Microbundle

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1 Introduction

The concept of Microbundle of dimension $n$${{Stub}} == Introduction == <wikitex>; The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister1964} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle. {{beginthm|Definition|{{cite|Milnor1964}} }} An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$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]^h \ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \ & U \times \Rr^n \ar[ur]_{p_1}} $$ {{endthm}} For any space $M$ define the diagonal embedding $$\Delta_M \colon M \to M \times M;x \mapsto (x,x)~.$$ If $M$ is a differentiable $n$-manifold the normal bundle of $\Delta_M$ is the tangent bundle $\tau_M$ of $M$. In the topological category we have: {{beginrem|Example|{{citeD|Milnor1964|Lemma 2.1}}}} Let $M$ be topological $n$-manifold, and let $p_1 \colon M \times M \to M$ be the projection onto the first factor. Then $$ (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}} } $$ {{endthm|Definition}} {{beginthm|Theorem|\cite{Kister1964|Theorem 2} }} Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: # $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. # The inclusion $E_1 \to E$ is a microbundle isomorphism # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Theory]]n$ was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister1964] showed that every microbundle uniquely determines a topological $\Rr^n$$\Rr^n$-bundle.

Definition 1.1 [Milnor1964] .

An $n$$n$-dimensional microbundle is a quadruple

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$(E,B,i,j)$ such that there is a sequence

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$$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$$

and the following conditions hold.

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   $j\circ i=\id_B$
2. for all

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   $x\in B$ there exist open neigbourhood

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   $U\subset B$, an open neighbourhood

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   $V\subset E$ of

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   $i(b)$ and a homeomorphism

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   $$\displaystyle h \colon V \to U\times \mathbb{R}^n$$

which makes the following diagram commute:

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For any space $M$$M$ define the diagonal embedding

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$$\displaystyle \Delta_M \colon M \to M \times M;x \mapsto (x,x)~.$$

If $M$$M$ is a differentiable $n$$n$-manifold the normal bundle of

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$\Delta_M$ is the tangent bundle

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$\tau_M$ of $M$$M$.

In the topological category we have:

Example 1.2 [Milnor1964, Lemma 2.1].

Let $M$$M$ be topological $n$$n$-manifold, and let

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$p_1 \colon M \times M \to M$ be the projection onto the first factor. Then

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

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

Example 1.3. Let

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$\pi \colon E \to B$ be a topological $\Rr^n$$\Rr^n$-bundle with zero section

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$s \colon B \to E$. Then

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$$\displaystyle (E, B, s, \pi)$$

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

Definition 1.4.

Two microbundles

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$(E_n,B,i_n,j_n)$,

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$n=1,2$ over the same space $B$$B$ are isomorphic if there exist neighbourhoods

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$V_1\subset E_1$ of

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$i_1(B)$ and

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$V_2\subset E_2$ of

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$i_2(B)$ and a homeomorphism

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$H\colon V_1\to V_2$ making the following diagram commute.

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Theorem 1.5 [Kister1964, Theorem 2] .

Let

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$(E, B, i, j)$ be an $n$$n$-dimensional microbundle. Then there is a neighbourhood of

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$i(B)$,

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$E_1 \subset E$ such that:

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   $E_1$ is the total space of a topological $\Rr^n$$\Rr^n$-bundle over $B$$B$.
2. The inclusion

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   $E_1 \to E$ is a microbundle isomorphism
3. If

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   $E_2 \subset E$ is any other such neighbourhood of

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   $i(B)$ then there is a $\Rr^n$$\Rr^n$-bundle isomorphism

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   $(E_1 \to B) \cong (E_2 \to B)$.

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