Talk:Microbundles (Ex)
Marek Kaluba (Talk | contribs) |
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− | + | In case of daubts You should get familiar with the definition of [[Microbundle|microbundle]]. | |
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{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} | ||
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To prove that the second condition is satisfied we need to use local chart around $x$. | 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 | 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 | ||
+ | |||
$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
− | U \ar[ | + | &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: $h(u,v)=(u,h(u)-h(v))$. | 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: $h(u,v)=(u,h(u)-h(v))$. | ||
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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. | 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 [[Microbundle|microbundles]] . | |
− | + | In our case we have | |
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$$ | $$ | ||
\xymatrix{ | \xymatrix{ | ||
− | + | & V\ar[dd]^H \ar[rd]^{p_1}&\\ | |
− | + | M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\\ | |
+ | & M\times M \ar[ru]&} | ||
$$ | $$ | ||
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− | + | where $V\subset TM$ is an open neighbourhood of the zero section. | |
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− | where $V\subset TM$ is an open neighbourhood of the zero section | + | |
{{endproof}} | {{endproof}} | ||
+ | We need to find a map $H\colon V\to | ||
</wikitex> | </wikitex> |
Revision as of 02:10, 30 May 2012
In case of daubts You should get familiar with the definition of microbundle.
Exercise 0.1 [Milnor1964, Lemma 2.1]. Let be a topological manifold. Show that is a microbundle.
Proof.
Let be a topological manifold. Then the composition sends , so the first condition in the definition is satisfied.
To prove that the second condition is satisfied we need to use local chart around . Choose to be one of the open sets coming from atlas of and let be associated chart. The obvious choice for neighbourhood is to take . The first naive candidate for would be map . However such fails to make the following diagram commute
since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .
Exercise 0.2 [Milnor1964, Theorem 2.2]. Let be a (paracompact!) smooth manifold. Show that and 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 and treating it just as a microbundle where is the zero section.
To fix the notation please consult the definition of microbundle isomorphism on page on microbundles .
In our case we have
where is an open neighbourhood of the zero section.
We need to find a map $H\colon V\to
$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$. {{endproof}} {{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}} Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. {{endthm}} {{beginproof}} 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 [[Microbundle|microbundles]] . In our case we have $$ \xymatrix{ & V\ar[dd]^H \ar[rd]^{p_1}&\ M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\ & M\times M \ar[ru]&} $$ where $V\subset TM$ is an open neighbourhood of the zero section. {{endproof}} We need to find a map $H\colon V\to M be a topological manifold. Show that is a microbundle.Proof.
Let be a topological manifold. Then the composition sends , so the first condition in the definition is satisfied.
To prove that the second condition is satisfied we need to use local chart around . Choose to be one of the open sets coming from atlas of and let be associated chart. The obvious choice for neighbourhood is to take . The first naive candidate for would be map . However such fails to make the following diagram commute
since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .
Exercise 0.2 [Milnor1964, Theorem 2.2]. Let be a (paracompact!) smooth manifold. Show that and 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 and treating it just as a microbundle where is the zero section.
To fix the notation please consult the definition of microbundle isomorphism on page on microbundles .
In our case we have
where is an open neighbourhood of the zero section.
We need to find a map $H\colon V\to