S-duality II (Ex)

Exercise 0.1. 


Let $X$$<wikitex>; {{beginthm|Exercise}} \label{exrcs:S-duality-of-Thom-class-is-fund-class}
 Let $X$ be an $n$-GPC with SNF $\nu_{X} \colon X \rightarrow \text{BSG} (k)$ and denote by $\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the $S$-duality. Choose the Thom class $u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class $[X] \in C_{n} (X)$. Show that
 $$
\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 without solution]]X$ be an $n$$n$-GPC with SNF

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$\nu_{X} \colon X \rightarrow \text{BSG} (k)$ and denote by

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$\alpha_X \colon S^{n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the $S$$S$-duality. Choose the Thom class

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$u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class

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$[X] \in C_{n} (X)$. Show that


$\displaystyle 
\alpha_X \backslash (u(\nu_{X})) \sim \pm [X].$ $$\displaystyle 
\alpha_X \backslash (u(\nu_{X})) \sim \pm [X].$$








Exercise 0.2. 


Let $X$$X$ be an $n$$n$-GPC with SNF

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$\nu_{X} \colon X \rightarrow \text{BSG} (k)$. Choose the Thom class

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$u(\nu_{X}) \in C^{k} (\text{Th} (\nu_{X})$ and the fundamental class~

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$[X] \in C_{n} (X)$ so that

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$\alpha_X \backslash (u (\nu_{X})) \sim [X]$. Prove that the following diagram commutes up to chain homotopy:


$\displaystyle 
\xymatrix{
 C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{- \cap [X]} & & C^{n+k-\ast} (\text{Th} (\nu_{X})) \ar[dl]^{\alpha_{X} \backslash -} \\
 & C(X)
 }$