Novikov additivity I (Ex)

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Let (Y,X) be a 2n-dimensional manifold with boundary, n = 2k. Consider the homomorphism \varphi \colon H^{n} (Y,X) \rightarrow H^{n} (Y) 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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