Talk:Microbundles (Ex)
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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. | ||
#$j\circ i=\id_B$ | #$j\circ i=\id_B$ | ||
− | #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 $$h\colon V\to U\times \mathbb{R}^n.$$ | + | #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: | Moreover, the homeomorphism above must make the following diagram commute: | ||
+ | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | ||
{{endthm}} | {{endthm}} | ||
Revision as of 19:49, 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.