# Talk:Microbundles (Ex)

From Manifold Atlas

Revision as of 19:25, 29 May 2012 by Marek Kaluba (Talk | contribs)

Let us begin with the definition of microbundle.

**Definition 0.1.**

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.