Gromov-Hausdorff convergence and homotopy type

Janko Latschev^{∗}

November 3, 1999

Abstract

We show that for every closed Riemannian manifold X there exists a positive number ε_{0} > 0 such that for all 0 < ε ≤ ε_{0} there exists some δ > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than δ the geometric ε-complex |Y_{ε}| is homotopy equivalent to X. In particular, this gives a positive answer to a question of Hausmann [Hau95].

1 Introduction

This note grew out of an attempt to understand which properties of a given Riemannian manifold X are reflected in metric spaces Y Gromov-Hausdorff close to it. Even thought the topology of Y will generally be quite complicated, we will prove that by a very simple procedure one can recover the homotopy type of X from it, provided it is close enough in a sense made explicit below.

Given a pseudo-metric space (X, d) and a positive number ε > 0, one can define an abstract simplicial complex X_{ε} whose vertices are the points of X and whose k-simplices consist of all (k+1)-tuples of points in X with pairwise distances less than ε. Apparently this complex was first introduced by Vietoris in the thirties in an attempt to define a homology theory for general metric spaces. It has more recently found applications in the theory of hyperbolic groups.

It was proved by Hausmann [Hau95] that for a closed Riemannian manifold X and ε sufficiently small the geometric realization |X_{ε}| of this complex is homotopy equivalent to X. In fact, for ε > 0 sufficiently small he constructed an explicit homotopy equivalence T : |X_{ε}| ⟶ X as follows. One chooses a total order on the points of X and defines the map T by induction on the dimension of the simplices, starting with the identity on the 0-skeleton. Assuming T is defined on all (k-1)-simplices, we define it on a k-simplex {x_{0}, ..., x_{k}} with x_{0} < x_{1} < ... < x_{k} by mapping the straight line between x_{k} and a point p ∈ {x_{0}, ..., x_{k−1}} in the front face to the geodesic between the (already known) image points T (x_{k}) and T (p). This works for example when ε is smaller than the convexity radius of X. Because of the total order, these maps are compatible with each other and so T is continuous.

∗Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland; currently at: Max-Planck-Institut für Mathematik, Bonn email: latschev@mpim-bonn.mpg.de

Hausmann then raised the question whether there would always be a finite subset F ⊂ X and an ε > 0 such that the finite simplicial complex |F_{ε}| is homotopy equivalent to X under the restriction of the map T. In this note, we give a positive answer to that question by proving the stronger

Theorem 1.1 Let X be a closed Riemannian manifold. Then there exists ε_{0} > 0 such that for every 0 < ε ≤ ε_{0} there exists a δ > 0 such that the geometric realization |Y_{ε}| of the ε-complex of any metric space Y which has Gromov-Hausdorff distance less than δ to X is homotopy equivalent to X.

When Y ⊂ X is a subset, a homotopy equivalence is given by the map T as described above, and in general it is similar in spirit.

It was already observed by Hausmann that one can easily find Y ⊂ X and ε > 0 such that the map T : |Y_{ε}| ⟶ X is a domination, i.e. it admits a right homotopy inverse. The main difficulty in showing that this map can be chosen to be a genuine homotopy inverse is to show that |Y_{ε}| is sufficiently locally contractible. The concept of ε-contraction as defined in the next section turns out to be central to our argument.

While the proof we present here is qualitative, it will be clear that it is possible to extract reasonable estimates for ε_{0} and δ = δ(ε_{0}) which depend only on the geometry of X. This observation should be useful in extending the result to apply to more general length spaces. We hope to return to this question elsewhere.

2 The flat case

In this section, we will prove the assertion in the very special case when X is a closed flat manifold and F ⊂ X is a finite subset. In the next section we will repeat the argument along the same lines, but dealing with various technical issues arising in the more general context.

ε-contraction

ε-contraction is a variation of collapse as introduced by Hausmann [Hau95]. Let Y be a metric space and let ε > 0 be fixed. Suppose we are given a subset Y′ ⊂ Y and a point y′ ∈ Y \ Y′ satisfying

(C) d(x, z) < ε implies d(y′, z) < ε for any x ∈ Y′ and z ∈ Y.

Then the map c : Y ⟶ Y \ Y′ mapping Y′ to the point y′ ∈ Y (and leaving all other points fixed) is called an ε-contraction. By (C), such a map induces a simplicial map on ε-complexes for Y and Y \ Y′, and using (C) with x = z one sees that this map is contiguous to the identity of Y_{ε}. Recall that two maps f, g : Y_{ε} ⟶ Y_{ε} are called contiguous if for every simplex σ the images f(σ) and g(σ) again span a simplex.

In particular |(Y \ Y′)_{ε}| is a strong deformation retract of |Y_{ε}| (cf. [Spa66], p. 130). A metric space Y will be called ε-contractible if it can be mapped to a space of diameter less than ε through a sequence of ε-contractions. Clearly in this case |Y_{ε}| is contractible in the usual sense.

In this section we will only deal with elementary ε-contractions, where Y′ consists of only one point. The first key observation can now be stated as follows:

Lemma 2.1 Let B(0, r) ⊂ ℝ^{n} be the standard open ball of radius r, and let 0 < ε < r be given. Then there exists δ_{1} > 0 such that every finite δ_{1}-dense subset F ⊂ B(0, r) is ε-contractible.

Proof: Let any subset F ⊂ B(0, r) be given. Let p ∈ F have maximal distance r′ = d(p, 0) to the origin, and consider the set

I_{p} = ⋂_{x∈\overline{B(p,ε)}∩\overline{B(0,r′)}} \overline{B(x, ε)}.

If we want to move p by an elementary ε-contraction, than the points of I_{p} are precisely the allowed image points. We claim that I_{p} contains an open ball of definite radius, which we are trying to compute. It is easy to see that the only constraints on I_{p} come from points x in the intersection S(p, ε) ∩ S(0, r′) of the ε-sphere around p with the r′-sphere around 0.

So let x ∈ S(p, ε) ∩ S(0, r′) be given. Then the distance of x to the line \overline{0p} is given by

h = ε · \sqrt{1 − \frac{ε^{2}}{4r′^{2}}} .

The point p′ ∈ \overline{0p} closest to x has distance

η = η(ε, r) = \frac{ε^{2}}{2r′}  (1)

from p and distance

δ_{1} = δ_{1}(ε, r′) = ε · (1 − \sqrt{1 − \frac{ε^{2}}{4r′^{2}}} )  (2)

from the boundary of B(x, ε). Notice that for ε ≤ 2r′ we have

δ_{1} ≤ ε(1 − (1 − \frac{ε^{2}}{4r′^{2}} )) ≤ η.  (3)

Combining these results, we see that the ball B(p′, δ_{1}) is completely contained in I_{p} and does not contain p.

Notice that both η and δ_{1} increase with decreasing r′. Hence if F is δ_{1}-dense in B(0, r) for δ_{1} = δ_{1}(ε, r), and p′ ∈ B(0, r) is constructed from some point p ∈ F of maximal distance to 0 as above, then there exists a point q ∈ F ∩ B(p′, δ_{1}). Now by the definition of I_{p} the map c_{pq} : F ⟶ F is an elementary ε-contraction. We can apply this same argument to F′ = F \ {p}, and, continuing inductively, we will eventually move through a sequence of elementary ε-contractions to a subset F̃ ⊂ F of diameter less than ε. Thus F is ε-contractible as claimed. □

Remark 2.2 Note that the same argument applies more generally to every finite set F δ_{1}-dense in some convex subset D ⊂ B(0, r) containing 0.

To be able to apply this remark, the following technical lemma is useful.

Lemma 2.3 Let B_{1}, ..., B_{N} be a finite cover of some Riemannian manifold X by open balls, and let δ_{1} be given. Then there exists a number δ_{2} such that every subset F ⊂ X which is δ_{2}-dense in X has the property that F ∩ (B_{i_{1}} ∩ ... ∩ B_{i_{k}}) is δ_{1}-dense in every non-empty intersection B_{i_{1}} ∩ ... ∩ B_{i_{k}}.

Proof: We will argue by contradiction. So suppose such a δ_{2} does not exist. Then for every δ_{2} > 0 there exists a set F_{δ_{2}} which is δ_{2}-dense in X and there exists some non-empty intersection B_{i_{1}} ∩ ... ∩ B_{i_{k}}such that F ∩ (B_{i_{1}} ∩ ... ∩ B_{i_{k}}) is not δ_{1}-dense in B_{i_{1}} ∩ ... ∩ B_{i_{k}}.

In particular, we can find a sequence δ_{2,k} ⟶^{k→∞} 0 such that the “offending” intersection B_{i_{1}} ∩ ... ∩ B_{i_{k}} is fixed. Thus we get a sequence of points p_{k} ∈ B_{i_{1}} ∩ ... ∩ B_{i_{k}} such that no point of F_{k} ∩ (B_{i_{1}} ∩ ... ∩ B_{i_{k}}) is closer than δ_{1} to p_{k}. Since X is compact, the p_{k} have some subsequence converging to a point p ∈ X.

Now B(p, \frac{δ_{1}}{2} )∩(B_{i_{1}}∩...∩B_{i_{k}}) is open and non-empty, since it contains infinitely many points of the sequence {p_{k}}. Therefore it contains a ball of definite size B(q, ρ). For δ_{2,k} < ρ there has to be some point x_{k} ∈ F_{k} in this ball. But then

d(x_{k}, p_{k}) ≤ d(x_{k}, p) + d(p, p_{k}) < \frac{δ_{1}}{2} + d(p, p_{k})

thus contradicting the assumption on p_{k} as soon as d(p, p_{k}) < \frac{δ_{1}}{2} . □

We are now ready to answer Hausmann’s question in the flat case.

Proposition 2.4 Let X be a flat closed manifold. Then there exists ε_{0} > 0 such that for every 0 < ε ≤ ε_{0} there is some δ > 0 such that the ε-complex of any finite δ-dense subset F ⊂ X is homotopy equivalent to X under the map T.

Proof: Consider a finite cover B(a_{1}, r) ∪ B(a_{2}, r) ∪ ... ∪ B(a_{N}, r) of X by balls of radius r < conv(X), and define ε_{0} to be the Lebesgue number of this cover. Here conv(X) denotes the convexity radius of X. By making r (and hence ε) smaller if necessary, we may assume that the distance of any two disjoint balls is strictly positive. For any ε ≤ ε_{0}, let δ_{1} = δ_{1}(ε, r) and δ_{2} be constructed as in the previous two lemmas. By making them smaller if necessary, we may assume that δ_{2} ≤ δ_{1} < \frac{ε}{2}. We claim that δ = δ_{2} satisfies the conclusion of the proposition.

Namely, consider any finite δ_{2}-dense set F ⊂ X and choose a total order on its points. As described in the introduction, the map T : |F_{ε}| ⟶ X is given by mapping each simplex to the geodesic suspension of its front face. Notice also that since the balls B(a_{i}, r) are strictly convex, T maps simplices of |(F ∩ B(a_{i}, r))_{ε}| into B(a_{i}, r).

To define a map S : X ⟶ |F_{ε}|, we use an idea found in [Gro96], §1.4.D’. Consider a continuous partition of unity {ρ_{x}}_{x∈F} subordinate to the cover of X by δ_{1}-balls centered at points of F, where we require that p ∈ B(a_{i}, r) and ρ_{x}(p) > 0 implies x ∈ B(a_{i}, r). For example, if B(a_{1}, r), ..., B(a_{k}, r) are the balls in the cover containing x ∈ F, we can define

ρ′_{x}(p) := d(p, ⋃_{a_{j} ∉{a_{1},...,a_{k}}} B(a_{j}, r)) · d(p, X \ B(x, δ_{1})).

By Lemma 2.3, our choice of δ_{2} guarantees that for every p ∈ X there is some x ∈ F with ρ′_{x}(p) > 0. As usual, ρ_{x} is now obtained by normalizing.

Next, we use the total order on F to define a bijective map from F to the vertices of the standard (N − 1)-simplex Δ = {t ∈ ℝ^{N} | ∑ t_{i} = 1} ∈ ℝ^{N}, where N is the number of points in F. We can identify |F_{ε}| with the appropriate subcomplex of this simplex. Now define S : X ⟶ |F_{ε}| ⊂ ℝ^{N} by

S(p) = (ρ_{x_{1}}(p), ρ_{x_{2}}(p), ..., ρ_{x_{N}} (p)).

The condition δ_{1} < \frac{ε}{2} guarantees that the image of this map is indeed contained in |F_{ε}|. Notice also that by construction S(B(a_{i}, r)) ⊂ |(F ∩ B(a_{i}, r))_{ε}|.

Now again by construction T ∘ S : X ⟶ X maps each of the balls B(a_{i}, r) into itself, so that this map is homotopic to the identity of X by a straight line homotopy. In the same way, S ∘ T : |F_{ε}| ⟶ |F_{ε}| maps each of the sets |(F ∩ B(a_{i}, r))_{ε}| into itself. Now by an induction on cells argument using Lemma 2.1 and Remark 2.2 we see that this map is homotopic to the identity of |F_{ε}|, and the proposition is established. □

3 The general case

We will start the treatment of the general case by presenting a stronger version of Lemma 2.1.

Lemma 3.1 Let 0 < ε ≤ \frac{r}{2} be given. Then there exists a number δ̃_{1} > 0 such that every metric space (Y, d) whose Gromov-Hausdorff distance to some convex subset D ⊂ B(0, r) ⊂ ℝ^{n} of the standard flat ball of radius r is less than δ̃_{1} is ε-contractible.

Here δ̃_{1} = δ̃_{1}(ε, r) can be chosen as δ_{1}(ε, r)/5 with δ_{1}(ε, r) given by equation (2).

Proof: Let δ̃_{1} be defined as stated, and let (Y, d) be a given metric space δ̃_{1}- close to the convex subset D of the standard ball B(0, r) ⊂ ℝ^{n}. Without loss of generality D contains 0.

Let d̃ be some pseudometric on Y ⊔ D extending d on Y and d_{eucl} on D such that Y is contained in the δ̃_{1}-neighborhood of D and vice versa. The existence of this pseudometric is equivalent to our assumption that D and Y are δ̃_{1}-close ([Gro99], ch. 3).

Fix a finite subset F ⊂ D which is δ̃_{1}-dense in D. Then the d̃-balls B̃ of radius 2δ̃_{1} around points of F will cover all of Y ⊔ D.

Now pick a point y ∈ F of maximal distance to 0, and construct y′ ∈ D as in the proof of Lemma 2.1. Then there is a point ỹ′ ∈ Y with d̃(ỹ′, y′) < δ̃_{1}.

We want to show that mapping Y′ = B̃(y, 2δ̃_{1}) ∩ Y to ỹ′ ∈ Y is an εcontraction. So fix any point ỹ ∈ Y′.

To verify condition (C), suppose that x̃ ∈ Y is a point with d(ỹ, x̃) < ε. Then there exists x ∈ D with d̃(x̃, x) < δ̃_{1} and therefore

d_{eucl}(y, x) ≤ d̃(y, ỹ) + d(ỹ, x̃) + d̃(x̃, x) < ε + 3δ̃_{1}.

So there has to be some point z ∈ \overline{yx} ⊂ D such that d_{eucl}(y, z) < ε and d_{eucl}(x, z) < 3δ̃_{1}. From the proof of Lemma 2.1 we also know that

d_{eucl}(z, y′) ≤ ε − δ_{1}(ε, r),

where δ_{1}(ε, r) is given by equation (2). Therefore

d(x̃, ỹ′) ≤ d̃(x̃, x) + d_{eucl}(x, z) + d_{eucl}(z, y′) + d̃(y′, ỹ′)
< ε − δ_{1}(ε, r) + 5δ̃_{1}.

Hence if δ̃_{1} ≤ δ_{1}/5, then condition (C) holds. Note also that

d(ỹ′, 0) ≤ d̃(ỹ′, y′) + d_{eucl}(y′, 0)
< δ̃_{1} + (d_{eucl}(y, 0) − η(ε, r))
< d(ỹ, 0) − η(ε, r) + 3δ̃_{1}
< d(ỹ, 0),

which shows that ỹ′ ∉ Y′ = B̃(y, 2δ̃_{1}) ∩ Y as required. The procedure can now be iterated just as in the proof of Lemma 2.1, and since everything is controlled by the finite set F we are assured that it will terminate after a finite number of steps. □

Now let some closed Riemannian manifold X be given. Since the assertion of Theorem 1.1 is invariant under scaling, it is sufficient to prove it for some rescaled X̂. By multiplying the metric on X by a large constant, we can achieve that any subset D ⊂ B(p, 1) of any ball B(p, 1) of radius 1 in the rescaled space X̂ has distance less than δ̃_{1}(\frac{1}{2}, 1)/2 to the subset D′ ⊂ T_{p}X̂ in the flat ball of radius 1 corresponding to it under the exponential map at p.

We can cover X̂ by finitely many balls B(a_{1}, 1), ..., B(a_{N}, 1) of radius 1 such that the centers are \frac{1}{2}-dense in X̂. It follows that the Lebesgue number ε_{0} of this cover is at least \frac{1}{2}.

For a metric space Y which is δ-close to X̂ for some δ > 0, again consider some metric d̃ on the disjoint union Y ⊔ X̂ as in the proof of Lemma 3.1 such that Y is contained in the δ-neighborhood of X̂ and vice versa. It is easy to see that if δ is smaller than the Lebesgue number of the cover ⋃ B(a_{i}, 1) of X̂, then the balls B̃(a_{i}, 1) of radius 1 in the metric d̃ on Y ⊔ X̂ will cover all of Y ⊔ X̂.

The analogue of Lemma 2.3 in the present setting is

Lemma 3.2 Given X̂ = ⋃ B(a_{i}, 1) as above and 0 < δ < \frac{1}{2}, there exists δ̃_{2} > 0 such that every metric space Y which is δ_{2}-close to X̂ has the property that Y ∩ (B̃(a_{i_{1}}, 1) ∩ ... ∩ B̃(a_{i_{k}}, 1)) is δ-close to B(a_{i_{1}}, 1) ∩ ... ∩ B(a_{i_{k}}, 1) whenever the later is non-empty.

The proof is essentially the same as for Lemma 2.3 and will be ommitted.

To prove Theorem 1.1 for X̂ with ε_{0} = \frac{1}{2}, we proceed just as in the proof of Proposition 2.4. Given any ε < ε_{0}, we first find δ̃_{1} = δ̃_{1}(ε, 1) as in Lemma 3.1 and then δ̃_{2} = δ̃_{2}(\frac{δ̃_{1}}{2} ) as in Lemma 3.2.

Given a metric space Y which is \frac{δ̃_{2}}{4} -close to X̂, we can find a finite subset F ⊂ Y which is \frac{δ̃_{2}}{4} -close to X̂ by covering X̂ with finitely many balls of radius \frac{δ̃_{2}}{4} and then picking one point from each of the corresponding balls in the metric d̃ on Y ⊔ X̂.

Notice also that the sets Y ∩ (B̃(a_{i_{1}}, 1) ∩ ... ∩ B̃(a_{i_{k}}, 1)) are ε-contractible by Lemma 3.1 and the construction of X̂ and δ̃_{2}. We now define the map T : |Y_{ε}| ⟶ X̂ as follows. Each point y ∈ Y is contained in a certain maximal intersection ⋂ B̃(a_{i}, 1) of Y ⊔ X̂. Now simply map y to a closest point x ∈ ⋂ B(a_{i}, 1) ⊂ X, then proceed by induction on cells as before. To define the inverse map S : X̂ ⟶ |Y_{ε}|, use a partition of unity on X̂ subordinate to the cover by balls B̃(y, δ̃_{1}) around the points of F ⊂ Y as in the proof of Proposition 2.4. Now the rest of the argument goes through just as in that case, using the Lemmas 3.1 and 3.2 in place of Lemmas 2.1 and 2.3.

Acknowlegdments

I would like to thank Th. Mautsch for pointing out [Gro96] to me and F. Schlenk for critical comments on an earlier version of this paper. I also wish to thank the Max-Planck-Institut für Mathematik in Bonn, where this work was completed, for its hospitality and support.

References

[Gro96] M. Gromov, Carnot-Carathéodory spaces seen from within, in: Sub-Riemannian Geometry (A. Belaiche and J.-J. Risler, eds.), Progress in Mathematics, vol. 144, Birkhäuser, 1996, pp. 85–324.

[Gro99] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progr. in Math. 152, Birkhäuser, 1999.

[Hau95] J.-C. Hausmann, On the Vietoris-Rips Complexes and a Cohomology Theory for Metric Spaces, in: Prospects in Topology: Proceedings of a conference in honour of William Browder, Princeton University Press, 1995, pp. 175–188.

[Spa66] E. Spanier, Algebraic topology, McGraw-Hill, 1966.