Hurewicz Theorem Research Articles

The Hurewicz theorem in homotopy type theory

The Hurewicz theorem in homotopy type theory

The Hurewicz theorem in Sh◦* and Sh◦*²

The Hurewicz theorem in Sh◦* and Sh◦*²

ON MORPHISMS KILLING WEIGHTS AND STABLE HUREWICZ-TYPE THEOREMS

AbstractFor a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case $w=w^{sph}$ (the spherical weight structure on $SH$ ), we deduce the following converse to the stable Hurewicz theorem: $H^{sing}_{i}(M)=\{0\}$ for all $i<0$ if and only if $M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$ , this is the case if there exists a distinguished triangle $LM\to M\to RM$ , where $RM$ is an n-connected spectrum and $LM$ is an $m-1$ -skeleton (of M) in the sense of Margolis’s definition; this happens whenever $H^{sing}_i(M)=\{0\}$ for $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$ ; those ‘send n-w-skeleta into $m-1$ -w-skeleta’.

Higher stabilization and higher Freudenthal suspension

We prove that the stabilization (resp. iterated suspension) functor participates in a derived adjunction comparing pointed spaces with certain (highly homotopy coherent) homotopy coalgebras, in the sense of Blumberg-Riehl, that is a Dwyer-Kan equivalence after restriction to 1-connected spaces, with respect to the associated enrichments. A key ingredient of our proof, of independent interest, is a higher stabilization theorem (resp. higher Freudenthal suspension theorem) for pointed spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to iterating the space level stabilization map (resp. Freudenthal suspension map)—these technical results provide, in particular, new proofs (with strong estimates) of the stabilization and iterated loop-suspension completion results of Carlsson and the subsequent work of Arone-Kankaanrinta, and Bousfield and Hopkins, respectively, for 1-connected spaces; this is the stabilization (resp. Freudenthal suspension) analog of Dundas’ higher Hurewicz theorem.

Digital Hurewicz theorem and digital homology theory

In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we present digital Hurewicz theorem for the fundamental group of digital images. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg-Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory.

Conceptual proofs of the Menger and Rothberger games

We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game.

On the complexity of Borel equivalence relations with some countability property

We study the class of Borel equivalence relations under continuous reducibility. In particular , we characterize when a Borel equivalence relation with countable equivalence classes is $\Sigma$ 0 $\xi$ (or $\Pi$ 0 $\xi$). We characterize when all the equivalence classes of such a relation are $\Sigma$ 0 $\xi$ (or $\Pi$ 0 $\xi$). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G $\delta$ .

Genus Integration, Abelianization, and Extended Monodromy

Abstract Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in [9]. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of [6], but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.

Functorial reducing pro⁎-Grp category to pro-Grp

We introduce a new functor R˜ from pro⁎-Grp to pro-Grp and use it to give a new characterisation of coarse shape groups. It is shown that lim←⁡∘R˜∘pro⁎-πk=πˇk⁎, thus R˜∘pro⁎-πk is a full analogue of pro-πk. Furthermore, the Hurewicz theorem in a new context is proposed. We complete with a corollary stating isomorphicity of the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum, assertion that does not hold for shape groups.

Homology groups of types in stable theories and the Hurewicz correspondence

We give an explicit description of the homology group Hn(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups Hi(q) are trivial for 2≤i<n. The group Hn(p) turns out to be isomorphic to the automorphism group of a certain part of the algebraic closure of n independent realizations of p; it follows from the authors' earlier work that such a group must be abelian. We call this the “Hurewicz correspondence” by analogy with the Hurewicz Theorem in algebraic topology.

Comparing Euler Classes

We establish the equality of two definitions of a Euler class in algebraic geometry: the first definition is as a ‘characteristic class’ with values in Chow–Witt theory, while the second definition is as an ‘obstruction class.’ Along the way, we refine Morel's relative Hurewicz theorem in A1-homotopy theory, and show how to define (twisted) Chow–Witt groups for geometric classifying spaces.

Directed Homology Theories and Eilenberg-Steenrod Axioms

In this paper, we define and study a homology theory, that we call “natural homology”, which associates a natural system of abelian groups to every space in a large class of directed spaces and precubical sets. We show that this homology theory enjoys many important properties, as an invariant for directed homotopy. Among its properties, we show that subdivided precubical sets have the same homology type as the original ones ; similarly, the natural homology of a precubical set is of the same type as the natural homology of its geometric realization. By same type we mean equivalent up to some form of bisimulation, that we define using the notion of open map. Last but not least, natural homology, for the class of spaces we consider, exhibits very important properties such as Hurewicz theorems, and most of Eilenberg-Steenrod axioms, in particular the dimension, homotopy, additivity and exactness axioms. This last axiom is studied in a general framework of (generalized) exact sequences.

Nonabelian algebraic topology

This is a survey of central results in nonabelian algebraic topology. We present how the homotopy category of homotopy \(n\)-types and a certain localization of the category of crossed \(n\)-cubes of groups are equivalent. The functor inducing this equivalence satisfy a generalized Seifert-van Kampen theorem, in that it preserves connectivity and colimits of certain diagrams of generalized fibrations. We show descriptions of certain colimits of crossed \(n\)-cubes of groups and show how they have been used to generalize the Blakers-Massey theorem, the Hurewicz theorem and Hopf’s formula for the homology of groups, as well as a combinatorial formula for the homotopy groups of the sphere \(\mathbb {S}^2\). We also study the wedge sum of Eilenberg-MacLane spaces.

Discrete homology theory for metric spaces

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a `fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

Homology Theory of Graphs

Higher homotopy of graphs has been defined in several articles. In Dochterman (Hom complexes and homotopy theory in the category of graphs. arXiv math/0605275 v2,28/09/2006, 2006), the authors asked for a companion homology theory. We define such a theory for the category of unoriented reflexive graphs; it exhibits a long exact sequence for a pair of graphs (G, A), satisfies an excision property and a Hurewicz theorem. This allows us to compute the top homology of the graphical n-spheres showing that the theory is not trivial and is able to detect n-dimensional holes in a graph. The long-term objective is to compare the homotopy of the topological and graphical spheres.

Dimension of the product and classical formulae of dimension theory

Let $f : X \lo Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that $\dim X \leq \dim Y +\dim f$ where $\dim f =\sup \{\dim f^{-1}(y): y \in Y \}$. The first author conjectured that {\em $\dim Y + \dim f$ in Hurewicz's theorem can be replaced by $\sup \{\dim (Y \times f^{-1}(y)): y \in Y \}$}. We disprove this conjecture. As a by-product of the machinery presented in the paper we answer in negative the following problem posed by the first author: {\em Can for compact $X$ the Menger-Urysohn formula $\dim X \leq \dim A + \dim B +1$ be improved to $\dim X \leq \dim (A \times B) +1$ ?} On a positive side we show that both conjectures holds true for compacta $X$ satisfying the equality $dim(X\times X)=2\dim X$.

Homotopy completion and topological Quillen homology of structured ring spectra

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes.

Weak partition properties on trees

We investigate the following weak Ramsey property of a cardinal ?: If ? is coloring of nodes of the tree ? \mathfrak{d}}$$ is regular, then $${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$$ and that $${\mathfrak{b}}$$ $${(\mathfrak{b})^{ < \omega}_{\omega}}$$ and $${\mathfrak{d}}$$ $${(\mathfrak{d})^{ < \omega}_{\omega}}$$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is ?-locally compact iff it does not contain a closed homeomorphic copy of irrationals.

NET BUNDLES OVER POSETS AND K-THEORY

We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fiber bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions of classical Hurewicz theorems. Focusing our attention on Hilbert net bundles, we define the K-theory of a poset and introduce functions over the homotopy groupoid satisfying the same formal properties as Chern classes. As when the given poset is a base for the topology of a space, our results apply to the category of locally constant bundles.

Complexity of Ramsey null sets

We show that the set of codes for Ramsey positive analytic sets is Σ21-complete. This is an analogue of a theorem of Hurewicz saying that the set of uncountable compact subsets of an uncountable Polish space is Σ11-complete. As a corollary, we get that the σ-ideal of Ramsey null sets is not ZFC-correct, which answers a question of Ikegami, Pawlikowski and Zapletal.