Surgery obstruction, Arf-invariant (Ex)

Let $(K,\lambda,\mu)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(K,\lambda,\mu)$ be a $(-1)$$(-1)$-symmetric unimodular quadratic form over $\Z$$\Z$ so that $(K,\lambda)\cong H_{-1}(\Z^r)$$(K,\lambda)\cong H_{-1}(\Z^r)$ with canonical basis $\{e_1,\ldots, e_r,f_1,\ldots, f_r\}$$\{e_1,\ldots, e_r,f_1,\ldots, f_r\}$. Recall that the quadratic refinement $\mu: K \to \Z_2$$\mu: K \to \Z_2$ is a function such that for all $x,y\in K$$x,y\in K$,

$\displaystyle \mu(x+y) = \mu(x)+\mu(y) + \lambda(x,y)\quad (\mathrm{mod}\;2).$ $$\displaystyle \mu(x+y) = \mu(x)+\mu(y) + \lambda(x,y)\quad (\mathrm{mod}\;2).$$

Define the Arf invariant of $(K,\lambda,\mu)$$(K,\lambda,\mu)$ by

$\displaystyle A(K,\lambda,\mu) := \sum_{i=1}^r\mu(e_i)\mu(f_i) \in \Z_2.$ $$\displaystyle A(K,\lambda,\mu) := \sum_{i=1}^r\mu(e_i)\mu(f_i) \in \Z_2.$$

Prove that the Arf invariant is well-defined and defines an isomorphism

$\displaystyle A:L_2(\Z)\cong \Z_2.$ $$\displaystyle A:L_2(\Z)\cong \Z_2.$$

Hint: Start by classifying quadratic forms on $\Z^2$$\Z^2$, use induction and also count the size of the sets $\mu^{-1}(0)$$\mu^{-1}(0)$ and $\mu^{-1}(1)$$\mu^{-1}(1)$.