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)

n

-microbundle over a topological space

B

(the "base space") consists of a triple

(E,i,p)

, where

E

is a topological space (the "total space"),

i:B\toE

and

p:E\toB

are continuous maps (respectively, the "zero section" and the "projection map") such that:
  1. the composition

p\circi

is the identity of

B

;
  1. for every

b\inB

, there are a neighborhood

U\subseteqB

of

b

and a neighbourhood

V\subseteqE

of

i(b)

such that

i(U)\subseteqV

,

p(V)\subseteqU

,

V

is homeomorphic to

U x \Rn

and the maps

p\mid:V\toU

and

i\mid:U\toV

commute with

pr1:U x Rn\toU

and

U\toU x Rn,x\mapsto(x,0)

.In analogy with vector bundles, the integer

n\geq0

is also called the rank or the fibre dimension of the microbundle. Similarly, note that the first condition suggests

i

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

E

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

p:E\toB

of rank

n

has an obvious underlying

n

-microbundle
, where

i

is the zero section.

B

, the cartesian product

B x Rn

(together with the projection on

B

and the map

x\mapsto(x,0)

) defines an

n

-microbundle, called the standard trivial microbundle of rank

n

. Equivalently, it is the underlying microbundle of the trivial vector bundle of rank

n

.

n

, the cartesian product

M x M

together with the projection on the first component and the diagonal map

\Delta:M\toM x M

defines an

n

-microbundle, called the tangent microbundle of

M

.

n

-microbundle

(E,i,p)

over

B

and a continuous map

f:A\toB

, the space

f*E:=\{(a,e)\inA x E\midf(a)=p(e)\}

defines an

n

-microbundle over

A

, called the pullback (or induced) microbundle by

f

, together with the projection

p:=pr1:f*E\toA

and the zero section

i:A\tof*E,x\mapsto(x,(i\circf)(x))

. If

p:E\toB

is a vector bundle, the pullback microbundle of its underlying microbundle is precisely the underlying microbundle of the standard pullback bundle.

n

-microbundle

(E,i,p)

over

B

and a subspace

A\subseteqB

, the restricted microbundle, also denoted by

E\mid=p-1(A)

, is the pullback microbundle with respect to the inclusion

A\hookrightarrowB

.

Morphisms

Two

n

-microbundles

(E1,i1,p1)

and

(E2,i2,p2)

over the same space

B

are isomorphic (or equivalent) if there exist a neighborhood

V1\subseteqE1

of

i1(B)

and a neighborhood

V2\subseteqE2

of

i2(B)

, together with a homeomorphism

V1\congV2

commuting with the projections and the zero sections.

More generally, a morphism between microbundles consists of a germ of continuous maps

V1\toV2

between neighbourhoods of the zero sections as above.

An

n

-microbundle is called trivial if it is isomorphic to the standard trivial microbundle of rank

n

. The local triviality condition in the definition of microbundle can therefore be restated as follows: for every

b\inB

there is a neighbourhood

U\subseteqB

such that the restriction

E\mid

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

\Rn

and structure group

\operatorname{Homeo}(\Rn,0)

, the group of homeomorphisms of

\Rn

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

(M x M,\Delta,pr)

gives the topological tangent bundle. Intuitively, this bundle is obtained by taking a system of small charts for

M

, letting each chart

U

have a fiber

U

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

  1. Milnor . John Willard . John Milnor . Microbundles. I . 10.1016/0040-9383(64)90005-9 . 0161346 . 1964 . . 3 . 53–80.
  2. Kister . James M. . Microbundles are fibre bundles . . 80 . 1 . 1964 . 10.2307/1970498 . 190–199 . 1970498 . 0180986.
  3. 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.

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