nLab microbundle

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).

References and Literature

The original reference is

• John Milnor, Microbundles. Part I, Topology 3 (1964), Supplement 1, 53–80. doi: https://doi.org/10.1016/0040-9383(64)90005-9, PDF.

Classic treatments of their elementary theory include:

• N. H. Kuiper and R. K. Lashof, Microbundles and Bundles I. Elementary Theory, Invent. Math., 1, (1966), 1 – 17.

• N. H. Kuiper and R. K. Lashof, Microbundles and Bundles II. Semisimplicial Theory, Invent. Math., 1, (1966), 243 – 259.

Useful references are for instance

• Jacob Lurie, Spring 2009, Topics in Geometric Topology

• S. Buoncristiano, 2003, Fragments of geometric topology from the sixties.

• Alexander Kupers, Lectures on diffeomorphisms groups of manifolds, (pdf)