Microbundle Explained
In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964.[1] It allows the creation of bundle-like objects in situations where they would not ordinarily be thought to exist. For example, the tangent bundle is defined for a smooth manifold but not a topological manifold; use of microbundles allows the definition of a topological tangent bundle.
Definition
A (topological)
-microbundle over a topological space
(the "base space") consists of a triple
, where
is a topological space (the "total space"),
and
are continuous maps (respectively, the "zero section" and the "projection map") such that:- the composition
is the identity of
;- for every
, there are a neighborhood
of
and a neighbourhood
of
such that
,
,
is homeomorphic to
and the maps
and
commute with
and
.In analogy with vector bundles, the integer
is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests
should be thought of as the zero section of a vector bundle, while the second mimics the local triviality condition on a bundle. An important distinction here is that "local triviality" for microbundles only holds near a neighborhood of the zero section. The space
could look very wild away from that neighborhood. Also, the maps gluing together locally trivial patches of the microbundle may only overlap the fibers.The definition of microbundle can be adapted to other categories more general than the smooth one, such as that of piecewise linear manifolds, by replacing topological spaces and continuous maps by suitable objects and morphisms.
Examples
of rank
has an obvious
underlying
-microbundle, where
is the zero section.
- Given any topological space
, the cartesian product
(together with the projection on
and the map
) defines an
-microbundle, called the standard trivial microbundle of rank
. Equivalently, it is the underlying microbundle of the trivial vector bundle of rank
.
, the cartesian product
together with the projection on the first component and the
diagonal map
defines an
-microbundle, called the
tangent microbundle of
.
-microbundle
over
and a continuous map
, the space f*E:=\{(a,e)\inA x E\midf(a)=p(e)\}
defines an
-microbundle over
, called the pullback (or induced) microbundle by
, together with the projection
and the zero section i:A\tof*E,x\mapsto(x,(i\circf)(x))
. If
is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard pullback bundle.
-microbundle
over
and a subspace
, the restricted microbundle, also denoted by
, is the pullback microbundle with respect to the inclusion
.Morphisms
Two
-microbundles
and
over the same space
are
isomorphic (or equivalent) if there exist a neighborhood
of
and a neighborhood
of
, together with a
homeomorphism
commuting with the projections and the zero sections.
More generally, a morphism between microbundles consists of a germ of continuous maps
between neighbourhoods of the zero sections as above.
An
-microbundle is called
trivial if it is isomorphic to the standard trivial microbundle of rank
. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every
there is a neighbourhood
such that the restriction
is trivial.
Analogously to parallelisable smooth manifolds, a topological manifold is called topologically parallelisable if its tangent microbundle is trivial.
Properties
A theorem of James Kister and Barry Mazur states that there is a neighborhood of the zero section which is actually a fiber bundle with fiber
and structure group
\operatorname{Homeo}(\Rn,0)
, the group of homeomorphisms of
fixing the origin. This neighborhood is unique up to isotopy. Thus every microbundle can be refined to an actual fiber bundle in an essentially unique way.
[2] Taking the fiber bundle contained in the tangent microbundle
gives the
topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for
, letting each chart
have a fiber
over each point in the chart, and gluing these trivial bundles together by overlapping the fibers according to the transition maps.
Microbundle theory is an integral part of the work of Robion Kirby and Laurent C. Siebenmann on smooth structures and PL structures on higher dimensional manifolds.[3]
References
- Milnor . John Willard . John Milnor . Microbundles. I . 10.1016/0040-9383(64)90005-9 . 0161346 . 1964 . . 3 . 53–80.
- Kister . James M. . Microbundles are fibre bundles . . 80 . 1 . 1964 . 10.2307/1970498 . 190–199 . 1970498 . 0180986.
- Book: Robion C. . Kirby . Robion Kirby. Laurent C.. Siebenmann. Laurent C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies. 88. Princeton University Press. Princeton, N.J.. 1977. 0-691-08191-3. 0645390.
- Gauld . David . Greenwood . Sina . Microbundles, manifolds and metrisability. . 128 . 9 . 2000 . 10.1090/s0002-9939-00-05343-0 . 2801–2808 . 1664358. free .
- Book: Switzer . Robert M. . Algebraic topology—homotopy and homology . . Berlin, New York . Classics in Mathematics . 978-3-540-42750-6 . 1886843 . 2002. See Chapter 14.
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