# 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 aroundTex syntax error.

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 ,Tex syntax errordoes and define

Tex syntax erroris non-vanishing along the zero section, so by the inverse function theorem there exist a neighbourhood of on which

Tex syntax erroris a diffeomorphism.

**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 aroundTex syntax error.

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 ,Tex syntax errordoes and define

Tex syntax erroris non-vanishing along the zero section, so by the inverse function theorem there exist a neighbourhood of on which

Tex syntax erroris a diffeomorphism.