(see also Chern-Weil theory, parameterized homotopy theory)
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).
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 $\xi$.
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.
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.
Microbundles were defined by John Milnor. The original paper can be found here.
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)
Last revised on January 14, 2020 at 09:43:56. See the history of this page for a list of all contributions to it.