# Talk:Microbundles (Ex)

First, You should get familiar with the definition of microbundle.

Exercise 0.1 [Milnor1964, Lemma 2.1]. Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$$\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.

Proof.

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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vxPwc4$$x$. Choose $U$$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 choice for neighbourhood $V\subset M\times M$$V\subset M\times M$ is to take $U\times U$$U\times U$. 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 such $h$$h$ fails to make the following diagram commute $\displaystyle \xymatrix{ &U\times U\ar[rd]^{p_1}\ar[dd]^h&\\ U\ar[ru]^{\Delta_M}\ar[rd]_{\id\times\{0\}} & & U\\ &U\times \Rr^n\ar[ru]_{p_1}&}$

since $(u,u)$$(u,u)$ is mapped to $(u,\phi(u))$$(u,\phi(u))$ and $\phi(u)$$\phi(u)$ doesn't necessarily be $0$$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))$$h(u,v)=(u,h(u)-h(v))$. $\square$$\square$

Exercise 0.2 [Milnor1964, Theorem 2.2]. Let $M$$M$ be a (paracompact!) smooth manifold. Show that $TM$$TM$ and $\xi_M$$\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$$TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$$(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$$M\xrightarrow{s_0} TM$ is the zero section.

To fix the notation please consult the definition of microbundle isomorphism on page on microbundles .

In our case we have $\displaystyle \xymatrix{ & V\ar[dd]^H \ar[rd]^{\pi}&\\ M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\\ & M\times M \ar[ru]_{p_1}&}$

where $V\subset TM$$V\subset TM$ is an open neighbourhood of the zero section.

We need to find a neighbourhood $V$$V$ and a map $H\colon V\to U\times U$$H\colon V\to U\times U$ such that points in the zero section ( $\{(x,0)\}$$\{(x,0)\}$ in local coordinates) are mapped to the diagonal $\{(x,x)\}$$\{(x,x)\}$.

At each point this is easy: Fix $b\in M$$b\in M$ and let $V'\subset TM$$V'\subset TM$ be a neighbourhood of $i(b)$$i(b)$ coming from the vector bundle structure. Choose a trivialization $V'\to M\times \Rr^n$$V'\to M\times \Rr^n$ and then set $H\colon M\times \Rr^n\to M\times M$$H\colon M\times \Rr^n\to M\times M$, $\displaystyle H(x,v)=(x,\exp(b,v)).$
By definition of $\exp$$\exp$ we have $H(b,0)=(b,\exp(b,0))=(b,b)$$H(b,0)=(b,\exp(b,0))=(b,b)$. However, we may now let $b$$b$ vary as $x$$x$ does and define $\displaystyle H(x,v)=(x,\exp(x,v)).$
As checked before this map maps the zero section to the diagonal. By definition of the expotential map the derivative of $H/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_I73PBF$$H$ is non-vanishing along the zero section, so by the inverse function theorem there exist a neighbourhood $V\subset TM$$V\subset TM$ of $M$$M$ on which $H$$H$ is a diffeomorphism. $\square$$\square$