Talk:Microbundles (Ex)
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Revision as of 19:49, 29 May 2012 by Diarmuid Crowley (Talk | contribs)
Let us begin with the definition of microbundle.
Definition 0.1.
An![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(E,B,i,j)](/images/math/7/1/0/71015335477690380a3d9e5627f000db.png)
![\displaystyle B\xrightarrow{i} E\xrightarrow{j} B](/images/math/4/0/a/40a68b194c05ed02a16adc8c005f25f3.png)
- for all
there exist open neigbourhood
and an open neighbourhood
of
and a homeomorphism
Moreover, the homeomorphism above must make the following diagram commute:
![\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}](/images/math/a/1/4/a1489c27410f30906c655cf1043aba2d.png)
Exercise 0.2 [Milnor1964, Lemma 2.1].
Let be a topological manifold. Show that
is a microbundle.
Let be a topological manifold. Then the composition
sends
, so the first condition in the definition is satisfied.
To prove that the second condition is satisfied we need to use local chart around .
Choose
to be one of the open sets coming from atlas of
and let
be associated chart. The obvious candidate for
is to take
. Now the first naive candidate for
would be map
. However
Exercise 0.3 [Milnor1964, Theorem 2.2].
Let be a smooth manifold. Show that
and
are isomorphic microbundles.