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Definition

Geometric

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

D n:={x n| i(x i) 21} n 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 n\mathbb{R}^n.

Its interior is the open nn-ball

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

Its boundary is the (n1)(n-1)-sphere.

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

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

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

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 DD in n\mathbb{R}^n with nonempty interior is homeomorphic to D nD^n.

Proof

Without loss of generality we may suppose the origin is an interior point of DD. We claim that the map ϕ:vv/v\phi: v \mapsto v/\|v\| maps the boundary D\partial D homeomorphically onto S n1S^{n-1}. By convexity, DD is homeomorphic to the cone on D\partial D, and therefore to the cone on S n1S^{n-1} which is D nD^n.

The claim reduces to the following three steps.

  1. The restricted map ϕ:DS n1\phi: \partial D \to S^{n-1} is continuous.

  2. It’s surjective: DD contains a ball B=B ε(0)B = B_{\varepsilon}(0) in its interior, and for each xBx \in B, the positive ray through xx intersects DD in a bounded half-open line segment. For the extreme point vv on this line segment, ϕ(v)=ϕ(x)\phi(v) = \phi(x). Thus every unit vector uS n1u \in S^{n-1} is of the form ϕ(v)\phi(v) for some extreme point vDv \in D, and such extreme points lie in D\partial D.

  3. It’s injective: for this we need to show that if v,wDv, w \in \partial D are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have w=tvw = t v for t>1t \gt 1, say. Let BB be a ball inside DD containing 00; then the convex hull of {w}B\{w\} \cup B is contained in DD and contains vv as an interior point, contradiction.

So the unit vector map, being a continuous bijection DS n1\partial D \to S^{n-1} between compact Hausdorff spaces, is a homeomorphism.

By slightly modifying this argument, we can prove that the closure of any open star-shaped region is homeomorphic to the nn-disk iff it is compact. For, we may assume every point in the closure is connected to the origin by a line segment in the closure. Steps 1 and 2 above hold without modification, and as for step 3, for any open neighborhood of the line segment between the origin and ww, any convex open subneighborhood of the segment contains vv as an interior point.

Corollary

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

Proof

DD has nonempty interior relative to its affine span which is some kk-plane, and then DD is homeomorphic to D kD^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 nn-ball is homeomorphic and even diffeomorphic to the Cartesian space n\mathbb{R}^n

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

For instance, the smooth map

xx1+|x| 2: n𝔹 n x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n

has smooth inverse

yy1|y| 2:𝔹 n n. y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.

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

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

Theorem

In dimension dd \in \mathbb{N} for d4d \neq 4 we have:

every open subset of d\mathbb{R}^d which is homeomorphic to 𝔹 d\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 4\mathbb{R}^4. See De Michelis-Freedman.

Theorem

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

Proof/Remark

This is a folk theorem . But explicit proofs in the literature are very hard to find. See the discussion at Refereces. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus).

Example

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

Parameterize the nn-simplex as

I(Δ n)={(x 1,,x n)|(i:x i>0)and( i=1 nx i<1)}. 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(Δ n) nf : I(\Delta^n) \to \mathbb{R}^n by

(x 1,,x n)(log(x 11x 1x n),,log(x n1x 1x n)). (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.)

Good covers by balls

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

References

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

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

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)

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 UMU \subset M are always diffeomorphic to n\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 n\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

  • M. 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 1C^1 class, not C C^\infty, so that this is not a proof, either.

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

See also the Math Overflow discussion here.

Combinatorial

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

Revised on July 17, 2014 05:54:23 by Colin Tan (182.55.187.4)