# Talk:Microbundles (Ex)

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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. | 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. | ||

#$j\circ i=\id_B$ | #$j\circ i=\id_B$ | ||

− | #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 $$h \colon V \to U\times \mathbb{R}^n.$$ | + | #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: | Moreover, the homeomorphism above must make the following diagram commute: | ||

− | $$ \xymatrix{ | + | $$ |

+ | \xymatrix{ | ||

+ | U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\ | ||

+ | V\ar[r]^{j} \ar[ur]^{h} & U,} | ||

+ | $$ | ||

+ | where $p_1$ is projection on the first factor and $U$ is included as a $0$-section in $U\times \mathbb{R}^n$. | ||

{{endthm}} | {{endthm}} | ||

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Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. | Let $M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. | ||

{{endthm}} | {{endthm}} | ||

+ | {{beginproof}} | ||

Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied. | Let $M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied. | ||

To prove that the second condition is satisfied we need to use local chart around $x$. | To prove that the second condition is satisfied we need to use local chart around $x$. | ||

− | Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. | + | Choose $U$ to be one of the open sets coming from atlas of $M$ and let $\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious candidate for $V\subset M\times M$ is to take $U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$. However such $h$ fails to make the following diagram commute |

+ | $$ | ||

+ | \xymatrix{ | ||

+ | U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\ | ||

+ | V\ar[r]^{p_1} \ar[ur]^{h} & U,} | ||

+ | $$ | ||

+ | since $(u,u)$ is mapped to $(u,\phi(u))$ and $\phi(u)$ doesn't necessarily be $0$ (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: $h(u,v)=(u,h(u)-h(v))$. | ||

+ | {{endproof}} | ||

+ | |||

{{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}} | {{beginthm|Exercise|{{citeD|Milnor1964|Theorem 2.2}}}} | ||

− | Let $M$ be a smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. | + | Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. |

{{endthm}} | {{endthm}} | ||

+ | {{beginproof}} | ||

+ | We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$ is the zero section. | ||

+ | |||

+ | However to show that these two definition agree we need a notion of microbundle isomorphism. | ||

+ | |||

+ | {{beginthm|Definition}} | ||

+ | Two microbundles $(E_n,X,i_n,j_n)$, $n=1,2$ over the same space $X$ 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{ | ||

+ | U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\ | ||

+ | V_1\ar[r]^{p_1} \ar[ur]^{H} & U,} | ||

+ | $$ | ||

+ | {{endthm|Definition}} | ||

+ | |||

+ | In our case | ||

+ | |||

+ | |||

+ | {{endproof}} | ||

+ | |||

</wikitex> | </wikitex> |

## Revision as of 20:25, 29 May 2012

Let us begin with the definition of microbundle.

**Definition 0.1.**

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [Milnor1964, Lemma 2.1]**.**

Tex syntax errorbe a topological manifold. Show that is a microbundle.

**Proof.**

Tex syntax errorbe a topological manifold. Then the composition sends , so the first condition in the definition is satisfied.

To prove that the second condition is satisfied we need to use local chart around .

Choose to be one of the open sets coming from atlas ofTex syntax errorand let be associated chart. The obvious candidate for is to take . The first naive candidate for would be map . However such fails to make the following diagram commute

since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .

**Exercise 0.3** [Milnor1964, Theorem 2.2]**.**

Tex syntax errorbe a (paracompact!) smooth manifold. Show that and are isomorphic microbundles.

**Proof.**
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on and treating it just as a microbundle where is the zero section.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [Milnor1964, Lemma 2.1]**.**

Tex syntax errorbe a topological manifold. Show that is a microbundle.

**Proof.**

Tex syntax errorbe a topological manifold. Then the composition sends , so the first condition in the definition is satisfied.

To prove that the second condition is satisfied we need to use local chart around .

Choose to be one of the open sets coming from atlas ofTex syntax errorand let be associated chart. The obvious candidate for is to take . The first naive candidate for would be map . However such fails to make the following diagram commute

since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .

**Exercise 0.3** [Milnor1964, Theorem 2.2]**.**

Tex syntax errorbe a (paracompact!) smooth manifold. Show that and are isomorphic microbundles.

**Proof.**
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on and treating it just as a microbundle where is the zero section.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case

- for all there exist open neigbourhood and an open neighbourhood of and a homeomorphism

Moreover, the homeomorphism above must make the following diagram commute:

where is projection on the first factor and is included as a -section in .

**Exercise 0.2** [Milnor1964, Lemma 2.1]**.**

Tex syntax errorbe a topological manifold. Show that is a microbundle.

**Proof.**

Tex syntax errorbe a topological manifold. Then the composition sends , so the first condition in the definition is satisfied.

To prove that the second condition is satisfied we need to use local chart around .

Choose to be one of the open sets coming from atlas ofTex syntax errorand let be associated chart. The obvious candidate for is to take . The first naive candidate for would be map . However such fails to make the following diagram commute

since is mapped to and doesn't necessarily be (well, it depends on our definition of an atlas of a manifold?). We need just a small adjustment: .

**Exercise 0.3** [Milnor1964, Theorem 2.2]**.**

Tex syntax errorbe a (paracompact!) smooth manifold. Show that and are isomorphic microbundles.

**Proof.**
We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on and treating it just as a microbundle where is the zero section.

However to show that these two definition agree we need a notion of microbundle isomorphism.

**Definition 0.4.**
Two microbundles , over the same space are isomorphic if there exist neighbourhoods of and of and a homeomorphism making the following diagram commute.

In our case