S-duality II (Ex)
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$$ \alpha_X \backslash (u(\nu_{X})) \sim \pm [X]. $$ | $$ \alpha_X \backslash (u(\nu_{X})) \sim \pm [X]. $$ | ||
{{endthm}}
| {{endthm}}
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{{beginthm|Exercise}} \label{exrcs:S-duality-and-Thom-iso-is-Poincare-duality}
| {{beginthm|Exercise}} \label{exrcs:S-duality-and-Thom-iso-is-Poincare-duality}
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− | 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 | + | 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:
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$$\xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -} \\ & C(X) & }$$ | $$\xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -} \\ & C(X) & }$$ | ||
{{endthm}} | {{endthm}} |
Latest revision as of 11:50, 1 June 2012
Exercise 0.1.
LetTex syntax errorbe an
Tex syntax error-GPC with SNF
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Tex syntax error-duality map. Choose the Thom class
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Exercise 0.2.
LetTex syntax errorbe an
Tex syntax error-GPC with SNF
Tex syntax error. Choose the Thom class
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Tex syntax error. Prove that the following diagram commutes up to chain homotopy:
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