Talk:Microbundles (Ex)

Definition 0.1.

$$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

1. $j\circ i=\id_B$
2. for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism
   $$\displaystyle h\colon V\to U\times \mathbb{R}^n.$$

Moreover, the homeomorphism above must make the following diagram commute:

where $p_1$ is projection on the first factor and $U$ is included as a $0$-section in $U\times \mathbb{R}^n$.

Exercise 0.2 [Milnor1964, Lemma 2.1]. Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.

Proof.

Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied.

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 choice for neighbourhood $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute

since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$.

$\square$

Exercise 0.3 [Milnor1964, Theorem 2.2]. Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ 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 $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section.

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

Definition 0.4. Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute.

In our case we have

where $V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$ via local trivialisation.

$\square$