Let us begin with the definition of microbundle.
Definition 0.1.An -dimensional microbundle is a quadruple such that there is a sequence
- 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 for
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Exercise 0.3 [Milnor1964, Lemma 2.1, Theorem 2.2]. Let be a smooth manifold. Show that and are isomorphic microbundles.