Talk:Microbundles (Ex)
(Difference between revisions)
m |
m |
||
Line 9: | Line 9: | ||
{{endthm}} | {{endthm}} | ||
− | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1 | + | {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} |
Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. | Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. | ||
{{endthm}} | {{endthm}} | ||
Line 18: | Line 18: | ||
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 $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However | 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 $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However | ||
− | {{beginthm|Exercise|{{citeD|Milnor1964| | + | {{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}} |
Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. | Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> |
Revision as of 19:44, 29 May 2012
Let us begin with the definition of microbundle.
Definition 0.1.
An -dimensional microbundle is a quadruple such that there is a sequence 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]. 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 for would be map . However
Exercise 0.3 [Milnor1964, Theorem 2.2]. Let be a smooth manifold. Show that and are isomorphic microbundles.