Thom spaces (Ex)

Exercise 0.1. Let $<wikitex>; {{beginthm|Exercise}} Let $X,X_1,X_2$ be $CW$-complexes and let $\xi,\xi_1,\xi_2$ be vector bundles over $X,X_1,X_2$ respectively. Denote by $\xi_1\times\xi_2$ the product bundle over $X_1\times X_2$. Find homeomorphisms $$ \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\cong\mathrm{Th}(\xi_1\times\xi_2),\quad \Sigma\mathrm{Th}(\xi):=S^1\wedge\mathrm{Th}(\xi)\cong\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}). $$ {{endthm}} {{beginthm|Exercise}} Let $\xi_k$ be the universal oriented vector bundle of rank $k$ and let $(j_k,\overline{j_k})$: $\xi_k\oplus\mathbb{R}\to\xi_{k+1}$ be a bundle map. {{endthm}} {{beginthm|Exercise}} Define $$ \gamma_k:=\mathrm{id}_X\times\xi_k,\quad (i_k,\overline{i_k}):=\mathrm{id}_X\times(j_k,\overline{j_k}). $$ Show that for all $k\geq0$ we have $V_{k+1}\circ\Omega_n(\overline{i_k})=V_k$. Define $$ \mathrm{Th}(\overline{i_k}):\quad \Sigma\mathrm{Th}(\gamma_k)\cong\mathrm{Th}(\gamma_k\oplus\mathbb{R})\to\mathrm{Th}(\gamma_{k+1}) $$ and $$ s_k:=\pi_{n+k+1}(\mathrm{Th}(\overline{i_k}))\circ\Sigma:\quad \pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\mathrm{Th}(\gamma_{k+1})), $$ where $\Sigma$ is the suspension homomorphism. Show that for all $k\geq0$ we have $P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k})=s_k\circ P_n(\gamma_k)$. {{endthm}} {{beginthm|Question}} Can we do similar things for unoriented manifolds, manifolds with spin structure,...? {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]X,X_1,X_2$ be $CW$-complexes and let $\xi,\xi_1,\xi_2$ be vector bundles over $X,X_1,X_2$ respectively. Denote by $\xi_1\times\xi_2$ the product bundle over $X_1\times X_2$. Find homeomorphisms

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

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

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

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

Exercise 0.3. Define

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

and

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

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

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