S-duality I (Ex)

Let $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ and $Y$$Y$ be a finite pointed CW-complexes. A map

$\displaystyle \alpha \colon S^N \to X \wedge Y$ $$\displaystyle \alpha \colon S^N \to X \wedge Y$$

is called an S-duality if the slant product induced by $\alpha$$\alpha$

$\displaystyle - \backslash \alpha \colon H^i(X; \Zz) \to H_{N-i}(Y;\Zz)$ $$\displaystyle - \backslash \alpha \colon H^i(X; \Zz) \to H_{N-i}(Y;\Zz)$$

is an isomorphism for all $i$$i$. In this case $X$$X$ and $Y$$Y$ are called an $S$$S$-duals of each other.

Exercise 0.1. Show that $S$$S$-duality satisfies the following"

• For every finite CW-complex $X$$X$ there exists an $N$$N$-dimensional S-dual, which we denote $X^\ast$$X^\ast$, for some large $N \geq 1$$N \geq 1$.
• If $X^\ast$$X^\ast$ is an $N$$N$-dimensional $S$$S$-dual of $X$$X$ then $\Sigma X^\ast$$\Sigma X^\ast$ is an $(N+1)$$(N+1)$-dimensional $S$$S$-dual of $X$$X$.
• For any space $Z$$Z$ we have isomorphisms

$\displaystyle S \co [X,Z] \cong [S^N,Z \wedge Y] \quad \gamma \mapsto S(\gamma) = (\gamma \wedge \id_Y) \circ \alpha,$ $$\displaystyle S \co [X,Z] \cong [S^N,Z \wedge Y] \quad \gamma \mapsto S(\gamma) = (\gamma \wedge \id_Y) \circ \alpha,$$

$\displaystyle S \co [Y,Z] \cong [S^N,X \wedge Z] \quad \gamma \mapsto S(\gamma) = (\id_X \wedge \gamma) \circ \alpha.$ $$\displaystyle S \co [Y,Z] \cong [S^N,X \wedge Z] \quad \gamma \mapsto S(\gamma) = (\id_X \wedge \gamma) \circ \alpha.$$

• A map $f \co X \rightarrow Y$$f \co X \rightarrow Y$ induces a map $f^\ast \co Y^\ast \rightarrow X^\ast$$f^\ast \co Y^\ast \rightarrow X^\ast$ for $N$$N$ large enough via the isomorphism

$\displaystyle [X,Y] \cong [S^N,Y \wedge X^\ast] \cong [Y^\ast,X^\ast].$ $$\displaystyle [X,Y] \cong [S^N,Y \wedge X^\ast] \cong [Y^\ast,X^\ast].$$

• If $X \rightarrow Y \rightarrow Z$$X \rightarrow Y \rightarrow Z$ is a cofibration sequence then $Z^\ast \rightarrow Y^\ast \rightarrow X^\ast$$Z^\ast \rightarrow Y^\ast \rightarrow X^\ast$ is a cofibration sequence for $N$$N$ large enough.