Deformations of Lipschitz Homeomorphisms

In the early 1970’s, Siebenmann [89] introduced a class of spaces, termed locally cone-like spaces (CS sets). Such spaces generalize the notion of a manifold. In particular, such spaces are not necessarily manifolds, but can always be stratified into manifolds. Although such a space can be quite singular, its space of open embeddings behaves as though it were the space of open embeddings of a manifold. More precisely, Siebenmann introduced a general deformation rule $\mathcal{D}(X)$, satisfied by $CS$ sets, that enjoys numerous desirable properties. For instance, the deformation rule allows one to deform open embeddings lying in a small enough neighborhood of the inclusion in a manner akin to the manifold case. In particular, the deformation rule yields an isotopy theory that generalizes the Edwards and Kirby isotopy theory [20]. Indeed, Siebenmann’s theory shows that the homeomorphism group of a reasonable compactum $X$ that is not necessarily a manifold, yet enjoys the rule $\mathcal{D}(X)$, is locally contractible (say, with the compact-open topology). The local contractibility of the homeomorphism group of a compact manifold is well known and was first proven by Cernavskii in 1968 [11], and later reproven, in the early 1970’s, by the use of the simple, yet powerful torus trick due to Kirby [49, 20].

The idea behind showing that $CS$ sets satisfy the rule $\mathcal{D}(X)$ is as follows. One notices that $CS$ sets of a certain “depth” $d$ can be covered by open sets of depth at most $d$, each of which is “equivalent” to some $\mathbb{R}^{m}\times cL$, where $cL$ denotes the cone over $L$. Thus, it suffices to show, through induction, that the principle $\mathcal{D}(\mathbb{R}^{m}\times cL)$ holds. The key point being that one must deform embeddings on $\mathbb{R}^{m}\times cL$ appropriately. Siebenmann’s deformation rule $\mathcal{D}(X)$, and more specifically, the theory of Edwards and Kirby, are topological and, although they offer many desirable properties, they do not necessarily satisfy the condition that the perturbed map inherits the regularity properties of the initial map. In the 1970’s, shortly after the announcement of the celebrated theory of Edwards and Kirby [20], Sullivan [93], using deep theory from etalé cohomology, offered, for Lipschitz manifolds, an alternative to the Kirby immersion device. Namely, by replacing the torus by a certain quotient of hyperbolic space, Sullivan obtained an analogue of the Edwards–Kirby theory on Lipschitz manifolds. Therefore, Sullivan’s construction allowed for the introduction of a Lipschitz deformation rule $\mathcal{L}(X)$.

Examples of sets admitting a CS stratification that are not necessarily manifolds include Alexandrov spaces (see [21] for a nice proof of this fact). Alexandrov spaces, introduced by Burago, Gromov and Perelman [9] in 1990 (see also [1, 8]) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. Indeed, they have played a major role in metric and Riemannian geometry (see for example [53, 46, 47, 38, 45, 37, 60, 61, 7, 5, 44, 27, 36, 35, 24, 83]).

By analyzing the local structure of Alexandrov spaces, Perelman in 1991 used Siebenmann’s theory of deforming homeomorphisms to establish the celebrated stability theorem (see [81, 43]). The stability theorem asserts that if a given sequence of compact $n$-dimensional Alexandrov spaces $\{X_{i}\}_{i\in\mathbb{N}}$ with a uniform lower curvature bound Gromov–Hausdorff converges to another compact Alexandrov space $X$ with no collapse, then for all sufficiently large $i$, $X_{i}$ and $X$ are homeomorphic. It is claimed that Perelman proved a bi-Lipschitz analogue, though no such proof has been published.

To generalize both Siebenmann’s theory and the topological techniques used by Perelman in his stability theorem to the Lipschitz category, one must, first, have appropriate deformation principles. Second, one must deform, appropriately, (open) Lipschitz embeddings on $\mathbb{R}^{m}\times(cone)$. Both of these points have been addressed and have been answered in [88, 93]. Lastly, one must obtain Lipschitz analogues of the relevant topological ingredients of Perelman’s stability theorem: the isotopy extension theorem (Theorem 6.5 in [89]) and the union lemma (Lemma 6.9 in [89]). Indeed, a corollary of both is the important fibration theorem (Corollary 6.14 in [89] and Theorem $A$ in [81]) that a closed topological submersion $p\colon E\rightarrow X$ with compact fibers, such that $\mathcal{D}(p^{-1}(x))$ is satisfied for all $x\in X$, and $p^{-1}(x)$ is a locally connected Hausdorff space, is a locally trivial fiber bundle.

Using the techniques in Siebenmann’s paper [89], Perelman obtained the fundamental gluing theorem (Theorem $B$ in [81] and Theorem 4.6 in [43]). The gluing theorem, roughly speaking, gives conditions as to when one can glue locally defined homeomorphisms near a global Hausdorff approximation to a homeomorphism that remains close to the approximation. Using reverse induction, and the notion of a “frame”, Perelman proved the stability theorem by essentially reducing it to the gluing theorem. Indeed, Perelman reduced the problem of stability, which is a global problem, to a local problem. We note that in addition to the stability theorem, the topological gluing theorem has found applications in the theory of $\mathrm{RCD}$ spaces (Theorem 3.5 in [45]).

In this paper we will first prove the Lipschitz analogue of the isotopy extension result in [89]. Further, we will under a canonical adaptation of the rule $\mathcal{L}(X)$ (see Definition 4.5), prove the Lipschitz analogue of the “union lemma”. The union lemma (Lemma 4.13), roughly speaking, ensures that under certain mild but rather important conditions, one can glue “product charts” on a neighborhood of a union. The union lemma has further applications other than Siebenmann’s theory of deforming homeomorphisms (see for instance page 227 in [50]). Using the new Lipschitz union lemma, we will show that provided that the fibers of a nice enough Lipschitz submersion satisfy the (adapted) Lipschitz deformation rule, then the Lipschitz submersion is indeed a locally trivial Lipschitz bundle. Moreover the adapted rule will yield a stronger and more desirable isotopy extension theorem (see Corollary 4.6).

Our first theorem is a generalization of Siebenmann’s isotopy extension principle (Theorem 6.5 in [89]).

In addition to Perelman’s proof of the stability theorem, the topological isotopy extension theorem has found applications in K-theory (see section 1 in [98]).

Note that our terminology might differ from other uses (see Definition 2.1). In particular, when we say “Lipschitz embeddings”, we mean embedding in the Lipschitz ”category”. In particular, there is more data than just a topological embedding that is Lipschitz. We do this so that we follow the conventions in the literature and thus make the paper easier to read.

Making the natural addendum to $\mathcal{L}(X)$ (see above), and denoting the refined deformation principle by $\mathcal{L}^{*}(X)$, we prove our second main result, a Lipschitz fibration theorem.

We note that the topological analogue of the above theorem is a crucial step in the proof that Alexandrov spaces are locally conical (Theorem 4.4 in [43], Theorem A in [81], Theorem 1.4 [80]).

Lastly, we prove the Lipschitz analogue of Perelman’s gluing theorem (Theorem $B$ in [81], Theorem 4.6 in [43]) under the rule $\mathcal{L}(X)$. Indeed, the topological gluing theorem is a crucial step in the topological stability theorem.

In what follows, denote by $\chi\colon\mathbb{R}^{\geq 0}\rightarrow\mathbb{R}^{\geq 0}$ increasing continuous functions such that $\chi(0)=0$.

Two important remarks are in order. First, in the statement of Perelman’s topological gluing theorem, the condition "MCS" can be replaced by the more general assumption that the spaces under consideration satisfy Siebenmann’s deformation principle. Therefore Theorem C is truly the Lipschitz analogue of Perelman’s gluing theorem. Second, the union lemma allowed Perelman to obtain the strong gluing theorem (Complement to Theorem $B$ [81], and Theorem $4.10$ in [43]).

Our article is organized as follows: In section 2, we discuss the preliminaries, give examples, and then define the Lipschitz deformation principle. In particular, we discuss the existence and non-existence of various Lipschitz structures on various spaces that admit different types of curvature bounds. We further include references which are related to the examples. In section 3, we discuss how to obtain a relative formulation of Siebenmann’s deformation principle. Then, we prove the Lipschitz isotopy extension theorem. In section 4, we discuss Lipschitz submersions and prove the Lipschitz Union lemma. In section 5 we prove the Lipschitz analogue of Perelman’s deformation lemma and the gluing theorem and give a few remarks concerning Alexander isotopies on cones. Finally, in Section 6, we quickly develop the results parallel to the results in earlier sections. However, in this case, we control the Lipschitz constants.

First, we convene that, throughout this paper, Lipschitz will be understood to be a local condition.

A useful result that we will use, along with its variations, often and tacitly is the following (see [89]).

Now consider the following deformation rule.

Now that we have given various examples arising from different contexts, we return to the theory of deforming Lipschitz homeomorphisms. To that end, we first make the following remark.

We conclude this section by discussing some conventions we will follow in this paper.

In this section, we shall prove a Lipschitz analogue of the isotopy extension theorem in [89]. Our proof is slightly more technical than Siebenmann’s. This is due to a couple of reasons: First, as stated, Sullivan’s deformation principle is slightly different than Siebenmann’s $\mathcal{D}(X)$. Second, our statement offers a mild strengthening of the corresponding one in Siebenmann’s paper. That is, Siebenmann proves the isotopy extension theorem for when $f_{b}$ is the inclusion and does not immediately address whether $F_{t}\circ f_{b}=f_{b}$ away from a neighborhood of $C$ (when $C$ is compact). Third, we employ a “thickening technique” by controlling the borders of the sets so as to ensure that we can glue Lipschitz maps to a larger Lipschitz map (recall that our definition of Lipschitz is local). This idea will be useful, and used throughout the paper. In particular, it will be used in the proof of the main theorems. To the best of our knowledge, this technique does not appear in the literature. Lastly, we note that we will also use ideas from [89].

To set the stage, we will first show that the Lipschitz deformation principle $\mathcal{L}(X)$ implies a relative deformation principle. Consequences of relative deformation principles are important and abound. For example, in the topological category, the "strong gluing theorem", a relative version of the important "gluing theorem" (see [43]) allowed Perelman to prove his stability theorem. Siebenmann used relative versions of his deformation principle to obtain deformation results, including but not limited to, the theory of foliations [89]. Edward’s and Kirby [20] used relative versions of their deformation theory, to show, among many things, that the homeomorphism groups of certain manifolds is locally contractible in a relative manner.

The general framework of the deformation theory in the Lipschitz category does not work as briefly as it does in the topological category. Let us clarify why this is the case. In the topological category, we have the following result [89] that is quite useful in the topological deformation theory.

In general, in Theorem A, we only have a continuous family of open embeddings $f_{t}$ that can be perturbed to another continuous family of Lipschitz isomorphisms $F_{t}$. Basic examples show that it need not be the case that the perturbed family will inherit stronger regularity properties as the following example more clearly articulates.

Due to the preceding example, in order to obtain a Lipschitz deformation theory that is parallel to Siebenmann’s, it is customary to refine the Lipschitz deformation deformation principle. To that end, one must define the notion of a Lipschitz isotopy [88].

Due to the issue raised above, one makes the following addendum to the deformation statement $\mathcal{L}(X)$ [88]:

The above property is natural. For example, it is satisfied by Lipschitz manifolds [93] and locally finite simplicial complexes [88]. The (refined) principle, which we will denote by $\mathcal{L}^{*}(X)$ further yields the following results, a strengthening of the isotopy extension principle and a deformation which respects the “bundle” structure (cf. Theorem 6.1 in [89]). The proof of Theorem A, combined with property $(P)$ yields the following corollaries.

Now we will introduce the notion of a Lipschitz submersion (c.f. Definition 6.8 in [89], and page 59 in [50]). Note that topological submersions enjoy many properties and have shown to be useful (see [50]). They are, in particular, a weaker form of a fiber bundle. We devote the rest of this paper to showing that a proper Lipschitz submersion with fibers satisfying $\mathcal{L}^{*}(X)$ is indeed a Lipschitz fiber bundle. In what follows, we will endow the product space with the sum metric (of course, one may also choose another equivalent one).

Now we shall give a proof of the Lipschitz analogue of Lemma 6.12 in [89]. The following lemma was used to give a proof of the Union lemma in [89]. We remark that there is no proof there. We give a proof here only to further illustrate the usefulness of property $(P)$. Our proof of the union lemma is for the most part different than the one in [89].

Now, we will prove Theorem B. The proof is similar to the corresponding one in [89]. We make a simplification.