# Regular homotopy group of immersions (Ex)

The goal of this exercise is to get more feeling for the regular homotopy group of $k$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}k$-immersions in $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vjOEtO$$M$, $I_k(M)$$I_k(M)$ and the intersection/self-intersection form on it. Below are two definitions of it. Both are used in [Wall1999].

Definition 0.1. In [Lück2001] the group is defined as follows. Elements of $I_k(M)$$I_k(M)$ are represented by pointed $k$$k$-immersions, i.e pairs $(f,w)$$(f,w)$ with $f:S^k\looparrowright M$$f:S^k\looparrowright M$ is an immersion which does not necessarily map the basepoint $1\in S^k$$1\in S^k$ to the basepoint $b\in M$$b\in M$ and $w:I\rightarrow M$$w:I\rightarrow M$ is a path from $b$$b$ to $g(1)$$g(1)$. Two pairs $(f,w), (f',w')$$(f,w), (f',w')$ are considered equivalent if they are pointed homotopic, i.e. if there exists a regular homotopy $H:S^k\times I\rightarrow M$$H:S^k\times I\rightarrow M$ between $f$$f$ and $f'$$f'$ such that $w\ast H(1,-)$$w\ast H(1,-)$ and $w'$$w'$ are homotopic relative endpoints. The sum of $[(f_0,w_0)]$$[(f_0,w_0)]$ and $[(f_1,w_1)]$$[(f_1,w_1)]$ is defined by forming the (class of the) connected sum immersion $f_1\# f_2$$f_1\# f_2$ along with the (class of the) path $w_1\ast w_0^{-1}$$w_1\ast w_0^{-1}$. The action of $\pi_1(M,b)$$\pi_1(M,b)$ is given by mapping $[(f,w)]$$[(f,w)]$ to $[(f,w\ast\omega)]$$[(f,w\ast\omega)]$ where $\omega$$\omega$ is a loop at $b$$b$ representing a $g\in\pi_1(M)$$g\in\pi_1(M)$.

The equivariant intersection of $(g,w), (g',w')$$(g,w), (g',w')$ is described as follows. Choose representavives with $g_0$$g_0$ and $g_1$$g_1$ transverse. For every double point $(x_0,x_1)$$(x_0,x_1)$ with $g_0(x_0)=g_1(x_1)=d$$g_0(x_0)=g_1(x_1)=d$ determine the sign $\epsilon(d)$$\epsilon(d)$ in the usual way, i.e. by comparing orientations of $T_{x_0}S^k\oplus T_{x_1}S^k$$T_{x_0}S^k\oplus T_{x_1}S^k$ and $T_dM$$T_dM$. The element $g(d)$$g(d)$ is given by the class of the loop
$\displaystyle w_1\ast f_1(u_1)\ast f_0(u_0)^{-1}\ast w_0^{-1}$
where $u_i$$u_i$ is a path in $S^k$$S^k$ from $1$$1$ to $x_i$$x_i$.

Definition 0.2. In [Ranicki2002] the group is defined as follows. Elements of $I_k(M)$$I_k(M)$ are represented by $(f,\widetilde{f})$$(f,\widetilde{f})$ with $f:S^k\looparrowright M$$f:S^k\looparrowright M$ a $k$$k$-immersion and $\widetilde{g}:S^k\looparrowright \widetilde{M}$$\widetilde{g}:S^k\looparrowright \widetilde{M}$ a lift of $f$$f$ to the universal cover of $M$$M$. Two pairs $(f,\widetilde{f}), (f',\widetilde{f'})$$(f,\widetilde{f}), (f',\widetilde{f'})$ are considered equivalent if they are regular homotopy equivalent. The sum is given by connected sum. The action of $\pi_1(M,b)$$\pi_1(M,b)$ is via deck transformations on the lift.

To determine the equivariant intersection of $(f_0,\widetilde{f_0})$$(f_0,\widetilde{f_0})$ and $(f_1,\widetilde{f_1})$$(f_1,\widetilde{f_1})$ choose $f_0$$f_0$ and $f_1$$f_1$ to be transverse. For every doublepoint $(x_0,x_1)$$(x_0,x_1)$ with $f_0(x_0)=f_1(x_1)=d$$f_0(x_0)=f_1(x_1)=d$ there exists an element $g(d)$$g(d)$ such that $g(d)f_0(x_0)=f_1(x_1)$$g(d)f_0(x_0)=f_1(x_1)$. Define the equivariant index of $f_0$$f_0$ and $f_1$$f_1$ at $d$$d$ to be $\epsilon(d)g(d)\in\pm\pi_1(M)$$\epsilon(d)g(d)\in\pm\pi_1(M)$ where $\epsilon(d)$$\epsilon(d)$ is determined by comparing orientations again.

1) Show that the above definitions of $\mathbb{Z}[\pi_1(M)]$$\mathbb{Z}[\pi_1(M)]$-modules are equivalent.

2) Show that the descriptions of the equivariant intersections of (regular homotopy classes of) $k$$k$-immersions coincide.

3) Show that the corresponding descriptions of Wall's $\mu$$\mu$-form (the self-intersection form) coincide up to possible conjugation by a fixed element $\alpha\in\pi_1(M)$$\alpha\in\pi_1(M)$.