# Talk:Microbundles (Ex)

Marek Kaluba (Talk | contribs) |
Marek Kaluba (Talk | contribs) (2nd exercise WIP) |
||

Line 1: | Line 1: | ||

<wikitex>; | <wikitex>; | ||

− | + | In case of daubts You should get familiar with the definition of [[Microbundle|microbundle]]. | |

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

{{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} | ||

Line 24: | Line 11: | ||

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: $h(u,v)=(u,h(u)-h(v))$. | ||

Line 39: | Line 28: | ||

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 | |

− | + | ||

$$ | $$ | ||

\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]&} | ||

$$ | $$ | ||

− | |||

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

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | + | ||

− | 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