Microbundle

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== Examples ==
== Examples ==
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For any space $X$ define the diagonal embedding
For any space $X$ define the diagonal embedding
$$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$
$$\Delta_X \colon X \to X \times X, \quad x \mapsto (x,x)~.$$

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1 Definition

The concept of a 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; i.e. a fibre bundle with structure group the homeomorphisms of \Rr^n fixing 0.

Definition 1.1 [Milnor1964] . Let B be a topological space. An n-dimensional microbundle over B is a quadruple (E,B,i,j) where E is a space, i and j are maps fitting into the following diagram

\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]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}.

The space E is called the total space of the bundle and B the base space.

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:

\displaystyle  \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}}. }

2 Examples

For any space X define the diagonal embedding

\displaystyle \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
Tex syntax error
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

\displaystyle  (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
\displaystyle (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:

  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
    .

3 References

$. {{beginthm|Definition|{{cite|Milnor1964}} }} Let $B$ be a topological space. An '''$n$-dimensional microbundle''' over $B$ is a quadruple $(E,B,i,j)$ where $E$ is a space, $i$ and $j$ are maps fitting into the following diagram $$B\xrightarrow{i} E\xrightarrow{j} B$$ 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: $$ \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \ & U \times \Rr^n \ar[ur]_{p_1}}. $$ The space $E$ is called the '''total space''' of the bundle and $B$ the '''base space'''. 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: $$ \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \ & V_2 \ar[ur]_{j_2|_{V_2}}. } $$ {{endthm}} == 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. {{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, the '''tangent microbundle''' $\tau_M$ of $M$. {{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|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:}} [[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-bundle; i.e. a fibre bundle with structure group the homeomorphisms of \Rr^n fixing 0.

Definition 1.1 [Milnor1964] . Let B be a topological space. An n-dimensional microbundle over B is a quadruple (E,B,i,j) where E is a space, i and j are maps fitting into the following diagram

\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]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}.

The space E is called the total space of the bundle and B the base space.

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:

\displaystyle  \xymatrix{ & V_1 \ar[dd]^{H} \ar[dr]^{j_1|_{V_1}} \\ B \ar[ur]^{i_1} \ar[dr]_{i_2} & & B \\ & V_2 \ar[ur]_{j_2|_{V_2}}. }

2 Examples

For any space X define the diagonal embedding

\displaystyle \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
Tex syntax error
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

\displaystyle  (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
\displaystyle (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:

  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
    .

3 References

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