# Talk:Microbundles (Ex)

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{{beginthm|Definition|}} | {{beginthm|Definition|}} | ||

An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold. | An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold. | ||

− | + | #$j\circ i=\id_B$ | |

− | + | #for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h\colon V\to U\times \mathbb{R}^n.$$ | |

+ | |||

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

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

+ | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} | ||

+ | Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. | ||

+ | {{endthm}} | ||

+ | 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 $x$. | ||

+ | 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 candidate for $V\subset M\times M$ is to take $U\times U$. Now the first naive candidate for $h\colon V=U\timesU\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However | ||

+ | |||

+ | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1, Theorem 2.2}}}} | ||

+ | Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. | ||

+ | {{endthm}} | ||

</wikitex> | </wikitex> |

## Revision as of 18:25, 29 May 2012

Let us begin with the definition of microbundle.

**Definition 0.1.**

and the following conditions hold.

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

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

**Exercise 0.2** [Milnor1964, Lemma 2.1, Theorem 2.2]**.**
Let be a topological manifold. Show that is a microbundle.

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 candidate for is to take . Now the first naive candidate forTex syntax errorwould be map . However

**Exercise 0.3** [Milnor1964, Lemma 2.1, Theorem 2.2]**.**
Let be a smooth manifold. Show that and are isomorphic microbundles.