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