Spanier-Whitehead duality


Stable Homotopy theory

Monoidal categories




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


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

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


The original references are

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

  • Spanier, E. H.; Whitehead, J. H. C. (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

Atiyah duality is due to

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

Further discussion of Atiyah duals is in

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

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)

Revised on February 27, 2014 10:13:37 by Urs Schreiber (