S-duality (homotopy theory)

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 . ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle A\wedge A^{*}}$ ${\displaystyle \mathbf {S} }$

A duality morphism or a duality between and is a morphism of spectra ${\displaystyle A}$ ${\displaystyle A^{*}}$

${\displaystyle u\colon \mathbf {S} \to A\wedge A^{*}}$

so that for each spectrum the through ${\displaystyle E}$

${\displaystyle u_{E}(\phi ):=(\phi \wedge id_{A^{*}})u}$

${\displaystyle u^{E}(\phi ):=(id_{A}\wedge \phi )u}$

defined images

${\displaystyle u_{E}\colon \left[A,E\right]\to \left[\mathbf {S} ,E\wedge A^{*}\right]}$

${\displaystyle u^{E}\colon \left[A^{*},E\right]\to \left[\mathbf {S} ,A\wedge E\right]}$

Are bijections .

The spectra and are called S-dual if there is a duality morphism . S-duality is a symmetrical relation . ${\displaystyle A}$ ${\displaystyle A^{*}}$ ${\displaystyle u\colon \mathbf {S} \to A\wedge A^{*}}$

Two spectra and are called -dual for if and S are dual. Here referred to the by -defined range. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle n}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle A}$ ${\displaystyle \Sigma ^{n}B}$ ${\displaystyle \Sigma ^{n}B}$ ${\displaystyle (\Sigma ^{n}B)_{k}=B_{k+n}}$

S-dual morphism

Be and two duality morphisms, then there is a morphism for each ${\displaystyle u\colon \mathbf {S} \to A\wedge A^{*}}$ ${\displaystyle v\colon \mathbf {S} \to B\wedge B^{*}}$

${\displaystyle f\colon A\to B}$

its S-dual morphism

${\displaystyle f^{*}\colon B^{*}\to A^{*}}$

defined as the image from under the isomorphism ${\displaystyle f}$

${\displaystyle (v^{A^{*}})^{-1}u_{B}\colon \left[A,B\right]\to \left[\mathbf {S} ,B\wedge A^{*}\right]\to \left[B^{*},A^{*}\right]}$.

( So it is well defined except for homotopy.) ${\displaystyle f^{*}}$

In particular, S-dual is if and only if . ${\displaystyle f\in \left[A,B\right]}$ ${\displaystyle g\in \left[B^{*},A^{*}\right]}$ ${\displaystyle u_{B}(f)=v_{A^{*}}(g)}$

Examples

• The canonical equivalence is an S-duality. ${\displaystyle u\colon \mathbf {S} \to \Sigma ^{n}\mathbf {S} \wedge \Sigma ^{-n}\mathbf {S} }$
• For a closed manifold with a hanging spectrum , the Milnor-Spaniard becomes S-duality ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle \Sigma ^{\infty }M}$

${\displaystyle u\colon \mathbf {S} \to Th(\nu _{M})\wedge \Sigma ^{-n}\Sigma ^{\infty }M_{+}}$

defined as follows: Choose an embedding for one and a tube environment with projection . Then we look at the composition ${\displaystyle M\subset S^{N}}$ ${\displaystyle N\gg n}$ ${\displaystyle M\subset U\subset S^{N}}$ ${\displaystyle p\colon {\overline {U}}\to M}$ ${\displaystyle Th(\nu _{M})\simeq {\overline {U}}/\partial {\overline {U}}}$

${\displaystyle f\colon S^{N}\to {\overline {U}}/\partial {\overline {U}}\to {\overline {U}}/\partial {\overline {U}}\wedge M_{+}}$,

where the first image collapses to a point and the second image is induced by. Then ${\displaystyle S^{N}-U}$ ${\displaystyle (id,p)}$

${\displaystyle u:=\Sigma ^{-N}\Sigma ^{\infty }f\colon \mathbf {S} \to Th(\nu _{M})\wedge \Sigma ^{-n}\Sigma ^{\infty }M_{+}}$

an S-duality.

If it is possible to orientate a ring spectrum , then the cohomological orientations (Thom classes) correspond to ${\displaystyle M}$ ${\displaystyle E}$ ${\displaystyle E}$

${\displaystyle u_{E}\colon \left[Th(\nu _{M}),E\right]\to \left[\mathbf {S} ,E\wedge \Sigma ^{-n}\Sigma ^{\infty }M_{+}\right]}$

the homological orientations (fundamental classes). ${\displaystyle E}$

literature

• YB Rudyak: On Thom spectra, orientability, and cobordism , Springer-Verlag, 1998, Corrected reprint 2008

Web links

• Spanier-Whitehead: Duality in homotopy theory
• Milnor-Spanier: Two remarks on fiber homotopy type