# Talk:Microbundles (Ex)

Marek Kaluba (Talk | contribs) |
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To prove that the second condition is satisfied we need to use local chart around $x$. | To prove that the second condition is satisfied we need to use local chart around $x$. | ||

− | Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. Now the first naive candidate for $h\colon V=U\ | + | Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. Now the first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However |

{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} |

## Revision as of 18:43, 29 May 2012

Let us begin with the definition of microbundle.

**Definition 0.1.**

- 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 for would be map . However

**Exercise 0.3** [Milnor1964, Lemma 2.1, Theorem 2.2]**.**
Let be a smooth manifold. Show that and are isomorphic microbundles.