# S-duality (homotopy theory)

In algebraic topology, a branch of mathematics, called S- duality is a duality between topological spectra and thus between generalized homology and cohomology theories -.

## Definition

Let and two spectra. We denote by her smash- product and with the sphere spectrum.

A Dualitätsmorphismus or a duality between and is a morphism of spectra

So that for each spectrum by

Defined pictures

Bijections are.

The spectra and are called S- dual if there is a Dualitätsmorphismus. S- duality is a symmetric relation.

Two spectra and hot - dual for when and S are dual. Here, the designated by defined range.

## S- dual morphism

Let and be two Dualitätsmorphismen, then for every morphism

Be S- dual morphism

Defined as the image of under the isomorphism

( Ie is well defined up to homotopy. )

In particular, S- dual is to exactly when.

## Examples

- The canonical equivalence is an S- duality.
- For a closed manifold with Einhängungsspektrum the Milnor -Spanier S- duality is