nLab
microbundle

Contents

Context

Bundles

Contents

Idea

A microbundle is something like an approximation to the notion of vector bundle: a locally trivial bundle EXE \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 nn is a 4-tuple ξ=(E,p,B,i)\xi = (E,p,B,i) where

  • EE is a topological space (the total space of ξ\xi),

  • BB is a topological space (the base space of ξ\xi,

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

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

such that

  • ii is a section of pp, i.e. pi=id Bp\circ i = id_B

  • the local triviality condition holds:

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

Morphisms of microbundles

A morphism of microbundles ϕ:ξξ\phi:\xi\to\xi' is a germ of maps from neighborhoods of the zero section of ξ\xi to ξ\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 VV of the zero section in ξ\xi to a neighborhood VV' of the zero section in ξ\xi' commuting with projections and inclusions of the zero sections.

Examples

Tangent microbundle

The main example is the tangent microbundle (M×M,p 1,M,i)(M\times M,p_1,M,i) of a topological manifold MM where p 1:M×MMp_1:M\times M\to M is the projection onto the first factor. If (U,f)(U,f) is a chart of the manifold MM around point xMx\in M (where xUMx\in U\subset M and f:UR nf:U\to R^n is a homeomorphism with h(x)=0h(x)=0) then define h:U×R nU×Uh:U\times R^n\to U\times U by h(u,v)=(u,f 1(f(v)u))h(u,v)=(u,f^{-1}(f(v)-u)).

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

References

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:

Useful references are for instance

Last revised on February 27, 2021 at 15:25:55. See the history of this page for a list of all contributions to it.