S-duality II (Ex)
(Difference between revisions)
m |
m |
||
Line 6: | Line 6: | ||
{{beginthm|Exercise}} \label{exrcs:S-duality-and-Thom-iso-is-Poincare-duality}
| {{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:
| 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:
| ||
− | $$\xymatrix{C^{n-\ast} (X) \ar[rr]^{- \cup u(\nu_{X})} \ar[dr]_{-\cap [X]} & & C^{n+k-\ast}(\text{Th}(\nu_{X})) \ar[dl]^{\alpha_{X}\backslash -}\\ | + | $$\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}} | ||
</wikitex> | </wikitex> |
Revision as of 15:58, 23 April 2012
Exercise 0.1.
LetTex syntax errorbe an
Tex syntax error-GPC with SNF
Tex syntax errorand denote by
Tex syntax errorthe canonical
Tex syntax error-duality map. Choose the Thom class
Tex syntax errorand the fundamental class
Tex syntax error. Show that
Tex syntax error
Exercise 0.2.
LetTex syntax errorbe an
Tex syntax error-GPC with SNF
Tex syntax error. Choose the Thom class
Tex syntax errorand the fundamental class~
Tex syntax errorso that
Tex syntax error. Prove that the following diagram commutes up to chain homotopy:
Tex syntax error