# nLab Spanier-Whitehead duality

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# 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

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.

Lecture notes include

Further discussion of Atiyah duals is in

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

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)

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