Novikov additivity I (Ex)

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Observe that the analogous statement is true if we replace manifols with boundary by Poincare pairs.
Observe that the analogous statement is true if we replace manifols with boundary by Poincare pairs.
Hint: section 7 of ASIT III
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Hint: section 7 of {{cite|Atiyah&Singer1968b}}
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== References ==
== References ==

Revision as of 10:31, 29 May 2012

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

\displaystyle  H^{n} (Y) \otimes H^{n} (Y) \rightarrow \mathbb{R}

is degenerate in general. Show that the intersection form B on \hat{H}^{n} (Y) defined by

\displaystyle  B(\varphi (a), \varphi (b)) = \langle a \cup b , [Y] \rangle

is a non-degenerate symmetric bilinear form and let us define the signature \textup{Sign} (Y) to be the signature of this form.

Suppose that we have also another 2n-dimensional manifold Y' with boundary -X. Form the closed manifold Z = Y' \cup_{X} Y. Show that

\displaystyle  \textup{Sign} (Z) = \textup{Sign} (Y') + \textup{Sign} (Y).

Observe that the analogous statement is true if we replace manifols with boundary by Poincare pairs.

Hint: section 7 of [Atiyah&Singer1968b]

References

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