Novikov additivity I (Ex)

Latest revision as of 12:28, 1 June 2012

Let $<wikitex>; Let $(Y,X)$ be a n$-dimensional manifold with boundary, $n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\widehat{H}^{n} (Y)$ the image of $\varphi$. Coefficients are understood to be in $\mathbb{R}$. The middle dimensional intersection form $$ H^{n} (Y) \otimes H^{n} (Y) \rightarrow \mathbb{R} $$ is degenerate in general. Show that the intersection form $B$ on $\widehat{H}^{n} (Y)$ defined by $$ B(\varphi (a), \varphi (b)) = \langle a \cup b , [Y] \rangle $$ 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 n$-dimensional manifold $Y'$ with boundary $-X$. Form the closed manifold $Z = Y' \cup_{X} Y$. Show that $$ \textup{Sign} (Z) = \textup{Sign} (Y') + \textup{Sign} (Y). $$ Hint: section 7 of ASIT III </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](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

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

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]

[edit] References

• [Atiyah&Singer1968b] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR0236952 (38 #5245) Zbl 0164.24301