Microbundle

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1 Introduction

The concept of Microbundle of dimension $n$$== Introduction == <wikitex>; The concept of Microbundle of dimension $n$ was first introduced in {{cite|Milnor1964}} to give a model for the tangent bundle of an n-dimensional [[topological manifold]]. Later \cite{Kister????} showed that every microbundle uniquely determines a topological $\Rr^n$-bundle. {{beginthm|Definition|{{cite|Milnor1964}} }} An $n$-dimensional microbundle is a quadruple $(E,B,i,j)$ such that there is a sequence $$B\xrightarrow{i} E\xrightarrow{j} B$$ and the following conditions hold. #$j\circ i=\id_B$ #for all $x\in B$ there exist open neigbourhood $U\subset B$ and an open neighbourhood $V\subset E$ of $i(b)$ and a homeomorphism $$h \colon V \to U\times \mathbb{R}^n.$$ Moreover, the homeomorphism above must make the following diagram commute: $$ \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)$$ is an $n$-dimensional microbundle. {{endrem}} To do: definition of microbundle isomorphism. {{beginthm|Theorem|\cite{Kister????| } }} Let $(E, B, i, j)$ be an $n$-dimensional microbundle. Then there is a neighbourhood of $i(B)$, $E_1 \subset E$ such that: # $E_1$ is the total space of a topological $\Rr^n$-bundle over $B$. # The inclusion $E_1 \to E$ is a microbundle isomorphism # If $E_2 \subset E$ is any other such neighbourhood of $i(B)$ then there is a $\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$. {{endthm}} </wikitex> == References == {{#RefList:}}n$ was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister????] showed that every microbundle uniquely determines a topological $\Rr^n$$\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

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

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$$U\subset B$ and an open neighbourhood $V\subset E$$V\subset E$ of $i(b)$$i(b)$ and a homeomorphism
   $\displaystyle h \colon V \to U\times \mathbb{R}^n.$ $$\displaystyle h \colon V \to U\times \mathbb{R}^n.$$

Moreover, the homeomorphism above must make 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

Tex syntax error

$M$ 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

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

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

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

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

To do: definition of microbundle isomorphism.

Theorem 1.4 [Kister????,] . 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$$\Rr^n$-bundle isomorphism $(E_1 \to B) \cong (E_2 \to B)$$(E_1 \to B) \cong (E_2 \to B)$.

2 References

• [Kister????] Template:Kister????
• [Milnor1964] J. Milnor, Microbundles. I, Topology 3 (1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404