### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

#### Monoidal categories

monoidal categories

duality

# Contents

## 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

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 $X$.

## References

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