Quadratic forms I (Ex)

We follow [Lück2001, Definition 4.22] Let $P$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}P$ be a finitely generated free $\Lambda$$\Lambda$ module, $\Lambda$$\Lambda$ a ring with involution. Let

$\displaystyle T \colon \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*), \quad f \mapsto f^* \circ e(P)$ $$\displaystyle T \colon \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*), \quad f \mapsto f^* \circ e(P)$$

be the involution where $e(P) \colon P \to P^{**}$$e(P) \colon P \to P^{**}$ is the canonical isomorphism given by evaluation. Define

$\displaystyle Q_\epsilon(P) : = \textup{coker}(1 - \epsilon T) \colon \left( \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*) \right).$ $$\displaystyle Q_\epsilon(P) : = \textup{coker}(1 - \epsilon T) \colon \left( \textup{Hom}_{\Lambda} (P, P^*) \to \textup{Hom}_{\Lambda} (P, P^*) \right).$$

Exercise 0.1. Show that $\theta \in Q_\epsilon(P)$$\theta \in Q_\epsilon(P)$ defines a unique $\epsilon$$\epsilon$-quadratic form on $P$$P$ and that every such form arises in this way.

Remark 0.2. See also [Ranicki1980, Section 2].