# Talk:Microbundles (Ex)

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Let us begin with the definition of microbundle.

Definition 0.1.

An $n$$; Let us begin with the definition of microbundle. {{beginthm|Definition|}} An n-dimensional microbundle is a quadruple (E,B,i,j) such that there is a sequence B\xrightarrow{i} E\xrightarrow{j} B and the following conditions hold. #j\circ i=\id_B #for all x\in B there exist open neigbourhood U\subset B and an open neighbourhood V\subset E of i(b) and a homeomorphism h\colon V\to U\times \mathbb{R}^n. Moreover, the homeomorphism above must make the following diagram commute: {{endthm}} {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} Let M be a topological manifold. Show that \xi_M : = (M \times M, M, \Delta_M, p_1) is a microbundle. {{endthm}} Let M be a topological manifold. Then the composition p_1\circ\Delta_M sends x\mapsto (x,x)\mapsto x, so the first condition in the definition is satisfied. To prove that the second condition is satisfied we need to use local chart around x. Choose U to be one of the open sets coming from atlas of M and let \phi\colon U\to \mathbb{R}^n be associated chart. The obvious candidate for V\subset M\times M is to take U\times U. Now the first naive candidate for h\colon V=U\timesU\to U\times\mathbb{R}^n would be map \id\times \phi. However {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} Let M be a smooth manifold. Show that TM and \xi_M are isomorphic microbundles. {{endthm}} n$-dimensional microbundle is a quadruple $(E,B,i,j)$$(E,B,i,j)$ such that there is a sequence
$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$
and the following conditions hold.
1. $j\circ i=\id_B$$j\circ i=\id_B$
2. for all $x\in B$$x\in B$ there exist open neigbourhood $U\subset B$$U\subset B$ and an open neighbourhood $V\subset E$$V\subset E$ of $i(b)$$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:

Exercise 0.2 [Milnor1964, Lemma 2.1, Theorem 2.2]. Let $M$$M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$$\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.

Let $M$$M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$$p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$$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$$x$. Choose $U$$U$ to be one of the open sets coming from atlas of $M$$M$ and let $\phi\colon U\to \mathbb{R}^n$$\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$$V\subset M\times M$ is to take $U\times U$$U\times U$. Now the first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$$h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$$\id\times \phi$. However

Exercise 0.3 [Milnor1964, Lemma 2.1, Theorem 2.2]. Let $M$$M$ be a smooth manifold. Show that $TM$$TM$ and $\xi_M$$\xi_M$ are isomorphic microbundles.