Novikov additivity I (Ex)

Let $(Y,X)$$<wikitex>; Let $(Y,X)$ be a n$-dimensional manifold with boundary, $n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\hat{H}^{n} (Y)$ the image of $\varphi$. Coefficients are understood to be in $\mathbb{R}$. The middle dimensional intersection form $$ H^{n} (Y) \otimes H^{n} (Y) \rightarrow \mathbb{R} $$ is degenerate in general. Show that the intersection form $B$ on $\hat{H}^{n} (Y)$ defined by $$ 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). $$ Observe that the analogous statement is true if we replace manifols with boundary by Poincare pairs. Hint: section 7 of {{cite|Atiyah&Singer1968b}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](Y,X)$ be a $2n$$2n$-dimensional manifold with boundary, $n = 2k$$n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$$\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (X)$ and denote $\hat{H}^{n} (Y)$$\hat{H}^{n} (Y)$ the image of $\varphi$$\varphi$. Coefficients are understood to be in $\mathbb{R}$$\mathbb{R}$.

The middle dimensional intersection form

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

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

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

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

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

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

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

Hint: section 7 of [Atiyah&Singer1968b]

References

• [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