The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $Xτ_h K\implies β(X)τK$ for all paracompact spaces $X$. Here are the main results of the paper:
Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite.
Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite.
Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$.
Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces.
Quasi-finite CW complexes lead naturally to the concept of $Xτ{\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.
We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.
In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are established. We calculate the set of all map degrees between certain two $(n-2)$-connected $(2n-1)$-dimensional torsion free Poincaré complexes. For low $n$, using knowledge of possible degrees of self maps, we classify, up to homotopy, torsion free $(n-2)$-connected $(2n-1)$-dimen…
The puzzle method was introduced by Choi and Park as an effective method for finding non-singular characteristic maps over wedged simplicial complexes $K(J)$ obtained from a given simplicial complex $K$. We study further the mod 2 case of the puzzle method. We firstly describe it completely in terms of linear algebraic language which allows us to develop a constructive puzzle algorithm. We also analyze our algorithm and compare its performances with other known algorithms including the Garrison and Scott algorithm.
We extend the crossing-change maps between grid complexes, defined by Ozsváth-Szabó-Stipsicz, to filtered grid complexes and give a combinatorial formulation of the Alishahi-Eftekhary $\mathfrak{l}(K)$ knot invariant.
This article introduces descriptive fixed sets and their properties in descriptive proximity spaces viewed in the context of planar ribbon complexes. These fixed sets are a byproduct of descriptive proximally continuous maps that spawn fixed subsets, eventual fixed subsets and almost fixed subsets of the maps. For descriptive continuous map $f$ on a descriptive proximity space $X$, a subset $A$ of $X$ is fixed, provided the description of $f(A)$ matches the desription of $A$. In terms ribbon complexes in a CW space, an Abelian group representation of a ribbon is Day-amenable and each amenable …
Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $λ: \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $λ$ is induced by a homeomorphism of $N$.
Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $λ:\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(α) = λ(α)$ for every vertex $α$ in $\mathcal{T}(N)$ where $H=[h]$.
We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $(g, n) \in \{(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\}$ or $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary components.
Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.
It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.
Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $λ: \mathcal{C}(S) \to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $α$ and $β$ are two vertices in $\mathcal{C}(S)$ and $i(α, β) \neq 0$, then $i(λ(α), λ(β)) \neq 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mod_S^*$ and $f: K \to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism …
Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $λ$ be a simplicial map of the complex of curves, $\mathcal{C}(N)$, on $N$ which satisfies the following: $[a]$ and $[b]$ are connected by an edge in $\mathcal{C}(N)$ if and only if $λ([a])$ and $λ([b])$ are connected by an edge in $\mathcal{C}(N)$ for every pair of vertices $[a], [b]$ in $\mathcal{C}(N)$. We prove that $λ$ is induced by a homeomorphism of $N$ if $(g, n) \in \{(1, 0), (1, 1), (2, 0)$, $(2, 1), (3, 0)\}$ or $g + n \geq 5$. Our result implies that superinjective simplicial maps …
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms betwe…
A countable CW complex $K$ is quasi-finite (as defined by A.Karasev) if for every finite subcomplex $M$ of $K$ there is a finite subcomplex $e(M)$ such that any map $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $XτK$, has an extension $g:X\to e(M)$. Levin's results imply that none of the Eilenberg-MacLane spaces $K(G,2)$ is quasi-finite if $G\ne 0$. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many nonzero Postnikov invariants. Here are the main results of the paper: Suppose $K$ i…
Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This…
In this paper we prove that, in the category of chain complexes, partial algebras can be functorially replaced by quasi-isomorphic algebras. In particular, partial algebras contain all of the important homological and homotopical information that genuine algebras do. Applying this result to McClure's partial algebra in [6] shows that the chains of a PL-manifold are quasi-isomorphic to an E-infinity algebra.
We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove t…
The Kakimizu complex is usually defined in the context of knots, where it is known to be quasi-Euclidean. We here generalize the definition of the Kakimizu complex to surfaces and 3-manifolds (with or without boundary). Interestingly, in the setting of surfaces, the complexes and the techniques turn out to replicate those used to study the Torelli group, {\it i.e.,} the "nonlinear" subgroup of the mapping class group. Our main results are that the Kakimizu complexes of a surface are contractible and that they need not be quasi-Euclidean. It follows that there exist (product) $3$-manifolds whos…
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial collapse of their associated simplicial complexes. Moreover, a simplicial collapse induces a collapse of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finit…
We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. Inspired by result…
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…
