Thom spaces (Ex)

Exercise 0.1. Let $X,X_1,X_2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X,X_1,X_2$ be $CW$$CW$-complexes and let $\xi,\xi_1,\xi_2$$\xi,\xi_1,\xi_2$ be vector bundles over $X,X_1,X_2$$X,X_1,X_2$ respectively. Denote by $\xi_1\times\xi_2$$\xi_1\times\xi_2$ the product bundle over $X_1\times X_2$$X_1\times X_2$. Find homeomorphisms

$\displaystyle \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\cong\mathrm{Th}(\xi_1\times\xi_2),\quad \Sigma\mathrm{Th}(\xi):=S^1\wedge\mathrm{Th}(\xi)\cong\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}).$ $$\displaystyle \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\cong\mathrm{Th}(\xi_1\times\xi_2),\quad \Sigma\mathrm{Th}(\xi):=S^1\wedge\mathrm{Th}(\xi)\cong\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}).$$

With the following exercises we work out the details of [Lück2001, page 58f].

Exercise 0.2. Let $\xi_k$$\xi_k$ be the universal oriented vector bundle of rank $k$$k$ and let $(j_k,\overline{j_k})$$(j_k,\overline{j_k})$: $\xi_k\oplus\underline{\mathbb{R}}\to\xi_{k+1}$$\xi_k\oplus\underline{\mathbb{R}}\to\xi_{k+1}$ be a bundle map. Define

$\displaystyle \gamma_k:=\mathrm{id}_X\times\xi_k,\quad (i_k,\overline{i_k}):=\mathrm{id}_X\times(j_k,\overline{j_k}).$ $$\displaystyle \gamma_k:=\mathrm{id}_X\times\xi_k,\quad (i_k,\overline{i_k}):=\mathrm{id}_X\times(j_k,\overline{j_k}).$$

Show that for all $k\geq0$$k\geq0$ we have $V_{k+1}\circ\Omega_n(\overline{i_k})=V_k$$V_{k+1}\circ\Omega_n(\overline{i_k})=V_k$.

Exercise 0.3. Define

$\displaystyle \mathrm{Th}(\overline{i_k}):\quad \Sigma\mathrm{Th}(\gamma_k)\cong\mathrm{Th}(\gamma_k\oplus\underline{\mathbb{R}})\to\mathrm{Th}(\gamma_{k+1})$ $$\displaystyle \mathrm{Th}(\overline{i_k}):\quad \Sigma\mathrm{Th}(\gamma_k)\cong\mathrm{Th}(\gamma_k\oplus\underline{\mathbb{R}})\to\mathrm{Th}(\gamma_{k+1})$$

and

$\displaystyle s_k:=\pi_{n+k+1}(\mathrm{Th}(\overline{i_k}))\circ\Sigma:\quad \pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\mathrm{Th}(\gamma_{k+1})),$ $$\displaystyle s_k:=\pi_{n+k+1}(\mathrm{Th}(\overline{i_k}))\circ\Sigma:\quad \pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\mathrm{Th}(\gamma_{k+1})),$$

where $\Sigma$$\Sigma$: $\pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\Sigma\mathrm{Th}(\gamma_k))$$\pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\Sigma\mathrm{Th}(\gamma_k))$ is the suspension homomorphism. Show that for all $k\geq0$$k\geq0$ we have $P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k})=s_k\circ P_n(\gamma_k)$$P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k})=s_k\circ P_n(\gamma_k)$.

Question 0.4. Can we do similar things for unoriented manifolds, manifolds with spin structure,...?