Fig 3 - uploaded by Shmuel Weinberger
Content may be subject to copyright.
The Lipschitz norm is uniformly bounded, yet the C 2 norm necessarily grows.
Source publication
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...
Similar publications
Recently, there has been interest in the question of whether a partial matrix in which many of the fully defined principal submatrices are PSD is approximately PSD completable. These questions are related to graph theory because we can think of the entries of a symmetric matrix as corresponding to the edges of a graph. We first introduce a family o...
![MWλ\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/320180730/figure/fig3/AS:960043285491724@1605903564907/MWldocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Knots Tproj(d,d-2)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/320180730/figure/fig5/AS:960043289686017@1605903565355/Knots-Tprojd-d-2documentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots and links in \(\mathbb {RP}^3\) which is not a topological isotopy invariant. In this paper we study real algebraic links of degree d with the maximal value of this invariant. We show that these links admit entirely topological description. In particular, these links are ch...
In this paper, we investigate automorphisms of compact K\"ahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^\infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected component...
Citations
... Another motivation for representing homotopy classes by simplicial maps and complexity bounds for such algorithms is the connection to quantitative questions in homotopy theory (Gromov 1999;Ferry and Weinberger 2013) and in the theory of embeddings (Freedman and Krushkal 2014). Given a suitable measure of complexity for the maps in question, typical questions are: What is the relation between the complexity of a given null-homotopic map f : X → Y and the minimum complexity of a nullhomotopy witnessing this? ...
... What is the minimum complexity of an embedding of a simplicial complex K into R d ? In quantitative homotopy theory, complexity is often quantified by assuming that the spaces are metric spaces and by considering Lipschitz constants (which are closely related to the sizes of the simplicial representatives of maps and homotopies Ferry and Weinberger 2013). For embeddings, the connection is even more direct: a typical measure is the smallest number of simplices in a subdivision K or K such that there exists a simplexwise linear-embedding K → R d . ...
A central problem of algebraic topology is to understand the homotopy groups \(\pi _d(X)\) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group \(\pi _1(X)\) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with \(\pi _1(X)\) trivial), compute the higher homotopy group \(\pi _d(X)\) for any given \(d\ge 2\). However, these algorithms come with a caveat: They compute the isomorphism type of \(\pi _d(X)\), \(d\ge 2\) as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \(\pi _d(X)\). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere \(S^d\) to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes \(\pi _d(X)\) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere \(S^d\) to X. For fixed d, the algorithm runs in time exponential in \(\mathrm {size}(X)\), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed \(d\ge 2\), we construct a family of simply connected spaces X such that for any simplicial map representing a generator of \(\pi _d(X)\), the size of the triangulation of \(S^d\) on which the map is defined, is exponential in \(\mathrm {size}(X)\).
... The cases q = 0 and q = 1 of this conjecture are proved in [FW13] and [CDMW16], respectively. Indeed, those results hold for homotopies and not only nullhomotopies. ...
... If F * ξ is a bounded distance from h(x), then Hopf invariants on boundaries of 4-cells can be determined by integrating a form a bounded distance from h(y). Combined with ideas from [FW13] and [CDMW16], this allows us to kill these Hopf invariants by modifying the map in a bounded way. ...
... Note that L times barycentric subdivision is not regular. Two known examples of regular subdivision schemes are the edgewise subdivision described in [EG00] and the cubical subdivision described in [FW13]. ...
In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemannian manifold $X$ and $Y$ a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected $Y$, asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.
... Subsequently, we hope to use these results to compute explicit embeddings of simplicial complexes into R d (as opposed to deciding embeddability). Moreover, the problems of computing maps representing homotopy classes or explicit embeddings are closely related to corresponding quantitative questions in homotopy theory [24,12] and in the theory of embeddings [19], see Section 2. 7 That is, they compute integers r, q 1 , . . . , q k such that π d (X) is isomorphic to Z r ⊕ Z q 1 ⊕ . . . ...
... We remark that the results of the present paper form an important stepping stone, since the algorithm for computing [X, Y ] computes the homotopy groups π k (Y ) of the target space, 2 ≤ k ≤ 2d, as intermediate results and uses these for further computations. 12 In a subsequent step, we then hope to generalize this further to the equivariant setting [X, Y ] G of [59], in which a finite group G of symmetries acts on the spaces X, Y and all maps and homotopies are required to be equivariant, i.e., to preserve the symmetries. Thus, the goal is to obtain an algorithm that, given an element of [X, Y ] G , computes an explicit equivariant map X → Y (represented as a simplicial map on a suitable subdivision of X, say). ...
... Similarly, algorithms and complexity bounds for representing homotopy classes by simplicial maps have a close connection to quantitative questions in homotopy theory [24,12]. Given a suitable measure of complexity for the maps in question, a typical question is: What is the relation between the complexity of a given nullhomotopic map f : X → Y and the minimum complexity of a nullhomotopy witnessing this? ...
... Moreover, extension results for ≥ are proved by patching a nearest point retraction of an extension together with a smooth extension of a smooth map [12, Theorems 1 and 2] for which there does not seem to be an immediate linear bound; when > a compactness argument leads to a nonlinear estimate of the norm of the extension by the norm of the trace which has no reason to be linear [57,Theorem 4]. When = 1 − 1∕ and is a compact Riemannian manifold such that either 1 In the limit case → 1 and → +∞, the problem of quantitative bounds has some analogy with the construction of controlled Lipschitz homotopies to constant maps [35], whose answer depends on the finiteness of the first homotopy groups of the target manifold [34]. ...
Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space $W^{s,p}(\mathcal{M}, \mathcal{N})$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach--Steinhaus uniform boundedness principle in linear Banach spaces.
... Quantitative homotopy theory. Another motivation for representing homotopy classes by simplicial maps and complexity bounds for such algorithms is the connection to quantitative questions in homotopy theory [21,13] and in the theory of embeddings [17]. Given a suitable measure of complexity for the maps in question, typical questions are: What is the relation between the complexity of a given null-homotopic map f : X → Y and the minimum complexity of a nullhomotopy witnessing this? ...
... What is the min-imum complexity of an embedding of a simplicial complex K into R d ? In quantitative homotopy theory, complexity is often quantified by assuming that the spaces are metric spaces and by considering Lipschitz constants (which are closely related to the sizes of the simplicial representatives of maps and homotopies [13]). For embeddings, the connection is even more direct: a typical measure is the smallest number of simplices in a subdivision K or K such that there exists a simplexwise linear-embedding K → R d . ...
... Algorithm from Lemma 6. 13. ...
A central problem of algebraic topology is to understand the \emph{homotopy groups} $\pi_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the \emph{fundamental group} $\pi_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is \emph{simply connected} (i.e., with $\pi_1(X)$ trivial), compute the higher homotopy group $\pi_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, \v{C}adek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $\pi_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $\pi_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $\pi_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $\pi_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$.
... (3) From a null-cobordism, transversality will greatly increase the size of a manifold. We deal with (1) and (2) simultaneously, in the same vein as [FW13]. In the latter paper, maps with target possessing finite homotopy groups are studied and it is shown that there are linear upper bounds on the size of the null-homotopy, and not only on the sizes of the maps comprising the null-homotopy. ...
... The obstruction in [FW13] has to do ultimately with homological filling functions. Isoperimetry likewise comes up in our result. ...
... The last author would like to thank Steve Ferry for a collaboration that began this work. Essentially, the polynomial bound in the non-oriented case can be obtained by combining [FW13] with some of the embedding arguments in this paper. We also thank MSRI for its hospitality during a semester (long ago) when we began working towards the results reported here. ...
For a given null-cobordant Riemannian n n -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n n . In the appendix the bound is improved to one that is O ( L 1 + ε ) O(L^{1+\varepsilon }) for every ε > 0 \varepsilon >0 .
This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y Y is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic L L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y .
Scalable spaces are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov’s long-standing conjecture on distortion in higher homotopy groups.
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.
We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring ideas from each of these domains to the other. Illustrative examples are given in approximation theory, variational calculus (closed geodesics), and quantitative cobordism theory.
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.
