S-duality II (Ex)

Exercise 0.1. 
 Let $X/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_H1pRUa$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ be an $n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_oQV69Q$$n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_D5SnNx$$\nu_{X} \colon X \rightarrow \text{BSG} (k)$ and denote by $\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_MSbcQe$$\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the canonical $S/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1u17iW$$S$-duality map. Choose the Thom class $u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Eexf9D$$u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class $[X] \in C_{n} (X)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_xjsrqm$$[X] \in C_{n} (X)$. Show that


$\displaystyle \alpha_X \backslash (u(\nu_{X})) \sim \pm [X]./var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gfVY54$ $$\displaystyle \alpha_X \backslash (u(\nu_{X})) \sim \pm [X].$$





Exercise 0.2. 
 Let $X/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_zJ2sbO$$X$ be an $n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_4vVqFx$$n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_RpL2xh$$\nu_{X} \colon X \rightarrow \text{BSG} (k)$. Choose the Thom class $u(\nu_{X}) \in C^{k} (\text{Th} (\nu_{X}))/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_a7VbV1$$u(\nu_{X}) \in C^{k} (\text{Th} (\nu_{X}))$ and the fundamental class~$[X] \in C_{n} (X)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_nNIZIM$$[X] \in C_{n} (X)$ so that $\alpha_X \backslash (u (\nu_{X})) \sim [X]/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_s9i1Wx$$\alpha_X \backslash (u (\nu_{X})) \sim [X]$. Prove that the following diagram commutes up to chain homotopy:


$\displaystyle \xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -} \\ & C(X) & }/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_NkkaAj$