nLab
microbundle

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.

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 xixi),

  • 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 xinUMx\inU\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)).

David Roberts: A couple of years ago I thought of importing topological groupoids to this concept for the following reason: The tangent microbundle M×MM\times M, when MM is a manifold, is the groupoid integrating the tangent bundle TMTM of MM. 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.

Useful references are for instance

and

Revised on June 26, 2012 22:47:14 by Urs Schreiber (82.169.65.155)