Microbundle

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(Definition of isomorphism)
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An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.
An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold.
#$j\circ i=\id_B$
#$j\circ i=\id_B$
#for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n.$$
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#for all $x\in B$ there exist open neigbourhood $U\subset B$, an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n$$
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which makes the following diagram commute:
Moreover, the homeomorphism above must make the following diagram commute:
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$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$
$$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}} $$
{{endthm}}
{{endthm}}
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{{endrem}}
{{endrem}}
To do: definition of microbundle isomorphism.
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{{beginthm|Definition}}
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Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ 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.
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$$
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\xymatrix{
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& V_1 \ar[dd]^H\ar[rd]^{j_1|_{V_1}} \\
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B\ar[ru]^{i_1}\ar[rd]_{i_2} & & B \\
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& V_2 \ar[ru]_{j_2|_{V_2}}
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}
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$$
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{{endthm|Definition}}
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}
{{beginthm|Theorem|\cite{Kister1964|Theorem 2} }}

Revision as of 22:37, 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 n 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 \Rr^n-bundle.

Definition 1.1 [Milnor1964] .

An n-dimensional microbundle is a quadruple (E,B,i,j) such that there is a sequence
\displaystyle B\xrightarrow{i} E\xrightarrow{j} B
and the following conditions hold.
  1. j\circ i=\id_B
  2. for all x\in B there exist open neigbourhood U\subset B, an open neighbourhood V\subset E of i(b) and a homeomorphism
    \displaystyle h \colon V \to U\times \mathbb{R}^n

which makes the following diagram commute:

\displaystyle  \xymatrix{& V \ar[dr]^{j|_V} \ar[dd] \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}

Example 1.2 [Milnor1964, Lemma 2.1]. Let M be topological n-manifold, let \Delta_M \colon M \to M \times M be the diagonal map and let p_1 \colon M \times M \to M be the projection onto the first factor. Then

\displaystyle  (M \times M, M, \Delta_M, p_1)

is an n-dimensional microbundle.

Example 1.3. Let \pi \colon E \to B be a topological \Rr^n-bundle with zero section s \colon B \to E. Then
\displaystyle (E, B, s, \pi)

is an n-dimensional microbundle.

Definition 1.4. Two microbundles (E_n,X,i_n,j_n), n=1,2 over the same space X 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.

Tex syntax error

Theorem 1.5 [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:

  1. E_1 is the total space of a topological \Rr^n-bundle over B.
  2. The inclusion E_1 \to E is a microbundle isomorphism
  3. If E_2 \subset E is any other such neighbourhood of i(B) then there is a \Rr^n-bundle isomorphism
    Tex syntax error
    .

2 References

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