Microbundle
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1 Definition
The concept of a microbundle of dimension 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 -bundle; i.e. a fibre bundle with structure group the homeomorphisms of fixing .
Definition 1.1 [Milnor1964] . Let be a topological space. An -dimensional microbundle over is a quadruple where is a space, and are maps fitting into the following diagram
and the following conditions hold:
- for all there exist open neigbourhood , an open neighbourhood of and a homeomorphism
which makes the following diagram commute:
The space is called the total space of the bundle and the base space.
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute:
2 Examples
For any space $X$ define the diagonal embedding $$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$ If $X = M$ 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 the following definition.
Example 2.1 [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, the tangent microbundle $\tau_M$ of $M$.
Example 2.2. 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.
Theorem 2.3 [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 $(E_1 \to B) \cong (E_2 \to B)$.
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3 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
Definition 1.1 [Milnor1964] . Let be a topological space. An -dimensional microbundle over is a quadruple where is a space, and are maps fitting into the following diagram
and the following conditions hold:
- for all there exist open neigbourhood , an open neighbourhood of and a homeomorphism
which makes the following diagram commute:
The space is called the total space of the bundle and the base space.
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute:
2 Examples
For any space $X$ define the diagonal embedding $$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$ If $X = M$ 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 the following definition.
Example 2.1 [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, the tangent microbundle $\tau_M$ of $M$.
Example 2.2. 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.
Theorem 2.3 [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 $(E_1 \to B) \cong (E_2 \to B)$.
</wikitex>
3 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