Microbundle
(Difference between revisions)
m (→Introduction) |
m (→Introduction) |
||
Line 1: | Line 1: | ||
{{Stub}} | {{Stub}} | ||
== Introduction == | == Introduction == | ||
− | <wikitex | + | <wikitex>; |
The concept of a <i>microbundle</i> 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. | The concept of a <i>microbundle</i> 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. | ||
Line 28: | Line 28: | ||
{{beginthm|Definition}} | {{beginthm|Definition}} | ||
− | Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute | + | Two microbundles $(E_n,B,i_n,j_n)$, $n=1,2$ over the same space $B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$ of $i_1(B)$ and $V_2\subset E_2$ of $i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$ making the following diagram commute: |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
{{endthm|Definition}} | {{endthm|Definition}} | ||
Line 45: | Line 37: | ||
# If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. | # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. | ||
{{endthm}} | {{endthm}} | ||
+ | |||
+ | $$ | ||
+ | \xymatrix{ | ||
+ | & V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\ | ||
+ | B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\ | ||
+ | & V_2 \ar[ru]_{j_2|_{V_2}} | ||
+ | } | ||
+ | $$ | ||
</wikitex> | </wikitex> | ||
Revision as of 12:22, 6 December 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. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Tex syntax error-bundle.
Definition 1.1 [Milnor1964] .
An -dimensional microbundle is a quadrupleTex syntax errorsuch that there is a sequence
Tex syntax error
Tex syntax error
- for all
Tex syntax error
there exist open neigbourhoodTex syntax error
, an open neighbourhoodTex syntax error
ofTex syntax error
and a homeomorphismTex syntax error
which makes the following diagram commute:
Tex syntax error
For any space define the diagonal embedding
Tex syntax error
Tex syntax erroris the tangent bundle
Tex syntax errorof .
In the topological category we have:
Example 1.2 [Milnor1964, Lemma 2.1].
Let be topological -manifold, and letTex syntax errorbe the projection onto the first factor. Then
Tex syntax error
Tex syntax errorof .
Example 1.3. Let
Tex syntax errorbe a topological
Tex syntax error-bundle with zero section
Tex syntax error. Then
Tex syntax error
is an -dimensional microbundle.
Definition 1.4.
Two microbundlesTex syntax error,
Tex syntax errorover the same space are isomorphic if there exist neighbourhoods
Tex syntax errorof
Tex syntax errorand
Tex syntax errorof
Tex syntax errorand a homeomorphism
Tex syntax errormaking the following diagram commute:
Theorem 1.5 [Kister1964, Theorem 2] .
LetTex syntax errorbe an -dimensional microbundle. Then there is a neighbourhood of
Tex syntax error,
Tex syntax errorsuch that:
-
Tex syntax error
is the total space of a topologicalTex syntax error
-bundle over . - The inclusion
Tex syntax error
is a microbundle isomorphism - If
Tex syntax error
is any other such neighbourhood ofTex syntax error
then there is aTex syntax error
-bundle isomorphismTex syntax error
.
Tex syntax error
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