# Talk:Microbundles (Ex)

First, You should get familiar with the definition of microbundle.

**Exercise 0.1** [Milnor1964, Lemma 2.1]**.**
Let be a topological manifold. Show that is a microbundle.

**Proof.**

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 choice for neighbourhood is to take . The first naive candidate for would be map . However such fails to make the following diagram commute

since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .

**Exercise 0.2** [Milnor1964, Theorem 2.2]**.**
Let be a (paracompact!) smooth manifold. Show that and are isomorphic microbundles.

**Proof.**
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on and treating it just as a microbundle where is the zero section.

To fix the notation please consult the definition of microbundle isomorphism on page on microbundles .

In our case we have

where is an open neighbourhood of the zero section.

We need to find a neighbourhood and a map such that points in the zero section ( in local coordinates) are mapped to the diagonal .

At each point this is easy: Fix and let be a neighbourhood of coming from the vector bundle structure. Choose a trivialization and then set ,