Thom spaces (Ex)
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(Created page with "<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 $\xi_1\times\xi_2$ th...") |
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$$ | $$ | ||
{{endthm}} | {{endthm}} | ||
+ | With the following exercises we work out the details of {{citeD|Lück2001|page 58f}}. | ||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
Let $\xi_k$ be the universal oriented vector bundle of rank $k$ | 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. | + | and let $(j_k,\overline{j_k})$: $\xi_k\oplus\underline{\mathbb{R}}\to\xi_{k+1}$ be a bundle map. Define |
− | + | ||
− | + | ||
− | Define | + | |
$$ | $$ | ||
\gamma_k:=\mathrm{id}_X\times\xi_k,\quad | \gamma_k:=\mathrm{id}_X\times\xi_k,\quad | ||
(i_k,\overline{i_k}):=\mathrm{id}_X\times(j_k,\overline{j_k}). | (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 | + | Show that for all $k\geq0$ we have $V_{k+1}\circ\Omega_n(\overline{i_k})=V_k$. |
+ | {{endthm}} | ||
+ | {{beginthm|Exercise}} | ||
+ | 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}) | + | \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 | and | ||
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\pi_{n+k}(\mathrm{Th}(\gamma_k))\to\pi_{n+k+1}(\mathrm{Th}(\gamma_{k+1})), | \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 $P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k})=s_k\circ P_n(\gamma_k)$. | + | 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 $P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k})=s_k\circ P_n(\gamma_k)$. | ||
{{endthm}} | {{endthm}} | ||
{{beginthm|Question}} | {{beginthm|Question}} | ||
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{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
− | == References == | + | <!-- == References == |
− | {{#RefList:}} | + | {{#RefList:}} --> |
[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises with solution]] |
Latest revision as of 09:47, 2 April 2012
Exercise 0.1. Let be -complexes and let be vector bundles over respectively. Denote by the product bundle over . Find homeomorphisms
With the following exercises we work out the details of [Lück2001, page 58f].
Exercise 0.2. Let be the universal oriented vector bundle of rank and let : be a bundle map. Define
Show that for all we have .
Exercise 0.3. Define
and
where : is the suspension homomorphism. Show that for all we have .
Question 0.4. Can we do similar things for unoriented manifolds, manifolds with spin structure,...?