# Microbundle

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− | Two microbundles $(E_n, | + | Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute. |

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## Revision as of 11:47, 30 May 2012

An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:20, 16 May 2013 and the changes since publication. |

## 1 Introduction

The concept of Microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister1964] showed that every microbundle uniquely determines a topological -bundle.

**Definition 1.1** [Milnor1964] **.**

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

which makes the following diagram commute:

**Example 1.2** [Milnor1964, Lemma 2.1]**.**
Let be topological -manifold, let be the diagonal map and let be the projection onto the first factor. Then

is an -dimensional microbundle.

**Example 1.3.**Let be a topological -bundle with zero section . Then

is an -dimensional microbundle.

**Definition 1.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

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**Theorem 1.5** [Kister1964, Theorem 2] **.**
Let be an -dimensional microbundle. Then there is a neighbourhood of , such that:

- is the total space of a topological -bundle over .
- The inclusion is a microbundle isomorphism
- If is any other such neighbourhood of then there is a -bundle isomorphism
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.

## 2 References

- [Kister1964] J. M. Kister,
*Microbundles are fibre bundles*, Ann. of Math. (2)**80**(1964), 190–199. MR0180986 (31 #5216) Zbl 0131.20602 - [Milnor1964] J. Milnor,
*Microbundles. I*, Topology**3**(1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404