# nLab ball

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

### Geometric

For $n \in \mathbb{N}$ a natural number, the $n$-dimensional ball or $n$-disk in $\mathbb{R}^n$ is the topological space

$D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\} \subset \mathbb{R}^n$

equipped with the induced topology as a subspace of the Cartesian space $\mathbb{R}^n$.

Its interior is the open $n$-ball

$\mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \} \subset \mathbb{R}^n \,.$

Its boundary is the $(n-1)$-sphere.

More generally, for $(X,d)$ a metric space then an open ball in $X$ is a subset of the form

$B(x,r) \coloneqq \{x \in X \;|\; d(x,y) \lt r \}$

for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$. (The collection of all open balls in $X$ form the basis of the metric topology on $X$.)

### Combinatorial

There are also combinatorial notions of disks. For instance that due to (Joyal), as entering the definition of the Theta-category. See for instance (Makkai-Zawadowski).

## Properties

### Closed balls

A simple result on the homeomorphism type of closed balls is the following:

###### Theorem

A compact convex subset $D$ in $\mathbb{R}^n$ with nonempty interior is homeomorphic to $D^n$.

###### Proof

Without loss of generality we may suppose the origin is an interior point of $D$. We claim that the map $\phi: v \mapsto v/{\|v\|}$ maps the boundary $\partial D$ homeomorphically onto $S^{n-1}$. By convexity, $D$ is homeomorphic to the cone on $\partial D$, and therefore to the cone on $S^{n-1}$ which is $D^n$.

The claim reduces to the following three steps.

1. The restricted map $\phi: \partial D \to S^{n-1}$ is continuous.

2. It’s surjective: $D$ contains a ball $B = B_{\varepsilon}(0)$ in its interior, and for each $x \in B$, the positive ray through $x$ intersects $D$ in a bounded half-open line segment. For the extreme point $v$ on this line segment, $\phi(v) = \phi(x)$. Thus every unit vector $u \in S^{n-1}$ is of the form $\phi(v)$ for some extreme point $v \in D$, and such extreme points lie in $\partial D$.

3. It’s injective: for this we need to show that if $v, w \in \partial D$ are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have $w = t v$ for $t \gt 1$, say. Let $B$ be a ball inside $D$ containing $0$; then the convex hull of $\{w\} \cup B$ is contained in $D$ and contains $v$ as an interior point, contradiction.

So the unit vector map, being a continuous bijection $\partial D \to S^{n-1}$ between compact Hausdorff spaces, is a homeomorphism.

###### Corollary

Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.

###### Proof

$D$ has nonempty interior relative to its affine span which is some $k$-plane, and then $D$ is homeomorphic to $D^k$ by the theorem.

### Open Balls

Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the smooth category:

###### Observation

The open $n$-ball is homeomorphic and even diffeomorphic to the Cartesian space $\mathbb{R}^n$

$\mathbb{B}^n \simeq \mathbb{R}^n \,.$
###### Proof

For instance, the smooth map

$x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n$

has smooth inverse

$y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.$

This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a (smooth) manifold. Hence, each (smooth) $n$-dimensional manifold is locally isomorphic to both ${\mathbb{R}}^n$ and $\mathbb{B}^n$.

From general existence results about smooth structures on Cartesian spaces we have that

###### Theorem

In dimension $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.

See the first page of (Ozols) for a list of references.

###### Remark

In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$. See De Michelis-Freedman.

###### Theorem

(star-shaped domains are diffeomorphic to open balls)

Let $C \subset \mathbb{R}^n$ be a star-shaped open subset of a Cartesian space. Then $C$ is diffeomorphic to $\mathbb{R}^n$.

###### Remark

Theorem is a folk theorem, but explicit proofs in the literature are hard to find. See the discussion at References. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus 07). A simpler proof is given in Gonnord-Tosel 98 reproduced here.

Here is another proof:

###### Proof

Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in Lee proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying bump functions we can assume that $f\le 1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin; by rescaling the ambient space we can assume $\epsilon=2$.

The smooth vector field $V\colon x\mapsto f(x)\cdot x/{\|x\|}$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le 1$ such that $b=1$ for ${\|x\|} \gt 1/2$ and $b=0$ in a neighborhood of 0. The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon \mathbb{R} \times T\to T$. (The parameter of the flow is all of $\mathbb{R}$ and not just some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt {\|x\|} \lt 2$ the vector field $V$ equals $x\mapsto x/{\|x\|}$. Also, all flow lines of $V$ are radial rays.

Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})$ for ${\|x\|} \gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms,

${\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},$

hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form $L: (A,B) \to T$, where $A$ and $B$ can be finite or infinite. If $B$ is finite and the limit of $L(t)$ as $t \to B$ exists, then the vector field $V$ vanishes at $B$. In our case $V$ can only vanish at the boundary of $T$, which is precisely what we want for surjectivity.)

Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for ${\|x\|} \lt 2$ and as $p$ for ${\|x\|}\gt 1/2$. The map $g$ is smooth because for $1/2\lt {\|x\|} \lt 2$ both definitions give the same value.

And here is another proof, due to Gonnord and Tosel, translated into English by Erwann Aubry and available on MathOverflow:

###### Theorem

Every open star-shaped set $\Omega$ in $\mathbb{R}^n$ is $C^\infty$-diffeomorphic to $\mathbb{R}^n$.

###### Proof

For convenience assume that $\Omega$ is star-shaped at $0$.

Let $F=\mathbf{R}^n\setminus\Omega$ and $\phi:\mathbf{R}^n\rightarrow\mathbb{R}_+$ (here $\mathbf{R}_+=[0,\infty)$) be a $C^\infty$-function such that $F=\phi^{-1}(\{0\})$. (Such $\phi$ exists by the Whitney extension theorem.)

Now we define $f:\Omega\rightarrow\mathbb{R}^n$ via the formula:

$f(x)=\overbrace{\left[1+\left(\int_0^1\frac{dv}{\phi(vx)}\right)^2\|x\|^2\right]}^{\lambda(x)}\cdot x=\left[1+\left(\int_0^{\|x\|}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot x.$

Clearly $f$ is smooth on $\Omega$.

We set $A(x)=\sup\{t\gt0\mid t\frac{x}{\|x\|}\in\Omega\}$. $f$ sends injectively the segment (or ray) $[0,A(x))\frac{x}{\|x\|}$ to the ray $\mathbf{R}_+\frac{x}{\|x\|}$. Moreover, $f(0\frac{x}{\|X\|})=0$ and

$\lim_{r\rightarrow A(x)}\left\|f(r\frac{x}{\|x\|})\right\|=\lim_{r\to A(x)}\left[1+\left(\int_0^{r}\frac{dt}{\phi\left(t\cdot\frac{rx}{\|x\|}\cdot\left\|\frac{\|x\|}{rx}\right\|\right)}\right)^2\right]\cdot r= \left[1+\left(\int_0^{A(x)}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot A(x)=+\infty.$

Indeed, if $A(x)=+\infty$, then it holds for obvious reason. If $A(x)\lt+\infty$, then by definitions of $\phi$ and $A(x)$ we get that $\phi(A(x)\frac{x}{\|x\|})=0$. Hence by the mean value theorem and the fact that $\phi$ is $C^1$ due to

$\phi\left(r\frac{x}{\|x\|}\right)\le M(A(x)-r)$

for some constant $M$ and every $r$. As a result,

$\int_0^{A(x)}\frac{dt}{\phi\left(t\frac{x}{\|x\|}\right)}$

diverges. Hence we infer that $f([0,A(x))\frac{x}{\|x\|})=\mathbf{R}_+\frac{x}{\|x\|}$ and so $f(\Omega)=\mathbf{R}^n$.

To end the proof we need to show that $f$ has a $C^\infty$-inverse. But as a corollary from the inverse function theorem we get that it is sufficient to show that $df$ vanishes nowhere.

Suppose that $d_x f(h)=0$ for some $x\in\Omega$ and $h\neq 0$. From definition of $f$ we get that

$d_x f(h)=\lambda(x)h+d_x \lambda(h)x.$

Hence $h=\mu x$ for some $\mu\neq 0$ and from that $x\neq 0$. As a result $\lambda(x)+d_x \lambda(x)=0$. But we have that $\lambda(x)\ge1$ and function $g(t):=\lambda(tx)$ is increasing, so $g'(1)=d_x \lambda(x)\gt0$, which gives a contradiction.

###### Example

Let $I(\Delta^n) \subset \mathbb{R}^n$ be the interior of the standard $n$-simplex. Then there is a diffeomorphism to $\mathbb{B}^n$ defined as follows:

Parameterize the $n$-simplex as

$I(\Delta^n) = \left\{ (x^1, \cdots, x^n) \in \mathbb{R} | (\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1) \right\} \,.$

Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by

$(x^1, \ldots, x^n) \mapsto (\log(\frac{x^1}{1 - x^1 - \ldots -x^n}), \ldots, \log(\frac{x^n}{1 - x^1 - \ldots - x^n})) \,.$

(Thanks to Todd Trimble.) One way to think about it is that $I(\Delta^n)$ is the positive orthant of an open $n$-ball in $l^1$ norm, so that in the opposite direction we have a chain of invertible maps

$\array{ \mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\ & & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1) }$

which we simply invert to get the map $f$ above.

### Good covers by balls

One central application of balls is as building blocks for coverings. See good open cover for some statements.

### Geometric

• V. Ozols, Largest normal neighbourhoods , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor)

That an open subset $U \subseteq \mathbb{R}^4$ homeomorphic to $\mathbb{R}^4$ equipped with the smooth structure inherited as an open submanifold of $\mathbb{R}^4$ might nevertheless be non-diffeomorphic to $\mathbb{R}^4$, see

• De Michelis, Stefano; Freedman, Michael H. (1992) “Uncountably many exotic $\mathbb{R}^4$‘s in standard 4-space”, J. Diff. Geom. 35, pp. 219-254.

### Star-shaped regions diffeomorphic to open ball

The proof that open star-shaped regions are diffeomorphic to a ball appears as theorem 237 in

It is a lengthy proof, due to Stefan Born.

A simpler version of the proof apparently appears on page 60 of

• Stéphane Gonnord, Nicolas Tosel, Calcul Différentiel, ellipses (1998)

and is reproduced in

Apparently this proof is little known. For instance in a remark below lemma 10.5.5 of

• Lawrence Conlon, Differentiable manifolds, Birkhäuser (last edition 2008)

it says:

It seems that open star shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.

And in

• Jeffrey Lee, Manifolds and differential geometry (2009)

one finds the statement:

Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to $\mathbb{R}^n$ is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).

Here “Grom” refers to

• Mikhail Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology. F. Tricerri ed., World Sci., Singapore,

(1990), 1-38. (pdf)

where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of $C^1$ class, not $C^\infty$, so this is not a proof either.

For a discussion of diffeomorphisms between geodesically convex regions and open balls see at good open cover.

• Mihaly Makkai, Marek Zawadowski, Duality for Simple $\omega$-Categories and Disks (TAC)