In algebraic topology , a branch of mathematics , S-duality denotes a duality between topological spectra and thus between generalized homology and cohomology theories .
definition
Let and two spectra . We denote with her Smash product and with the spectrum of spheres .
A duality morphism or a duality between and is a morphism of spectra
so that for each spectrum the through
defined images
Are bijections .
The spectra and are called S-dual if there is a duality morphism . S-duality is a symmetrical relation .
Two spectra and are called -dual for if and S are dual. Here referred to the by -defined range.
S-dual morphism
Be and two duality morphisms, then there is a morphism for each
its S-dual morphism
defined as the image from under the isomorphism
-
.
( So it is well defined except for homotopy.)
In particular, S-dual is if and only if .
Examples
- The canonical equivalence is an S-duality.
- For a closed manifold with a hanging spectrum , the Milnor-Spaniard becomes S-duality
- defined as follows: Choose an embedding for one and a tube environment with projection . Then we look at the composition
-
,
- where the first image collapses to a point and the second image is induced by. Then
- an S-duality.
- If it is possible to orientate a ring spectrum , then the cohomological orientations (Thom classes)
correspond to
- the homological orientations (fundamental classes).
literature
- YB Rudyak: On Thom spectra, orientability, and cobordism , Springer-Verlag, 1998, Corrected reprint 2008
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