Talk:Microbundles (Ex)

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

Exercise 0.1 [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/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_vxPwc4. 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

\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) 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 0.2 [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.

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 is an open neighbourhood of the zero section.

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

At each point this is easy: Fix b\in M and let V'\subset TM be a neighbourhood of i(b) coming from the vector bundle structure. Choose a trivialization V'\to M\times \Rr^n and then set H\colon M\times \Rr^n\to M\times M,
\displaystyle H(x,v)=(x,\exp(b,v)).
By definition of \exp we have H(b,0)=(b,\exp(b,0))=(b,b). However, we may now let b vary as 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 is non-vanishing along the zero section, so by the inverse function theorem there exist a neighbourhood V\subset TM of M on which H is a diffeomorphism.
\square


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