Novikov additivity I (Ex)
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Tibor Macko (Talk | contribs) (Created page with "<wikitex>; Let $(Y,X)$ be a $2n$-dimensional manifold with boundary, $n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\wideha...") |
Tibor Macko (Talk | contribs) |
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<wikitex>; | <wikitex>; | ||
− | Let $(Y,X)$ be a $2n$-dimensional manifold with boundary, $n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\ | + | Let $(Y,X)$ be a $2n$-dimensional manifold with boundary, $n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\hat{H}^{n} (Y)$ the image of $\varphi$. Coefficients are understood to be in $\mathbb{R}$. |
The middle dimensional intersection form | The middle dimensional intersection form | ||
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H^{n} (Y) \otimes H^{n} (Y) \rightarrow \mathbb{R} | H^{n} (Y) \otimes H^{n} (Y) \rightarrow \mathbb{R} | ||
$$ | $$ | ||
− | is degenerate in general. Show that the intersection form $B$ on $\ | + | is degenerate in general. Show that the intersection form $B$ on $\hat{H}^{n} (Y)$ defined by |
$$ | $$ | ||
B(\varphi (a), \varphi (b)) = \langle a \cup b , [Y] \rangle | B(\varphi (a), \varphi (b)) = \langle a \cup b , [Y] \rangle |
Revision as of 10:05, 29 May 2012
Let be a -dimensional manifold with boundary, . Consider the homomorphism and denote the image of . Coefficients are understood to be in .
The middle dimensional intersection form
is degenerate in general. Show that the intersection form on defined by
is a non-degenerate symmetric bilinear form and let us define the signature to be the signature of this form.
Suppose that we have also another -dimensional manifold with boundary . Form the closed manifold . Show that
Hint: section 7 of ASIT III