# Talk:Microbundles (Ex)

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− | + | First, 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]^{\pi}&\\ | |

− | + | M\ar[dr]_{\Delta_M}\ar[ur]^{s_0} & & M\\ | |

+ | & M\times M \ar[ru]_{p_1}&} | ||

$$ | $$ | ||

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− | + | 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 $\{(x,x)\}$. | |

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− | U\times U\ | + | 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 $H\colon M\times \Rr^n\to M\times M$, $$H(x,v)=(x,\exp(b,v)).$$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 $$H(x,v)=(x,\exp(x,v)).$$As checked before this map maps the zero section to the diagonal. By definition of the expotential map the derivative of $H$ 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. | |

{{endproof}} | {{endproof}} | ||

</wikitex> | </wikitex> |

## Latest revision as of 09:16, 30 May 2012

First, 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 neighbourhood and a map such that points in the zero section ( in local coordinates) are mapped to the diagonal .

At each point this is easy: Fix and let be a neighbourhood of coming from the vector bundle structure. Choose a trivialization and then set ,

**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 neighbourhood and a map such that points in the zero section ( in local coordinates) are mapped to the diagonal .

At each point this is easy: Fix and let be a neighbourhood of coming from the vector bundle structure. Choose a trivialization and then set ,

**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 neighbourhood and a map such that points in the zero section ( in local coordinates) are mapped to the diagonal .

At each point this is easy: Fix and let be a neighbourhood of coming from the vector bundle structure. Choose a trivialization and then set ,