Microbundle
(Created page with "== Introduction == <wikitex>; The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensiona...") |
m |
||
Line 1: | Line 1: | ||
== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{ | + | The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle. |
{{beginthm|Definition|{{cite|Milnor1964}} }} | {{beginthm|Definition|{{cite|Milnor1964}} }} | ||
Line 23: | Line 23: | ||
To do: definition of microbundle isomorphism. | To do: definition of microbundle isomorphism. | ||
− | {{beginthm|Theorem|\cite{ | + | {{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} |
Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: | Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: | ||
# $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. | # $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. |
Revision as of 20:54, 29 May 2012
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. |
1 Introduction
The concept of Microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister1964] showed that every microbundle uniquely determines a topological -bundle.
Definition 1.1 [Milnor1964] .
An -dimensional microbundle is a quadruple such that there is a sequence- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism
Moreover, the homeomorphism above must make the following diagram commute:
Example 1.2 [Milnor1964, Lemma 2.1]. Let be topological -manifold, let be the diagonal map and let be the projection onto the first factor. Then
is an -dimensional microbundle.
is an -dimensional microbundle.
To do: definition of microbundle isomorphism.
Theorem 1.4 [Kister1964, Theorem 2] . Let be an -dimensional microbundle. Then there is a neighbourhood of , such that:
- is the total space of a topological -bundle over .
- The inclusion is a microbundle isomorphism
- If is any other such neighbourhood of then there is a -bundle isomorphism .
2 References
- [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