S-duality II (Ex)

Exercise 0.1. 
 Let $X/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qzViTQ$$<wikitex refresh>; {{beginthm|Exercise}} \label{exrcs:S-duality-of-Thom-class-is-fund-class}
 Let $X$ be an $n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)$ and denote by $\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the canonical $S$-duality map. Choose the Thom class $u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class $[X] \in C_{n} (X)$. Show that
 $$ \alpha_X \backslash (u(\nu_{X})) \sim \pm [X]. $$ {{endthm}}

 {{beginthm|Exercise}} \label{exrcs:S-duality-and-Thom-iso-is-Poincare-duality}
 Let $X$ be an $n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)$. Choose the Thom class $u(\nu_{X}) \in C^{k} (\text{Th} (\nu_{X})$ and the fundamental class~$[X] \in C_{n} (X)$ so that $\alpha_X \backslash (u (\nu_{X})) \sim [X]$. Prove that the following diagram commutes up to chain homotopy:
 $$\xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -} \ & C(X) & }$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]X$ be an $n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_KnLl2n$$n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_o1bbBV$$\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_uRZjAt$$\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the canonical $S/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_wZ1Q81$$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_kvKc6A$$u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class $[X] \in C_{n} (X)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_YHyhta$$[X] \in C_{n} (X)$. Show that


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





Exercise 0.2. 
 Let $X/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_mbWPrk$$X$ be an $n/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_65VY1U$$n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_YBIL6v$$\nu_{X} \colon X \rightarrow \text{BSG} (k)$. Choose the Thom class $u(\nu_{X}) \in C^{k} (\text{Th} (\nu_{X}))/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_KYPO37$$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_IwwiqK$$[X] \in C_{n} (X)$ so that $\alpha_X \backslash (u (\nu_{X})) \sim [X]/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_gxs3an$$\alpha_X \backslash (u (\nu_{X})) \sim [X]$. Prove that the following diagram commutes up to chain homotopy:


$\displaystyle \xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -} \\ & C(X) & }/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_K3gLk0$