Surgery obstruction, signature (Ex)

1) Let $(f, \overline{f}):M^{2n} \to X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(f, \overline{f}):M^{2n} \to X$ be an $n$$n$-connected degree one normal map. Let $(K_n(M),\lambda, \mu)$$(K_n(M),\lambda, \mu)$ be the associated kernel form. Prove that signature satisfies the relation

$\displaystyle \mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$ $$\displaystyle \mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$$

2) Prove that if $\mathrm{sign}(K_n(M),\lambda)=0$$\mathrm{sign}(K_n(M),\lambda)=0$ then we can find a Lagrangian with respect to $\lambda$$\lambda$. (This is proven in [Milnor1961, Lemma 8 and 9 of Milnor1961]

You may assume the following lemma:

Lemma 0.1 [Milnor1961, Lemma 8]. A quadratic form $\varphi$$\varphi$ with integer coefficients and determinant $\pm 1$$\pm 1$ has a non-trivial zero if and only if it is indefinite.

References

• [Milnor1961] J. Milnor, A procedure for killing homotopy groups of differentiable manifolds, Proc. Sympos. Pure Math, Vol. III (1961), 39–55. MR0130696 (24 #A556) Zbl 0118.18601