Talk:Microbundles (Ex)
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Revision as of 19:25, 29 May 2012 by Marek Kaluba (Talk | contribs)
Let us begin with the definition of microbundle.
Definition 0.1.
An -dimensional microbundle is a quadruple such that there is a sequence and the following conditions hold.
- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism
Moreover, the homeomorphism above must make the following diagram commute:
Exercise 0.2 [Milnor1964, Lemma 2.1, Theorem 2.2]. 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 forTex syntax errorwould be map . However
Exercise 0.3 [Milnor1964, Lemma 2.1, Theorem 2.2]. Let be a smooth manifold. Show that and are isomorphic microbundles.