Simple Homotopy Research Articles

Examples and cofibrant generation of effective Kan fibrations

We will make two contributions to the theory of effective Kan fibrations, which are a more explicit version of the notion of a Kan fibration, a notion which plays a fundamental role in simplicial homotopy theory. We will show that simplicial Malcev algebras are effective Kan complexes and that the effective Kan fibrations can be seen as the right class in an algebraic weak factorization system. In addition, we will introduce two strengthenings of the notion of an effective Kan fibration, the symmetric effective and degenerate-preferring Kan fibrations, and show that the previous results hold for these strengthenings as well.

Internal sums for synthetic fibered (∞,1)-categories

We give structural results about bifibrations of (internal) (∞,1)-categories with internal sums. This includes a higher version of Moens' Theorem, characterizing cartesian bifibrations with extensive aka stable and disjoint internal sums over lex bases as Artin gluings of lex functors. We also treat a generalized version of Moens' Theorem due to Streicher which does not require the Beck–Chevalley condition. Furthermore, we show that also in this setting the Moens fibrations can be characterized via a condition due to Zawadowski.Our account overall follows Streicher's presentation of fibered category theory à la Bénabou, generalizing the results to the internal, higher-categorical case, formulated in a synthetic setting. Namely, we work inside simplicial homotopy type theory, which has been introduced by Riehl and Shulman as a logical system to reason about internal (∞,1)-categories, interpreted as Rezk objects in any given Grothendieck–Rezk–Lurie (∞,1)-topos.

Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

Abstract We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n \mathbb{Z}_{n} grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 \pi_{1} -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.

Morse theory for group presentations

We introduce a novel combinatorial method to study Q ∗ ∗ Q^{**} -transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews–Curtis conjecture do satisfy the conjecture.

Random simple-homotopy theory

We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions. For triangulated d-manifolds with d≤6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 6$$\\end{document}, we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, (14k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(14k+1)$$\\end{document}-vertex triangulations of a new series of Bing’s houses with k rooms, k≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 3$$\\end{document}, which all can be deformed to a point using only six pure elementary expansions.

A new method to h-regularize finite topological spaces

The most useful methods for computing homology of finite topological spaces require that certain regularity conditions are satisfied. In particular, the most suitable algorithmic methods, from the point of view of complexity, are only applicable to h-regular spaces. In this paper, we show a procedure to h-regularize a finite topological space X of height at most 2, that is, to construct an h-regular space that is simple homotopy equivalent to X. This enables us to carry out some topological computations (homology groups) on the h-regular space obtained from this h-regularization process.

Minimal graphs for contractible and dismantlable properties

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be I-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of I-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of k-dismantlable graphs and k-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n\\ge 6$$\\end{document} with finite fundamental group, we show that the classifying space BDiff(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\ extrm{Diff}(M)$$\\end{document} of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\subset M$$\\end{document} of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

Homotopic and Geometric Galois Theory

In his “Letter to Faltings”, Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group \operatorname{G}_{\mathbb{Q}} of the field of rational numbers that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions – e.g. simplicial homotopy, Tannaka perversity, automorphic forms – that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory.

Towards a constructive simplicial model of Univalent Foundations

We provide a partial solution to the problem of defining a constructive version of Voevodsky's simplicial model of Univalent Foundations. For this, we prove constructive counterparts of the necessary results of simplicial homotopy theory, building on the constructive version of the Kan-Quillen model structure established by the second-named author. In particular, we show that dependent products along fibrations with cofibrant domains preserve fibrations, establish the weak equivalence extension property for weak equivalences between fibrations with cofibrant domain and define a univalent fibration that classifies small fibrations between bifibrant objects. These results allow us to define a comprehension category supporting identity types, Σ $\Sigma$ -types, Π $\Pi$ -types and a univalent universe, leaving only a coherence question to be addressed.

Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings N↪M of one topological manifold into another. More specifically, we describe an algorithm for computing the homology Hn(X,A) and cohomology Hn(X,A) of a finite connected CW-complex X with local coefficients in a Zπ1X-module A when A is finitely generated over Z. It can be used, in particular, to compute the integral cohomology Hn(X˜H,Z) and induced homomorphism Hn(X,Z)→Hn(X˜H,Z) for the covering map p:X˜H→X associated to a finite index subgroup H<π1X, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree 2 homology group H2(X˜H,Z) distinguishes between the homotopy types of the complements X⊂R4 of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree 1 homology homomorphism H1(p−1(B),Z)→H1(X˜H,Z) distinguishes between the homeomorphism types of the complements X⊂R3 of the granny knot and the reef knot, where B⊂X is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology). Our open source implementation allows the user to experiment with further examples of knots, knotted surfaces, and other embeddings of spaces. We conclude the paper with an explanation of how the cohomology algorithm also provides an approach to computing the set [W,X]ϕ of based homotopy classes of maps f:W→X of finite CW-complexes over a fixed group homomorphism π1f=ϕ in the case where dim⁡W=n, π1X is finite and πiX=0 for 2≤i≤n−1.

Simple homotopy theory and nerve theorem for categories

We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss an application to the topological data analysis for data-sets lying in a non-convex field.

Manifolds Homotopy Equivalent to Certain Torus Bundles over Lens Spaces

AbstractWe compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces Sl/(ℤ/p) with fiber Tn for an odd prime p and l ≥ 3 provided that the induced ℤ/p‐action on π1(Tn) = ℤn is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN‐extensions. We give a collection of classical surgery invariants such as splitting obstructions and ρ‐invariants that decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC

Exposed circuits, linear quotients, and chordal clutters

A graph G is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on IG‾, the edge ideal of the complement graph. Known results imply that IG‾ has linear quotients if and only if G is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higher-dimensional analogues of this result, and show that in fact linear quotients for more general circuit ideals of d-clutters can be characterized in terms of removing exposed circuits in the complement clutter. Restricting to properly exposed circuits can be characterized by a homological condition. This leads to a notion of higher dimensional chordal clutters which borrows from commutative algebra and simple homotopy theory. The interpretation of linear quotients in terms of shellability of simplicial complexes also has applications to a conjecture of Simon regarding the extendable shellability of k-skeleta of simplices. Other connections to combinatorial commutative algebra, chordal complexes, and hierarchical clustering algorithms are explored.

Some remarks on PL collapsible covers of 2-dimensional polyhedra

We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL collapsible subpolyhedra in terms of their simple homotopy type and certain local properties.

A differentiable homotopy method to compute perfect equilibria

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

On the stable Cannon Conjecture

The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable version that for any closed manifold N of dimension greater or equal to 2 there exists a closed manifold M together with a simple homotopy equivalence from M to the cartesian product of N and BG. If N is aspherical and pi_1(N) satisfies the Farrell-Jones Conjecture, then M is unique up to homeomorphism.

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

A simple homotopy proximal mapping algorithm for compressive sensing

In this paper, we present novel yet simple homotopy proximal mapping algorithms for reconstructing a sparse signal from (noisy) linear measurements of the signal or for learning a sparse linear model from observed data, where the former task is well-known in the field of compressive sensing and the latter task is known as model selection in statistics and machine learning. The algorithms adopt a simple proximal mapping of the $$\ell _1$$ norm at each iteration and gradually reduces the regularization parameter for the $$\ell _1$$ norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithms for recovering the sparse signal under three different settings (i) sparse signal recovery under noiseless measurements, (ii) sparse signal recovery under noisy measurements, and (iii) nearly-sparse signal recovery under sub-Gaussian noisy measurements. In particular, we show that when the measurement matrix satisfies restricted isometric properties (RIP), one of the proposed algorithms with an appropriate setting of a parameter based on the RIP constants converges linearly to the optimal solution up to the noise level. In addition, in setting (iii), a practical variant of the proposed algorithms does not rely on the RIP constants and our results for sparse signal recovery are better than the previous results in the sense that our recovery error bound is smaller. Furthermore, our analysis explicitly exhibits that more observations lead to not only more accurate recovery but also faster convergence. Finally our empirical studies provide further support for the proposed homotopy proximal mapping algorithm and verify the theoretical results.

From simplicial homotopy to crossed module homotopy in modified categories of interest

Abstract We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.