Finite CW-complex Research Articles

Homotopy transfer and rational models for mapping spaces

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_\infty$-coalgebra structure in $H_*(X;\mathbb{Q})\cong H_*(C)$ induced by the higher Massey coproducts provides the construction of the Quillen minimal model of $X$. We also describe an explicit $L_\infty$-structure on the complex of linear maps ${\rm Hom}(H_*(X; \mathbb{Q}), \pi_*(\Omega Y)\otimes\mathbb{Q})$, where $X$ is a finite nilpotent CW-complex and $Y$ is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping space ${\rm map}(X, Y)$. As an application we give conditions on the source and target in order to detect rational $H$-space structures on the components.

Homological finiteness of abelian covers

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H^1(X,\Q). Inside this parameter space, there is a subset \Omega_A^i(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer-Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts.

The homotopy fixed point set of Lie group actions on elliptic spaces

Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set $X^{hG}$, and the natural injection $k\colon X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi_*(k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

Infinite CW-complexes, Brauer groups and phantom cohomology

Expanding a result of Serre on finite CW-complexes, we show that the Brauer group coincides with the cohomological Brauer group for arbitrary compact spaces. Using results from the homotopy theory of classifying spaces for Lie groups, we give another proof of the result of Antieau and Williams that equality does not hold for Eilenberg-MacLane spaces of type K(ℤ/nℤ, 2). Employing a result of Dwyer and Zabrodsky, we show the same for the classifying spaces BG where G is an infinite-dimensional F p -vector space. In this context, we also give a formula expressing phantom cohomology in terms of homology.

Homotopy pullback of [formula omitted]-spaces and its applications to [formula omitted]-types of gauge groups

We construct the homotopy pullback of An-spaces and show some universal property of it. As the first application, we review Zabrodsky's result which states that for each prime p, there is a finite CW complex which admits an Ap−1-form but no Ap-form. As the second application, we investigate An-types of gauge groups. In particular, we give a new result on An-types of the gauge groups of principal SU(2)-bundles over S4, which is a complete classification when they are localized away from 2.

Kirchhoff’s theorems in higher dimensions and Reidemeister torsion

Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff's network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an application, we give a formula expressing Reidemeister torsion as an enumeration of higher dimensional spanning trees.

Cuts and flows of cell complexes

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe, and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.

Brown representability for exterior cohomology and cohomology with compact supports

AbstractIt is well known that cohomology with compact supports is not ahomotopy invariant but only a proper homotopy one. However, as theproper category lacks of general categorical properties, a Brown repre-sentability theorem type does not seem reachable. However, by provingsuch a theorem for the so called exterior cohomology in the completeand cocomplete exterior category, we show that the n-th cohomologywith compact supports of a given countable, locally finite, finite dimen-sional relative CW-complex (X,R + ) is naturally identified with the set[X,K n ] R + ofexteriorbasedhomotopyclassesfrom a“classifyingspace”K n . We also show that this space has the exterior homotopy type ofthe exterior Eilenberg-MacLane space for Brown-Grossman homotopygroups of type (R ∞ ,n), Rbeing the fixed coefficient ring. Introduction Properhomotopy theoryis designed to studynon compact topological spacesmodulo deformations which respect the behaviour of these spaces at infinity.In this paper we are concerned with the classification of cohomology invari-ants of proper homotopy. Among them, classical cohomology with compactsupports and locally finite cohomology [20, §3] are specially well adaptedfunctors to study locally compact spaces up to proper homotopy.

The period-index problem for twisted topologicalK–theory

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d, we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K-theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/l,2), where l is a prime, we construct a sequence of spaces with an order l class in Br, but whose indices tend to infinity.

English

We develop an unsuspended version of connective $E$-theory and prove connective versions of results by D\u{a}d\u{a}rlat--Loring and Shulman. As a corollary, we see that two separable $C^*$-algebras of the form $C_0(X) \otimes A$, where $X$ is a based, connected, finite CW-complex and $A$ is a unital, properly infinite algebra, are connective $E$-theory equivalent if and only if they are asymptotic matrix homotopy equivalent.

Novikov homology, jump loci and Massey products

AbstractLet X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple.If α ∈ H 1(X, ℝ) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H 1(X, ℝ).

Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes

We study global attractors $\mathcal{A}_f$ of scalar partial differential equations $u_t=u_{xx}+f(x,u,u_x)$ on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call $\mathcal{A}_f$ a Sturm global attractor. We assume all equilibria $v$ to be hyperbolic. Due to a gradient-like structure we can then write \begin{equation} \mathcal{A}_f = \bigcup\limits_{v}\, W^u(v) (*) \end{equation} as a dynamic decomposition into finitely many disjoint invariant sets: the unstable manifolds $W^u(v)$ of the equilibria $v$. Based on our previous Schoenflies result [17], we prove that the dynamic decomposition $(*)$ is in fact a regular finite CW-complex with cells $W^u(v)$, in the Sturm case. We call this complex the regular dynamic complex or Sturm complex of the Sturm attractor $\mathcal{A}_f$. We characterize the planar Sturm complexes by bipolar orientations of their 1-skeletons. We also show that any regular finite CW-complex which is the closure of a single 3-cell arises as a Sturm complex. We include a preliminary discussion of the tetrahedron and the octahedron as Sturm complexes.

Regularity in the growth of the loop space homology of a finite CW complex

To any path connected topological space X X , such that rk ⁡ H i ( X ) > ∞ \operatorname {rk} H_i(X) >\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .

The topological period-index problem over 6-complexes

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah–Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period–index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period–index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

Geometric K-homology with coefficients II: The Analytic Theory and Isomorphism

AbstractWe discuss the analytic aspects of the geometric model for K-homology with coefficients in ℤ/kℤ constructed in [12]. In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of spinc manifolds, geometric K-homology, and Atiyah-Singer index theorem.

Regular cell complexes in total positivity

Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over \({\mathbb{R}}\) is a regular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) intervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology.

Cohomology in 3D Magneto-Quasistatics Modeling

AbstractElectromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for defining potentials. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problemswhich involve conductive regionswith holes, cuts are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature.The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.

Alexander varieties and largeness of finitely presented groups

Let \(X\) be a finite CW complex. We say that \(X\) is large if \(X\) has a finite cover \(Y\) whose fundamental group surjects to a non-abelian free group. We show that the fundamental group of \(X\) is large if and only if there is a finite cover \(Y\) of \(X\) and a sequence of finite abelian covers \(\{Y_N\}\) of \(Y\) which satisfy \(b_1(Y_N)\ge N\). We give some applications of this result to the study of hyperbolic \(3\)-manifolds, mapping classes of surfaces and combinatorial group theory.

Action-induced nested classes of groups and jump (co)homology

Abstract Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes of groups as well as new characterizations for some well-known classes. We extend some of the properties of the cohomological dimension of a group to groups with jump (co)homology and study their implications to cellular actions on finite dimensional CW-complexes that satisfy a quite general homological condition.

Polyhedral representation of discrete Morse functions

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on posets.