# Microbundle

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## 1 Introduction

The concept of Microbundle of dimension $n$$== Introduction == ; 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}} }} 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 #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 which makes the following diagram commute: \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}} {{beginrem|Example|{{citeD|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 (M \times M, M, \Delta_M, p_1) is an n-dimensional microbundle. {{endrem}} {{beginrem|Example}} Let \pi \colon E \to B be a topological \Rr^n-bundle with zero section s \colon B \to E. Then (E, B, s, \pi) is an n-dimensional microbundle. {{endrem}} {{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. \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}} } {{endthm|Definition}} {{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: # E_1 is the total space of a topological \Rr^n-bundle over B. # The inclusion E_1 \to E is a microbundle isomorphism # 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}} == References == {{#RefList:}}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$$\Rr^n$-bundle.

Definition 1.1 [Milnor1964] .

An $n$$n$-dimensional microbundle is a quadruple $(E,B,i,j)$$(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$$j\circ i=\id_B$
2. for all $x\in B$$x\in B$ there exist open neigbourhood $U\subset B$$U\subset B$, an open neighbourhood $V\subset E$$V\subset E$ of $i(b)$$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$$M$ be topological $n$$n$-manifold, let $\Delta_M \colon M \to M \times M$$\Delta_M \colon M \to M \times M$ be the diagonal map and let $p_1 \colon M \times M \to M$$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$$n$-dimensional microbundle.

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

is an $n$$n$-dimensional microbundle.

Definition 1.4. Two microbundles $(E_n,B,i_n,j_n)$$(E_n,B,i_n,j_n)$, $n=1,2$$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$$V_1\subset E_1$ of $i_1(B)$$i_1(B)$ and $V_2\subset E_2$$V_2\subset E_2$ of $i_2(B)$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$$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)$$(E, B, i, j)$ be an $n$$n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$$i(B)$, $E_1 \subset E$$E_1 \subset E$ such that:

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