Thom spaces (Ex)
From Manifold Atlas
Revision as of 09:47, 2 April 2012 by Andreas Hermann (Talk | contribs)
Exercise 0.1.
Let be
-complexes and let
be vector bundles
over
respectively.
Denote by
the product bundle over
.
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}}).](/images/math/e/7/f/e7f14912a00d70354c944c4bb6fa09fa.png)
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
![\displaystyle \gamma_k:=\mathrm{id}_X\times\xi_k,\quad (i_k,\overline{i_k}):=\mathrm{id}_X\times(j_k,\overline{j_k}).](/images/math/6/7/f/67f317aac8ce8bcecb21621d3116f58b.png)
Show that for all we have
.
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})](/images/math/5/c/b/5cb72adca900f9b63746b0592a2e3c4a.png)
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})),](/images/math/d/b/d/dbd0e226b3976f563598f44734761688.png)
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,...?