Lipschitz Homotopy Research Articles

The uniform homotopy category

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associated uniform homotopy category of uniform spaces. Bounded cubical cohomology generalizes to a representable theory on the Lipschitz homotopy category. Bounded singular cohomology on path-connected spaces generalizes to a representable theory on the uniform homotopy category. Along the way, this paper develops a cubical analogue of Kan's Ex∞ functor and proves a cubical approximation theorem for uniform maps.

Lipschitz Homotopy Groups of Contact 3-Manifolds

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb H^n$ with its Carnot-Carathéodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of $\mathbb H^1$ to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(\pi,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

Converting uniform homotopies into Lipschitz homotopies via moduli of continuity

We give a simple procedure to convert a uniform homotopy F into a Lipschitz homotopy H for a large class of domain and target spaces. In this procedure, we use the modulus of continuity of F to estimate the Lipschitz constant of H. For Lipschitz homotopy in the time direction, our result is related to a theorem due to A. Calder and J. Siegel. As an application of our result, a Lipschitz version of a theorem of J. West is given.

Plato’s cave and differential forms

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of differential algebras are closely shadowed, in a technical sense, by maps between the corresponding spaces. As a concrete application, we prove the following conjecture of Gromov: if $X$ and $Y$ are finite complexes with $Y$ simply connected, then there are constants $C(X,Y)$ and $p(X,Y)$ such that any two homotopic $L$-Lipschitz maps have a $C(L+1)^p$-Lipschitz homotopy (and if one of the maps is a constant, $p$ can be taken to be $2$.) We hope that it will lead more generally to a better understanding of the space of maps from $X$ to $Y$ in this setting.

Lipschitz Homotopy Convergence of Alexandrov Spaces

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.

On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

We study the question: When are Lipschitz mappings dense in the Sobolev space W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) ? Here M M denotes a compact Riemannian manifold with or without boundary, while H n \mathbb {H}^n denotes the n n th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) for all 1 ≤ p > ∞ 1\le p>\infty if dim ⁡ M ≤ n \dim M \le n , but that Lipschitz maps are not dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) if dim ⁡ M ≥ n + 1 \dim M \ge n+1 and n ≤ p > n + 1 n\le p>n+1 . The proofs rely on the construction of smooth horizontal embeddings of the sphere S n \mathbb {S}^n into H n \mathbb {H}^n . We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the n n th Lipschitz homotopy group of H n \mathbb {H}^n . We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S n ,Y ) are not dense in the Sobolev space W 1,n (S n ,Y ). On the other hand we show that if a met- ric space Y is Lipschitz (n 1)-connected, then Lipschitz mappings Lip(X,Y ) are dense in N 1,p (X,Y ) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincare inequality.

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $${\pi_m^{\rm Lip}(\mathbb{H}_n)}$$ , in terms of properties of the classical homotopy group of the sphere, $${\pi_m(\mathbb{S}^n)}$$ . As an application we provide a new simplified proof of the fact that $${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$$ , and we prove a new result that $${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$$ when $${\dim \mathcal{M} \geq 4n}$$ and 4n−1 ≤ p < 4n.

Quantitative algebraic topology and Lipschitz homotopy

We consider when it is possible to bound the Lipschitz constant a priori in a homotopy between Lipschitz maps. If one wants uniform bounds, this is essentially a finiteness condition on homotopy. This contrasts strongly with the question of whether one can homotop the maps through Lipschitz maps. We also give an application to cobordism and discuss analogous isotopy questions.

Asymptotic coarsening of proper metric spaces and index problems

We introduce an asymptotic coarse structure on proper metric spaces and study the associated C * -algebras and assembly maps. We establish an asymptotic Lipschitz homotopy invariance theorem for the K-theory of these C * -algebras and the K-homology of the metric space, and show that the assembly map is an isomorphism over an asymptotically scaleable space.

From the Boundary of the Convex Core to the Conformal Boundary

If N is a hyperbolic 3-manifold with finitely generated fundamental group, then the nearest point retraction is a proper homotopy equivalence from the conformal boundary of N to the boundary of the convex core of N. We show that the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse and that one may bound the Lipschitz constants in terms of the length of the shortest compressible curve on the conformal boundary.

Uniform and Lipschitz homotopy classes of maps

If X is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line R into X an element of H 1 (X, U/U 0 ), where U is the space of uniformly continuous functions from R to R and U 0 is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to H 1 (X, U/U 0 ) is surjective. If X is the n-dimensional torus, it is bijective, while if X is a compact orientable surface of genus > 1, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds P to compact smooth manifolds X. With each such Lipschitz homotopy class we associate an element of H n (X, B + /B + 0 ), where n is the dimension of P, B is the space of bounded continuous functions from the positive real axis to R, and B + 0 is the set of all f ∈ B + such that lim t→ ∞ f(t) = 0.

Homotopies of bounded width are almost Lipschitz

Let Y be a compact Riemannian manifold. Let X be locally compact and paracompact. It is shown that every uniform homotopy and every homotopy of bounded width from X to Y may be approximated by a Lipschitz homotopy. Applications to the study of uniform homotopy and homotopies of bounded width are given.