Homotopy Type Research Articles

Characteristic cohomology II: Matrix singularities

AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

Involutions, links, and Floer cohomologies

AbstractWe develop a version of Seiberg–Witten Floer cohomology/homotopy type for a 4‐manifold with boundary and with an involution that reverses the structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3‐manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov‐type inequalities that relate topological quantities of 4‐manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4‐manifolds, and nonsmoothable unoriented surfaces in 4‐manifolds.

Non‐accessible localizations

AbstractIn a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe . When specialized to an appropriate family, this produces a localization which when interpreted in the ‐topos of spaces agrees with the localization corresponding to . Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any ‐topos. Second, while the local objects produced by Casacuberta et al. are always 1‐types, our construction can produce ‐types, for any . This is new, even in the ‐topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.

From samples to persistent stratified homotopy types

From samples to persistent stratified homotopy types

Stabilisation, scanning, and handle cancellation

In this note, we describe a family of arguments that link the homotopy type of (a) the diffeomorphism group of the disc D^{n} , (b) the space of co-dimension one embedded spheres in S^{n} , and (c) the homotopy type of the space of co-dimension two trivial knots in S^{n} . We also describe some natural extensions to these arguments. We begin with Cerf’s “upgraded” proof of Smale’s theorem, showing that the diffeomorphism group of S^{2} has the homotopy type of the isometry group. This entails a cancelling-handle construction, related to recently studied “scanning” maps of spaces of embeddings \operatorname{Emb}(D^{n-1}, S^{1}\times D^{n-1}) \to \Omega^{j} \operatorname{Emb}(D^{n-1-j}, S^{1} \times D^{n-1}) . We further give a Bott-style variation on Cerf’s construction and a related embedding calculus framework for these constructions. We use these arguments to prove that the monoid of Schönflies spheres \pi_{0} \operatorname{Emb}(S^{n-1}, S^{n}) is a group with respect to the connected-sum operation for all n \geq 2 . This last result is perhaps only interesting when n=4 , as when n \neq 4 , it follows from the resolution of the various generalised Schönflies problems.

Khovanov homology, wedges of spheres and complexity

Our main result has topological, combinatorial and computational flavor. It is motivated by a fundamental conjecture stating that computing Khovanov homology of a closed braid of fixed number of strands has polynomial time complexity. We show that the independence simplicial complex I(w) associated to the 4-braid diagram w (and therefore its Khovanov spectrum at extreme quantum degree) is contractible or homotopy equivalent to either a sphere, or a wedge of two spheres (possibly of different dimensions), or a wedge of three spheres (at least two of them of the same dimension), or a wedge of four spheres (at least three of them of the same dimension). On the algorithmic side we prove that finding the homotopy type of I(w) can be done in polynomial time with respect to the number of crossings in w. In particular, we prove the wedge of spheres conjecture for circle graphs obtained from 4-braid diagrams. We also introduce the concept of Khovanov adequate diagram and discuss criteria for a link to have a Khovanov adequate braid diagram with at most 4 strands.

Curvature Sets Over Persistence Diagrams

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0$$\\end{document} and n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 1$$\\end{document} we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as Dn,kVR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{n,k}^{\ extrm{VR}}$$\\end{document}. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which D4,1VR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{4,1}^{\ extrm{VR}}$$\\end{document} fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2k+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+2$$\\end{document} with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2

The homotopy decomposition of the suspension of a non-simply-connected five-manifold

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $five$ -manifolds. As applications, we compute the reduced $K$ -groups of $M$ and show that the suspension map between the third cohomotopy set $\pi ^3(M)$ and the fourth cohomotopy set $\pi ^4(\Sigma M)$ is a bijection.

On planarity of graphs in homotopy type theory

AbstractIn this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

How do spaces in physics emerge from pregeometric discrete building blocks governed by computational rules? To address this question we investigate non-deterministic rewriting systems (so-called multiway systems) of the Wolfram model. We express these rewriting systems as homotopy types. Using this new formulation of the model, we motivate how spatial structures are functorially inherited from pregeometric type-theoretic constructions. We show how higher homotopies are constructed from rewriting rules. These correspond to morphisms of an n-fold category. Subsequently, the n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\rightarrow \\infty $$\\end{document} limit of the Wolfram model rulial multiway system is identified as an ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}-groupoid, with the latter being relevant in light of Grothendieck’s homotopy hypothesis. We then go on to show how this construction extends to the classifying space of rulial multiway systems, which forms a multiverse of multiway systems and carries the formal structure of an ∞,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\left( \\infty , 1\\right) }$$\\end{document}-topos. This correspondence to higher categorical structures potentially offers a new way to understand how the kinds of spaces relevant to physics result from pregeometric combinatorial models. The key issue we have addressed in this work is to relate abstract non-deterministic rewriting systems to higher homotopy spaces. A consequence of constructing spaces and geometry synthetically is that it removes the need to make ad hoc assumptions about geometric attributes of a model such as an a priori background or any pre-assigned geometric data. Instead, geometry is inherited functorially from higher structures. This can be particularly useful for formally justifying different choices of underlying spacetime discretization schemes adopted by various models of quantum gravity. We conclude with comments on how our framework of higher category-theoretic combinatorial constructions closely corroborates with other approaches investigating higher categorical structures relevant to the foundations of physics.

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.

Homotopical foundations of parametrized quantum spin systems

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space [Formula: see text] to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space [Formula: see text] at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to [Formula: see text]-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

Homotopy type through homology groups

We show that for certain pairs of nonnegative integers, if the reduced integral homology groups of a space are nonzero only in that pair of degrees and those groups are finitely generated free abelian groups, then the space must have the weak homotopy type of a wedge of spheres of whose dimensions belong to the pair.

Strange new universes: Proof assistants and synthetic foundations

Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages present a barrier to entry for many mathematicians. Large language models have the potential to lower this barrier, enabling mathematicians to interact with proof assistants in a more familiar vernacular. Among other advantages, this may allow mathematicians to explore radically new kinds of mathematics using an LLM-powered proof assistant to train their intuitions as well as ensure their arguments are correct. Existing proof assistants have already played this role for fields such as homotopy type theory.

The compatibility of the minimalist foundation with homotopy type theory

The Minimalist Foundation, MF for short, is a two-level foundation for constructive mathematics ideated by Maietti and Sambin in 2005 and then fully formalized by Maietti in 2009. MF serves as a common core among the most relevant foundations for mathematics in the literature by choosing for each of them the appropriate level of MF to be translated in a compatible way, namely by preserving the meaning of logical and set-theoretical constructors. The two-level structure consists of an intensional level, an extensional one, and an interpretation of the latter in the former in order to extract intensional computational content from mathematical proofs involving extensional constructions used in everyday mathematical practice.In 2013 a completely new foundation for constructive mathematics appeared in the literature, called Homotopy Type Theory, for short HoTT, which is an example of Voevodsky's Univalent Foundations with a computational nature.So far no level of MF has been proved to be compatible with any of the Univalent Foundations in the literature. Here we show that both levels of MF are compatible with HoTT. This result is made possible thanks to the peculiarities of HoTT which combines intensional features of type theory with extensional ones by assuming Voevodsky's Univalence Axiom and higher inductive quotient types. As a relevant consequence, MF inherits entirely new computable models.

On the ∞$\infty$‐topos semantics of homotopy type theory

AbstractMany introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set‐based foundations. This expository article, written as lecture notes to accompany a three‐part mini course delivered at the Logic and Higher Structures workshop at CIRM‐Luminy, attempt to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any ‐topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.

Homotopy equivalent boundaries of cube complexes

A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document} (which depends on the CAT(0) metric), the Roller boundary∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} (which depends only on the combinatorial structure), and the simplicial boundary∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document} (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ∂RX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _R}X$$\\end{document} to define a simplicial Roller boundary R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document}. Then, we show that ∂TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _TX$$\\end{document}, ∂▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\ riangle X$$\\end{document}, and R▵X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {R}}_\ riangle X$$\\end{document} are all homotopy equivalent, Aut(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Aut}(X)$$\\end{document}-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.

Almost complex structures on homotopy complex projective spaces

We show that all homotopy CPns, smooth closed manifolds with the oriented homotopy type of CPn, admit almost complex structures for 3≤n≤6, and classify these structures by their Chern classes for n=4,6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP4s.

Homotopy type of the independence complex of some categorical products of graphs

Homotopy type of the independence complex of some categorical products of graphs