams, part III, section 5 of Stable homotopy and generalised homology, 1974

• Michael Hopkins (notes by Akhil Mathew), Lecture 22 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

Further discussion of Atiyah duals is in

• Ralph Cohen, Multiplicative properties of Atiyah duality (arXiv:0403486)

For equivariant stable homotopy theory Spanier-Whitehead duality is discussed on pages 23 onwards of

• John Greenlees, Peter May, Equivariant stable homotopy theory (pdf)

The Spanier-Whitehead duality has been studied in the new category of cospectra with ideas close to the later ones of shape theory in

• Elon L. Lima, The Spanier-Whitehead duality in new homotopy categories, Summa Brasil. Math. 4 1959 91–148 (1959) MR0116332

See also

• Roy Joshua, Spanier-Whitehead duality in etale homotopy Transactions of the American Mathematical Society Vol. 296, No. 1 (Jul., 1986), pp. 151-166 (article consists of 16 pages) (jstor)