# Talk:Microbundles (Ex)

Let us begin with the definition of microbundle.

Definition 0.1.

An $n$$; Let us begin with the definition of [[Microbundle|microbundle]]. {{beginthm|Definition|}} An n-dimensional microbundle is a quadruple (E,B,i,j) such that there is a sequence B\xrightarrow{i} E\xrightarrow{j} B and the following conditions hold. #j\circ i=\id_B #for all x\in B there exist open neigbourhood U\subset B and an open neighbourhood V\subset E of i(b) and a homeomorphism h\colon V\to U\times \mathbb{R}^n. Moreover, the homeomorphism above must make the following diagram commute: \xymatrix{ U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\ V\ar[r]^{j} \ar[ur]^{h} & U,} where p_1 is projection on the first factor and U is included as a -dimensional microbundle is a quadruple $(E,B,i,j)$$(E,B,i,j)$ such that there is a sequence
$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$
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
$\displaystyle h\colon V\to U\times \mathbb{R}^n.$

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

$\displaystyle \xymatrix{ U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{j} \ar[ur]^{h} & U,}$

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

Exercise 0.2 [Milnor1964, Lemma 2.1]. Let $M$$M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$$\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.

Proof.

Let $M$$M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$$p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$$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$$x$. Choose $U$$U$ to be one of the open sets coming from atlas of $M$$M$ and let $\phi\colon U\to \mathbb{R}^n$$\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$$V\subset M\times M$ is to take $U\times U$$U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$$h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$$\id\times \phi$. However such $h$$h$ fails to make the following diagram commute

$\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{p_1} \ar[ur]^{h} & U,}$

since $(u,u)$$(u,u)$ is mapped to $(u,\phi(u))$$(u,\phi(u))$ and $\phi(u)$$\phi(u)$ doesn't necessarily be $0$$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))$$h(u,v)=(u,h(u)-h(v))$.

$\square$$\square$

Exercise 0.3 [Milnor1964, Theorem 2.2]. Let $M$$M$ be a (paracompact!) smooth manifold. Show that $TM$$TM$ and $\xi_M$$\xi_M$ 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 $TM$$TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$$(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$$M\xrightarrow{s_0} TM$ is the zero section.

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

Definition 0.4. Two microbundles $(E_n,X,i_n,j_n)$$(E_n,X,i_n,j_n)$, $n=1,2$$n=1,2$ over the same space $X$$X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$$V_1\subset E_1$ of $i_1(B)$$i_1(B)$ and $V_2\subset E_2$$V_2\subset E_2$ of $i_2(B)$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$$H\colon V_1\to V_2$ making the following diagram commute.

$\displaystyle \xymatrix{ U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\ V_1\ar[r]^{p_1} \ar[ur]^{H} & U,}$

In our case we have

$\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\\ U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,}$

where $V\subset TM$$V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$$V\cong U\times \mathbb{R}^n$ via local trivialisation.

$\square$$\square$

$-section in$U\times \mathbb{R}^n$. {{endthm}} {{beginthm|Exercise|{{citeD|Milnor1964|Lemma 2.1}}}} Let$M$be a topological manifold. Show that$\xi_M : = (M \times M, M, \Delta_M, p_1)$is a microbundle. {{endthm}} {{beginproof}} Let$M$be a topological manifold. Then the composition$p_1\circ\Delta_M$sends$x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied. To prove that the second condition is satisfied we need to use local chart around$x$. Choose$U$to be one of the open sets coming from atlas of$M$and let$\phi\colon U\to \mathbb{R}^n$be associated chart. The obvious choice for neighbourhood$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 $n$-dimensional microbundle is a quadruple $(E,B,i,j)$$(E,B,i,j)$ such that there is a sequence $\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$ and the following conditions hold. 1. $j\circ i=\id_B$$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 $\displaystyle h\colon V\to U\times \mathbb{R}^n.$ Moreover, the homeomorphism above must make the following diagram commute: $\displaystyle \xymatrix{ U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{j} \ar[ur]^{h} & U,}$ where $p_1$$p_1$ is projection on the first factor and $U$$U$ is included as a $0$$0$-section in $U\times \mathbb{R}^n$$U\times \mathbb{R}^n$. Exercise 0.2 [Milnor1964, Lemma 2.1]. Let $M$$M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$$\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle. Proof. Let $M$$M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$$p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$$x\mapsto (x,x)\mapsto x$, so the first condition in the definition is satisfied. To prove that the second condition is satisfied we need to use local chart around $x$$x$. Choose $U$$U$ to be one of the open sets coming from atlas of $M$$M$ and let $\phi\colon U\to \mathbb{R}^n$$\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$$V\subset M\times M$ is to take $U\times U$$U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$$h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$$\id\times \phi$. However such $h$$h$ fails to make the following diagram commute $\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{p_1} \ar[ur]^{h} & U,}$ since $(u,u)$$(u,u)$ is mapped to $(u,\phi(u))$$(u,\phi(u))$ and $\phi(u)$$\phi(u)$ doesn't necessarily be $0$$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))$$h(u,v)=(u,h(u)-h(v))$. $\square$$\square$ Exercise 0.3 [Milnor1964, Theorem 2.2]. Let $M$$M$ be a (paracompact!) smooth manifold. Show that $TM$$TM$ and $\xi_M$$\xi_M$ are isomorphic microbundles. Proof. We have two concurring definitions of micro-tangent bundle. The first is given by the exercise above, the second by forgeting about the vector bundle structure on $TM$$TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$$(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$$M\xrightarrow{s_0} TM$ is the zero section. However to show that these two definition agree we need a notion of microbundle isomorphism. Definition 0.4. Two microbundles $(E_n,X,i_n,j_n)$$(E_n,X,i_n,j_n)$, $n=1,2$$n=1,2$ over the same space $X$$X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$$V_1\subset E_1$ of $i_1(B)$$i_1(B)$ and $V_2\subset E_2$$V_2\subset E_2$ of $i_2(B)$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$$H\colon V_1\to V_2$ making the following diagram commute. $\displaystyle \xymatrix{ U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\ V_1\ar[r]^{p_1} \ar[ur]^{H} & U,}$ In our case we have $\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\\ U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,}$ where $V\subset TM$$V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$$V\cong U\times \mathbb{R}^n$ via local trivialisation. $\square$$\square$$ (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}}}} Let $M$ be a (paracompact!) smooth manifold. Show that $TM$ and $\xi_M$ are isomorphic microbundles. {{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 we have $$\xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\ U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,}$$ where $V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$ via local trivialisation. {{endproof}} n-dimensional microbundle is a quadruple $(E,B,i,j)$$(E,B,i,j)$ such that there is a sequence
$\displaystyle B\xrightarrow{i} E\xrightarrow{j} B$
and the following conditions hold.
1. $j\circ i=\id_B$$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
$\displaystyle h\colon V\to U\times \mathbb{R}^n.$

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

$\displaystyle \xymatrix{ U \ar[d]^{i}\ar[r]& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{j} \ar[ur]^{h} & U,}$

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

Exercise 0.2 [Milnor1964, Lemma 2.1]. Let $M$$M$ be a topological manifold. Show that $\xi_M : = (M \times M, M, \Delta_M, p_1)$$\xi_M : = (M \times M, M, \Delta_M, p_1)$ is a microbundle.

Proof.

Let $M$$M$ be a topological manifold. Then the composition $p_1\circ\Delta_M$$p_1\circ\Delta_M$ sends $x\mapsto (x,x)\mapsto x$$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$$x$. Choose $U$$U$ to be one of the open sets coming from atlas of $M$$M$ and let $\phi\colon U\to \mathbb{R}^n$$\phi\colon U\to \mathbb{R}^n$ be associated chart. The obvious choice for neighbourhood $V\subset M\times M$$V\subset M\times M$ is to take $U\times U$$U\times U$. The first naive candidate for $h\colon V=U\times U\to U\times\mathbb{R}^n$$h\colon V=U\times U\to U\times\mathbb{R}^n$ would be map $\id\times \phi$$\id\times \phi$. However such $h$$h$ fails to make the following diagram commute

$\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{\id\times \{0\}}& U\times\mathbb{R}^n \ar[d]^{p_1}\\ V\ar[r]^{p_1} \ar[ur]^{h} & U,}$

since $(u,u)$$(u,u)$ is mapped to $(u,\phi(u))$$(u,\phi(u))$ and $\phi(u)$$\phi(u)$ doesn't necessarily be $0$$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))$$h(u,v)=(u,h(u)-h(v))$.

$\square$$\square$

Exercise 0.3 [Milnor1964, Theorem 2.2]. Let $M$$M$ be a (paracompact!) smooth manifold. Show that $TM$$TM$ and $\xi_M$$\xi_M$ 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 $TM$$TM$ and treating it just as a microbundle $(TM, M, \pi,s_0)$$(TM, M, \pi,s_0)$ where $M\xrightarrow{s_0} TM$$M\xrightarrow{s_0} TM$ is the zero section.

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

Definition 0.4. Two microbundles $(E_n,X,i_n,j_n)$$(E_n,X,i_n,j_n)$, $n=1,2$$n=1,2$ over the same space $X$$X$ are isomorphic if there exist neighbourhoods $V_1\subset E_1$$V_1\subset E_1$ of $i_1(B)$$i_1(B)$ and $V_2\subset E_2$$V_2\subset E_2$ of $i_2(B)$$i_2(B)$ and a homeomorphism $H\colon V_1\to V_2$$H\colon V_1\to V_2$ making the following diagram commute.

$\displaystyle \xymatrix{ U \ar[d]^{i_1}\ar[r]^{i_2}& V_2 \ar[d]^{p_2}\\ V_1\ar[r]^{p_1} \ar[ur]^{H} & U,}$

In our case we have

$\displaystyle \xymatrix{ U \ar[d]^{\Delta_M}\ar[r]^{s_0}& V \ar[d]^{p_2}\\ U\times U\ar[r]^{p_1} \ar[ur]^{H} & U,}$

where $V\subset TM$$V\subset TM$ is an open neighbourhood of the zero section. Because of the vector bundle structure we may identify $V\cong U\times \mathbb{R}^n$$V\cong U\times \mathbb{R}^n$ via local trivialisation.

$\square$$\square$