Talk:Microbundles (Ex)

Let us begin with the definition of microbundle.

Definition 0.1.

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

$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qyqIyE$ $$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

and the following conditions hold.

1. $j\circ i=\id_B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_sk47qm$$j\circ i=\id_B$
2. for all $x\in B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ka1hP5$$x\in B$ there exist open neigbourhood $U\subset B/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_yv0iBP$$U\subset B$ and an open neighbourhood $V\subset E/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_eEwQKz$$V\subset E$ of $i(b)/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_aARWik$$i(b)$ and a homeomorphism
   $\displaystyle h\colon V\to U\times \mathbb{R}^n.$ $$\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]. 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, Theorem 2.2]. Let $M$$M$ be a smooth manifold. Show that $TM$$TM$ and $\xi_M$$\xi_M$ are isomorphic microbundles.