# Microbundle

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. |

The users responsible for this page are: Diarmuid Crowley, Matthias Kreck. No other users may edit this page at present. |

## Contents |

## 1 Definition

The concept of a **microbundle** of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological -bundle; i.e. a fibre bundle with structure group the homeomorphisms of fixing .

**Definition 1.1** [Milnor1964] **.**
Let be a topological space. An **-dimensional microbundle** over is a quadruple
where is a space, and are maps fitting into the following diagram

and the following conditions hold:

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

which makes the following diagram commute:

The space is called the **total space** of the bundle and the **base space**.

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

## 2 Examples

An important example of a microbundle is the **tangent microbundle** of a topological (or similarly ) manifold .
Let

be the diagonal map for .

**Example 2.1** [Milnor1964, Lemma 2.1]**.**
Let be topological (or PL) -manifold, and let be the projection onto the first factor. Then

is an -dimensional microbundle, the **tangent microbundle** of .

**Remark 2.2.**
An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.

Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.

**Example 2.3.** Let be a topological -bundle with zero section . Then the quadruple

is an -dimensional microbundle.

## 3 The Kister-Mazur Theorem

A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.

**Theorem 3.1** [Kister1964, Theorem 2]**.**
Let be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex .
Then there is a neighbourhood of , such that the following hold.

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

**Remark 3.2.**
Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and -structures on higher dimensional topological manifolds.

## 4 References

- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004 - [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

## 5 External links

- The Wikipedia page about microbundles.

**Definition 1.1** [Milnor1964] **.**
Let be a topological space. An **-dimensional microbundle** over is a quadruple
where is a space, and are maps fitting into the following diagram

and the following conditions hold:

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

which makes the following diagram commute:

The space is called the **total space** of the bundle and the **base space**.

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

## 2 Examples

An important example of a microbundle is the **tangent microbundle** of a topological (or similarly ) manifold .
Let

be the diagonal map for .

**Example 2.1** [Milnor1964, Lemma 2.1]**.**
Let be topological (or PL) -manifold, and let be the projection onto the first factor. Then

is an -dimensional microbundle, the **tangent microbundle** of .

**Remark 2.2.**
An atlas of gives a product atlas of which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold in is isomorphic to its tangent bundle.

Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.

**Example 2.3.** Let be a topological -bundle with zero section . Then the quadruple

is an -dimensional microbundle.

## 3 The Kister-Mazur Theorem

A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.

**Theorem 3.1** [Kister1964, Theorem 2]**.**
Let be an -dimensional microbundle over a locally finite, finite dimensional simplicial complex .
Then there is a neighbourhood of , such that the following hold.

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

**Remark 3.2.**
Microbundle theory is an important part of the work by Kirby and Siebenmann [Kirby&Siebenmann1977] on smooth structures and -structures on higher dimensional topological manifolds.

## 4 References

- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004 - [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

## 5 External links

- The Wikipedia page about microbundles.