In this section, we complete the proof of our main result Theorem1.4. We assume the notation and conventions made in5.1. In addition, we assume now thatρ∈H1,2(X).
We consider maps Fq:X→Rnas in (3.5), for q∈X. Recall however that Fq(x)is defined for all x∈O, where O is a fixed convexRN-neighborhood of p. By the remarks made in Section3.3, to prove Theorem1.4it suffices to show that p does not belong to the branch set BFp of Fp. We suppose towards a contradiction that p∈BFp. To achieve a contradiction, we will analyze limits of rescaled maps and spaces near p by using the results in the previous section.
As earlier, we require several lemmas.
Lemma7.1. — There exists a sequence(qk)of points in the branch set BFpconverging to p such that each point qk is a point of vanishing mean oscillation forρ.
Proof. — It follows from the Sobolev assumption ρ ∈ H1,2(X) and from the Poincaré inequality (2.6) that standard covering theorems as in [20, Chapter 1] or [15, Theorem 3, p. 77], for example,
we obtain thatHn−2-almost every point is a point of vanishing mean oscillation forρ. On the other hand, as discussed in Section 3.3, we have thatHn−2(BFp ∩V) >0 for every neighborhood V of p. The lemma follows from these remarks.
To continue the proof of Theorem1.4, fix a sequence(qk)as in Lemma7.1. We may assume that B(qk,|p−qk|)⊂O for all k. By Proposition6.1, we can find for each k≥1 a sequence (rk,i)of positive reals, converging to nought as i→ ∞, such that the maps
(Fqk)qk,rk,i :Xqk,rk,i →Rn
converge, as i → ∞, uniformly on bounded sets to a bijective linear map Lk :Pk → Rn, where Pk is an n-plane in RN with Xqk,rk,i →Pk. Each of the n-planes Pk carry a natural limiting orientation inherited from X, and the linear maps Lk :Pk → Rn are orientation preserving and bi-Lipschitz with constant independent of k. (See Remark5.3 and Lemma6.2.) We also assume that each Lkis defined in all ofRNand is Lipschitz with uniform Lipschitz constant.
From now on, we understand the sequences(rk,i)as fixed.
Lemma7.2. — For every R>0 and every x∈Xqk,rk,i with|x| ≤R we have that (7.2) |(Fqk)qk,rk,i(x)−(Fp)qk,rk,i(x)| ≤ dρ ∞|x||p−qk|
whenever rk,iR<|p−qk|.
Proof. — Note that it follows from the choices that both maps in (7.2) are defined for every R and x as required. Thus, fix R>0 and x∈Xqk,rk,i such that |x| ≤R, and assume that rk,iR<|p−qk|. Then
|(Fqk)qk,rk,i(x)−(Fp)qk,rk,i(x)| = 1 rk,i
[qk,qk+rk,ix]
ρ−
[p,qk+rk,ix]
ρ+
[p,qk]
ρ
= 1 rk,i
[p,qk,qk+rk,ix]dρ
≤ 1 rk,i
dρ ∞|rk,ix||p−qk| = dρ ∞|x||p−qk|, where [p,qk,qk+rk,ix] denotes the (oriented) triangle spanned by the three points. The
lemma follows.
Next, fix linear isometriesτk:RN→RNsuch thatτk|Rn:Rn→Pkis an orientation preserving surjection. Here and later we viewRnas a subset ofRNin a natural way. The maps Lk are uniformly bi-Lipschitz, so after passing to a subsequence we can further
assume that the maps Lk ◦τk|Rn:Rn→Rn converge to an invertible sensepreserving linear map L:Rn→Rn locally uniformly. Then the maps(Fp)qk,rk,i ◦τk are defined for every x∈B(0,R)∩τk−1(Xqk,rk,i)such that rk,iR<|p−qk|.
Lemma7.3. — After passing to a subsequence of the sequence (qk), there exists a sequence of positive real numbers(sk)converging to zero such that the sets
Yk :=B(0,k)∩τk−1(Xqk,sk)
converge locally in the Hausdorff distance toRnand such that the maps (Fp)qk,sk ◦τk:Yk →Rn
are defined and converge locally uniformly to L as k→ ∞.
Proof. — We may assume that the numbers rk,i satisfy rk,ik<|p−qk|for all k and i, so in particular the maps(Fp)qk,rk,i◦τk are always defined on B(0,k)∩τk−1(Xqk,rk,i). Fix m≥1. Because Lk◦τk→L locally uniformly, there exists km≥m such that
(7.3) |L(y)−Lk◦τk(y)|< 1 m
whenever k≥km and y∈B(0,m)∩Rn. Moreover, for every k there exist ik,m andδm>0 such that
(7.4) |Lk◦τk(y)−(Fqk)qk,rk,i ◦τk(yk,i)|< 1 m whenever i≥ik,mand
(7.5) y∈B(0,m)∩Rn, yk,i∈B(0,m)∩τk−1(Xqk,rk,i)
satisfy|y−yk,i|< δm. We can also assume, by Lemma7.2, that kmand im,k are large enough so that in addition to (7.3) and (7.4) we have
(7.6) |(Fqk)qk,rk,i◦τk(yk,i)−(Fp)qk,rk,i◦τk(yk,i)|< 1 m
whenever k ≥km, i≥ik,m, and yk,i is as in (7.5). Now consider the subsequence (qkm) of (qk), and the concomitant numbers sm:=rkm,ikm,m converging to zero. By passing to a yet new subsequence of (km) and enlarging ik,m, if necessary, we may assume that the sets B(0,m)∩τk−1
m (Xqkm,sm)converge as required to Rn. In consequence, it follows from (7.3)–(7.6) that (upon obvious renaming) we have sequences as asserted. The lemma
follows.
We use the subsequence provided by the preceding lemma, and set (7.7) fk:=L−1◦(Fp)qk,sk ◦τk.
Thus, fk : Yk →Rn is a BLD-mapping with fk(0)=0. By construction, the sets Yk
converge locally in the Hausdorff distance to Rn, and the mappings fk converge lo-cally uniformly to the identity on Rn. For s>0, denote by Dk(s) the 0-component of B(0,s)∩Yk ⊂Yk. Then the local degree μ(0,fk,Dk(s)) is defined for every sufficiently large k. (Compare the discussion before Lemma5.4.)
Lemma7.4. — Given s>0, the local degree satisfies (7.8) μ(0,fk,Dk(s))=1
for all sufficiently large k.
Proof. — Fix s>0. As explained in Proposition4.2, for all sufficiently large k there exist mappingsψk :B(0,4s)→Yk such that
|ψk(x)−x| ≤C dist(x,Yk)
for every x∈B(0,4s), where C≥1 is independent of k. By pushing the obvious radial homotopies inRNto Yk by using the mapsψk, we easily infer that the maps
ψk◦fk|B(0,2s)∩Yk:B(0,2s)∩Yk→Yk
are all homotopic to the identity through homotopies Htksuch that 0∈/Htk(∂Dk(s)), for all large k. The preceding understood, it follows from Corollary5.5and from the properties
of the local degree that (7.8) holds.
Proof of Theorem1.4. — Recall that qk ∈BFp, where qk is given in Lemma 7.1. In particular, we have that for each k the mapping fk :Yk→Rn as defined in (7.7) is BLD and has branching at 0. This result contradicts the statement of Lemma7.4for large k.
The proof of Theorem1.4is thereby complete.
7.1. Remarks. — As discussed in the introduction, the requirement that a Cartan-Whitney presentationρbelongs to H1,2loc(X)in Theorems1.2,1.5, and1.4is a kind of sec-ond order requirement (for the coordinates). The proof here and in earlier sections shows that this hypothesis could be changed to a VMO-type condition. Indeed, the Sobolev condition was used only in order to have the conclusion of Lemma 7.1. For the same conclusion, it would suffice to assume that
(7.9) Hn−2-almost every point in X is a VMO-point forρ.
(Recall the definition for VMO-points from (5.1).) In particular, all the main results of this paper would remain valid if the hypothesis that ρ is in H1,2 is changed to require-ment (7.9).
8. Smoothability
In this section, we first formulate and prove the following general smoothability result.
Theorem 8.1. — Let X⊂RN be a locally Ahlfors n-regular, connected, locally linearly contractible, and oriented homology n-manifold. Assume that there is a Cartan-Whitney presentation ρ=(ρ1, . . . , ρn)for X that is defined in anRN-neighborhood W of X. If the restriction ofρto X belongs to the local Sobolev space H1,2loc(X), then X is a smoothable topological n-manifold.
The proof of Theorem8.1is a combination of our Theorem1.4and the argument from [57, p. 336]. For completeness, we present the details. We begin by describing a canonical metric on X induced by a globally defined Cartan-Whitney presentation. Such metrics were employed in [57], and the crucial point is to produce local bi-Lipschitz homeomorphisms with respect to this metric that are near isometries. A theorem of Shikata [54] (and its noncompact version in [17]) then gives smoothability.
8.1. Metrics induced by Cartan-Whitney presentations. — Let X⊂W⊂RNandρbe as in Theorem 8.1, except that no Sobolev regularity is required for the Cartan-Whitney presentationρ. Recall that for every flat 1-formωin an open set V inRN, and for every Lipschitz mapγ : [0,L] →V, the action
(8.1) ω(γ (t)), γ(t) ≤ ω ∞|γ(t)|
is defined for almost every t∈ [0,L](see [66, Theorem 9A, p. 303] and Subsection3.4).
In particular, we can define, for a,b∈X, (8.2) dρ(a,b):=inf
L
0
n
i=1
ρi(t), γ(t)2 1/2
dt,
where the infimum is taken over all Lipschitz embeddings γ : [0,L] →X such that γ (0)=a andγ (1)=b.
To see that dρ defines a metric, observe first that dρ(a,b) <∞because X is rectifi-ably connected (see2.5and [11, Section 17]). It remains to show that dρ(a,b) >0 if a=b.
Fix a Lipschitz embeddingγ : [0,L] →X. We writeγ also for the imageγ ([0,L]). Now (see [16, 3.2.19])H1-almost every point q∈γ has an approximate tangent 1-plane, and satisfies
(8.3) H1(γ ∩B(q,r))
2r ≤2
for every 0<r<r(q), where B(q,2r(q))⊂W. Next, fix x∈B(q,r(q))∩γ such that the open subarcγ (q,x)ofγ from q to x lies in B(q,|q−x|). Thenγ (q,x)is naturally oriented
and can be viewed as a flat 1-chain in W. Consider the map Fq:B(q,2r(q))→Rngiven in (3.5). By [16, 4.2.9, 4.2.10, and 4.5.14], we can find a flat 2-chain S in B(q,2r(q))such that∂S=γ (q,x)− [q,x]and that
(8.4) 4π|S| ≤ |γ (q,x)− [q,x]|2
(cf. the proof of Lemma6.5). Therefore, by the Stokes formula (3.2), and by (8.4), (8.3), we obtain On the other hand, it follows from the properties of BLD-mappings (see [25, Proposition 4.13 and Theorem 6.8] or [26, (4.4)]) that
(8.6) c|q−x| ≤ |Fq(q)−Fq(x)| = |Fq(x)|,
where c>0 depends only on the local data near γ. (This requires that we choose r(q) initially small enough so that we have uniform Ahlfors regularity and local linear con-tractibility for points and radii in question, and moreover such that Fq|B(q,2r(q))∩X is a BLD-mapping with uniform data; see3.3.)
Now let x∈B(q,r(q))∩γ be arbitrary. Denote by x the point on γ between q
where C>0 is independent of q and x. Since H1almost every point q onγ satisfies the preceding condition, we easily obtain that
It follows that dρ is a metric as desired.
Note that (8.7), (8.1), and the local quasiconvexity of X give that the metric dρ is locally bi-Lipschitzly equivalent to the Euclidean metric on X; that is, for every point in X there is a neighborhood and a constant C≥1 such that
(8.8) C−1|a−b| ≤dρ(a,b)≤C|a−b| for all pairs of points a,b in the neighborhood.
Proof of Theorem 8.1. — As mentioned earlier, the proof is similar to that in [57];
only our framework is slightly different. Let dρ be the metric on X given in (8.2). By
the Sobolev assumption, Theorem1.4gives that the map Fp:B(p,r)∩X→Rngiven in (1.4) is bi-Lipschitz for every p∈X and r<r(p). Here it is immaterial whether we use the Euclidean metric on X or the metric dρ(see the remark just before the proof). On the other hand, the asymptotic estimate in (3.7) implies that Fp is close to an isometry with respect to dρ for points near p.
To establish the last claim, fix x,y∈B(p,r)for r>0 small, and letγxybe any rec-tifiable arc joining the two points; by (8.7) and (8.8), we may assume thatγ ⊂B(p,r(p)) and that H1(γ )≤C|x−y|for some constant C≥1 independent of x and y. As earlier,
where C>0 depends only on the local data. Thus, (8.9) |Fp(x)−Fp(y)| ≤(1+(r))dρ(x,y),
In conclusion, we have shown that X can be covered by open sets such that both (8.9) and (8.10) hold for every pair of points in the neighborhood. By the results in [17,53], we obtain that X has a compatible smooth structure as desired.
The proof of Theorem8.1is complete.
8.2. Remarks. — (a) The Sobolev condition forρin Theorem8.1can be replaced by the VMO-condition in (7.9) as explained in7.1. More generally, it suffices to assume thatρ is such that the local BLD-mappings Fq as in (3.5) are always local homeomor-phisms.
(b) It is not sufficient in Theorem8.1 to only assume the existence of a globally defined Cartan-Whitney presentation. Namely, for each n≥3 there are homology n-manifolds X that satisfy all the hypotheses in Theorem8.1except the Sobolev regularity forρ, but that are not even topological manifolds, cf. [25, 9.1], [26, 2.4]
(c) As a further example, let X⊂RN be a compact n-dimensional polyhedron that is also an orientable PL-manifold (with respect to the given triangulation), but not smoothable; such polyhedra exist for many n [39]. Then we can map X onto the stan-dard n-sphere by a BLD-mapping and pull back the stanstan-dard coframe to obtain a global Cartan-Whitney presentation on X [26, 2.4]. (Note that the mapping has a Lipschitz ex-tension to an open neighborhood of X in RN.) We suspect that by using an example of this kind, one could demonstrate the sharpness of the Sobolev condition in Theorem8.1.
We do not know if analogous examples exist in dimensions where every PL-manifold is smoothable, e.g. in dimension n=4 [65]. In particular, we recall that it is not known whether every Lipschitz 4-manifold is smoothable.
8.3. Sullivan’s cotangent structures. — Theorem 1.1 is a essentially a direct conse-quence of Theorem8.1. For completeness, we review the concept of a cotangent structure from [57] and include the proof.
Let X be a Lipschitz n-manifold as defined in Subsection1.1, and let {(Uα, ϕα)} be a Lipschitz atlas. Thus,{Uα}is an open cover of X and
ϕα :Uα→ϕα(Uα)=:Vα⊂Rn are homeomorphisms such that the transition maps
ϕαβ=ϕβ ◦ϕ−α1:ϕα(Uα∩Uβ)→ϕβ(Uα∩Uβ)
are bi-Lipschitz. A Whitney or flat k-formω on X is by definition a collection {ωα} of flat k-formsωα defined in Vα such that the compatibility conditions
ϕαβ∗ ωβ =ωα
hold inϕα(Uα ∩Uβ)⊂Vα. (Compare [60].) Flat k-forms on X obviously form a vector space, that is moreover a module over the ring Lip(X) of (locally) Lipschitz functions on X.
It makes sense to talk about Lipschitz vector bundles over X; the transition func-tions are Lipschitz maps into a linear group GL(·,R). By using a Lipschitz partition of unity, one can construct metrics, or Lipschitzly varying inner products, in Lipschitz vector bundles over X.
We say that X admits a cotangent structure if there exists pair(E, ι), where E is an oriented Lipschitz vector bundle of rank n over X and ι is a module map over Lip(X) from Lipschitz sections of E to flat 1-forms on X satisfying the following:
(*) If s1, . . . ,sn:X→E are Lipschitz sections such that the induced section s1∧
· · · ∧sndetermines the chosen orientation on the bundle∧nE, then for everyα the flat n-formι(s1)α∧ · · · ∧ι(sn)α=ταdx1∧ · · · ∧dxnin Vα satisfies
(8.11) ess infKτα >0
for every compact subset K of Vα. Here ess inf refers to Lebesgue n-measure.
The above description of a cotangent structure is somewhat simpler than that in [57]; in particular, we avoid considerations about local (torsion free) connections. In our setting, the existence of the local BLD-mappings is established in [26, 4.6]. On the other hand, we consider oriented bundles for simplicity.
If X is a smoothable, orientable n-manifold, then X obviously admits a cotangent structure, namely we can put E=T∗X.
Proof of Theorem1.1. — Let a Lipschitz n-manifold X admit a cotangent structure (E, ι). The Sobolev hypothesis in Theorem 1.1 means that ι(si)α ∈H1,2loc(Vα) for every α and for every orthonormal frame s1, . . . ,sn of Lipschitz sections in Uα. In particular, by Theorem1.4the local BLD-mappings considered by Sullivan in [57, Theorem 1] all have local degree equal to one. Theorem1.1now follows as in the proof of Theorem 2 in [57, p. 336]. (The proof there is analogous to the proof of Theorem8.1: one uses a global frame to define a metric on X such that the local BLD-mappings are near isometries with
respect to this metric.)
8.4. Remark. — We stipulated in the preceding that the Sobolev hypothesis in The-orem 1.1 means that ι(si)α ∈H1,2loc(Vα) for every α and for every orthonormal frame s=(s1, . . . ,sn)of Lipschitz sections in Uα. This regularity hypothesis is independent of the frame, and is thus a property of the bundle only. Indeed, any two frames s,sdiffer by a Lipschitz sectionσ :X→Aut E, andι is a module map over Lip(X). Therefore, the mapping from (ι(s1)α, . . . , ι(sn)α)to(ι(si)α, . . . , ι(sn)α)preserves the first order Sobolev spaces.
Acknowledgements
The authors wish to thank Mario Bonk, Seppo Rickman, Karen E. Smith, and Dennis Sullivan for useful discussions. The first named author in particular wishes to express his sincere gratitude to Dennis Sullivan who taught him so much.
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J. H.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
S. K.
Sydney, Australia stephjk@umich.edu
Sydney, Australia stephjk@umich.edu