# Talk:Microbundles (Ex)

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:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [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 candidate for 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 0.3** [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.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case