# nLab microbundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Idea

A microbundle is something like an approximation to the notion of vector bundle: a locally trivial bundle $E \to X$ of topological spaces that has a section. Indeed, as observed by Milnor, every vector bundle gives an example of a microbundle (for a modern treatment see Kupers18, Example 27.2.3 or Lurie’s course, Topics in Geometric Topology, Lecture 10).

## Definition

### Microbundles

A real microbundle of dimension $n$ is a 4-tuple $\xi = (E,p,B,i)$ where

• $E$ is a topological space (the total space of $\xi$),

• $B$ is a topological space (the base space of $\xi$,

• and $p:E\to B$ a continuous map (projection),

• $i:B\hookrightarrow E$ another continuous map (inclusion of base space)

such that

• $i$ is a section of $p$, i.e. $p\circ i = id_B$

• the local triviality condition holds:

for all $b\in B$, there are neighborhoods $U\ni b$ and $V\ni i(b)$ and a homeomorphism $h:U\times R^n\to V\cap p^{-1}(U)$ such that $p(h(u,v))=u$ and $h(u,0)=i(u)$ for all $u\in U$. The open subspace $i(B)$ is called the zero section of $\xi$.

### Morphisms of microbundles

A morphism of microbundles $\phi:\xi\to\xi'$ is a germ of maps from neighborhoods of the zero section of $\xi$ to $\xi'$, which commutes with projections and inclusions, with composition defined for representatives as composition of functions on smaller neighborhoods.

In particular, an isomorphism of microbundles can be represented by a homeomorphism from a neighborhood $V$ of the zero section in $\xi$ to a neighborhood $V'$ of the zero section in $\xi'$ commuting with projections and inclusions of the zero sections.

## Examples

### Tangent microbundle

The main example is the tangent microbundle $(M\times M,p_1,M,i)$ of a topological manifold $M$ where $p_1:M\times M\to M$ is the projection onto the first factor. If $(U,f)$ is a chart of the manifold $M$ around point $x\in M$ (where $x\in U\subset M$ and $f:U\to R^n$ is a homeomorphism with $h(x)=0$) then define $h:U\times R^n\to U\times U$ by $h(u,v)=(u,f^{-1}(f(v)-u))$.

If $M$ is a smooth manifold, then the tangent microbundle is equivalent to the tangent bundle (Kupers18, Example 27.2.3).

David Roberts: A couple of years ago I thought of importing topological groupoids to this concept for the following reason: The tangent microbundle $M\times M$, when $M$ is a manifold, is the groupoid integrating the tangent bundle $TM$ of $M$. If we have a general Lie groupoid, we can form the Lie algebroid, which is a very interesting object. If we have a topological groupoid, it seems to me that there should be a microbundle-like object that acts like the algebroid of that groupoid. This should reduce to the tangent microbundle in the case of the codiscrete groupoid = pair groupoid. Perhaps not all topological groupoids would have an associated algebroid, but those wih source and target maps that are topological submersions probably will.

## References

Microbundles were defined by John Milnor. The original paper can be found here.

Classic treatments of their elementary theory include:

Useful references are for instance

Last revised on January 14, 2020 at 09:43:56. See the history of this page for a list of all contributions to it.