# Talk:Microbundles (Ex)

Marek Kaluba (Talk | contribs) (WiP) |
Marek Kaluba (Talk | contribs) |
||

Line 23: | Line 23: | ||

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 | + | 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{ | ||

Line 50: | Line 50: | ||

{{endthm|Definition}} | {{endthm|Definition}} | ||

− | In our case | + | In our case we have |

− | + | $$ | |

− | + | \xymatrix{ | |

+ | U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\\ | ||

+ | U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,} | ||

+ | $$ | ||

+ | where $V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$ via local trivialisation. | ||

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

</wikitex> | </wikitex> |

## Revision as of 19:42, 29 May 2012

Let us begin with the definition of microbundle.

**Definition 0.1.**

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [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 0.3** [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.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case we have

where is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify via local trivialisation.

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [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 0.3** [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.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case we have

where is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify via local trivialisation.

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [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 0.3** [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.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case we have

where is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify via local trivialisation.