Thom spaces (Ex)

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Exercise 0.1. 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

\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 be the universal oriented vector bundle of rank k and let (j_k,\overline{j_k}): \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}).

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\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})),

where \Sigma: \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 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,...?

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