Novikov additivity I (Ex)

Let $(Y,X)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(Y,X)$ be a $2n$$2n$-dimensional manifold with boundary, $n = 2k$$n = 2k$. Consider the homomorphism $\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (Y)$$\varphi \colon H^{n} (Y,X) \rightarrow H^{n} (Y)$ 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

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

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