Microbundle
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:20, 16 May 2013 and the changes since publication. |
This page has not been refereed. The information given here might be incomplete or provisional. |
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
Contents |
1 Definition
The concept of a microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later Kister [Kister1964], and independently Mazur, showed that every microbundle uniquely determines a topological
-bundle; i.e. a fibre bundle with structure group the homeomorphisms of
fixing
.
Definition 1.1 [Milnor1964] .
Let be a topological space. An
-dimensional microbundle over
is a quadruple
where
is a space,
and
are maps fitting into the following diagram
![\displaystyle B\xrightarrow{i} E\xrightarrow{j} B](/images/math/4/0/a/40a68b194c05ed02a16adc8c005f25f3.png)
and the following conditions hold:
.
- For all
there exist open neigbourhood
, an open neighbourhood
of
and a homeomorphism
which makes the following diagram commute:
![\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}.](/images/math/9/2/8/92848a11983524dd53282be31bba8921.png)
The space is called the total space of the bundle and
the base space.
Two microbundles ,
over the same space
are isomorphic if there exist neighbourhoods
of
and
of
and a homeomorphism
making the following diagram commute:
![\displaystyle \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}} }](/images/math/6/5/0/650cf2c11076ca7ab0a7ba68e744a9fa.png)
2 The tangent microbundle
An important example of a microbundle is the tangent microbundle of a topological (or similarly ) manifold
.
Let
![\displaystyle \Delta_M \colon M \to M \times M, \quad x \mapsto (x,x)~](/images/math/8/9/f/89f01af4443a45d5f80f540a29e2b551.png)
be the diagonal map for .
Example 2.1 [Milnor1964, Lemma 2.1].
Let be topological (or PL)
-manifold, and let
be the projection onto the first factor. Then
![\displaystyle (M \times M, M, \Delta_M, p_1)](/images/math/0/8/f/08febb705b78edb17c9c3a6116f9b713.png)
is an -dimensional microbundle, the tangent microbundle
of
.
Remark 2.2.
An atlas of gives a product atlas of
which shows that the second condition of a microbundle is fulfilled. Actually the definition of the micro tangent bundle looks a bit more like a normal bundle to the diagonal, a view which fits to the fact that the normal bundle of the diagonal of a smooth manifold
in
is isomorphic to its tangent bundle.
Another important example of a microbundle is the micro-bundle defined by a topological topological -bundle.
Example 2.3. Let be a topological
-bundle with zero section
.
Then the quadruple
![\displaystyle (E, B, s, \pi)](/images/math/8/d/e/8de96595ed823c69d3a289b186d86d41.png)
is an -dimensional microbundle.
A fundamental fact about microbundles is the following theorem, often called the Kister-Mazur theorem, proven independently by Kister and Mazur.
Theorem 2.4 [Kister1964, Theorem 2].
Let be an
-dimensional microbundle over a locally finite, finite dimensional simplicial complex
.
Then there is a neighbourhood of
,
such that the following hold.
-
is the total space of a topological
-bundle over
.
-
is a microbundle and the the inclusion
is a microbundle isomorphism.
- If
is any other such neighbourhood of
then there is a
-bundle isomorphism
.
Remark 2.5.
Microbundle theory is an important part of the Kirby and Siebenmann [Kirby&Siebenmann1977] work on smooth structures and -structures on higher dimensional manifolds.
3 References
- [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
- [Kister1964] J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR0180986 (31 #5216) Zbl 0131.20602
- [Milnor1964] J. Milnor, Microbundles. I, Topology 3 (1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404
4 External links
- The Wikipedia page about microbundles.