# Microbundle

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

− | + | which makes the following diagram commute: | |

− | + | ||

$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$ | ||

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

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{{endrem}} | {{endrem}} | ||

− | + | {{beginthm|Definition}} | |

+ | Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ 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. | ||

+ | |||

+ | $$ | ||

+ | \xymatrix{ | ||

+ | & V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\ | ||

+ | B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\ | ||

+ | & V_2 \ar[ru]_{j_2|_{V_2}} | ||

+ | } | ||

+ | $$ | ||

+ | |||

+ | {{endthm|Definition}} | ||

{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} | {{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} |

## Revision as of 22:37, 29 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