CW Complex Research Articles

Cohomological dimension and metrizable spaces. II

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A$, $B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{\mathbf {R} }(A\cup B)\le \dim _{\mathbf {R} }A+\dim _{\mathbf {R} }B+1 \end{equation*} for any ring $\mathbf {R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then [a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. [b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. [c.] There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z’$ with $K_{i}\in AE(Z’)$ for all $i\ge 1$ embeds in $Z$ as a closed subset.

Dehydrogenation Polymer−Cell Wall Complexes as a Model for Lignified Grass Walls

p-Hydroxycinnamyl alcohols were efficiently polymerized into maize walls by wall-bound peroxidases and in vitro generated hydrogen peroxide to produce dehydrogenation polymer−cell wall (DHP−CW) complexes. Electron microscopy of KMnO4-stained sections revealed that DHPs were distributed throughout the cell wall matrix. DHP−CW complexes were structurally similar to natural grass lignins according to pyrolysis, thioacidolysis, and 13C-NMR spectroscopy. Nonlignified walls were rapidly degraded by rumen microorganisms and by commercial fungal hydrolases, whereas DHP−CW complexes had a reduced rate and extent of degradation. This system is useful for modeling matrix interactions in lignified walls and for identifying means of improving the utilization of lignocellulosic materials for nutritional and industrial purposes. Keywords: Gramineae; Zea mays; cell wall; fiber; peroxidase; p-hydroxycinnamyl alcohols; lignin; dehydrogenation polymer; transmission electron microscopy; pyrolysis; thioacidolysis; NMR; fiber ...

On the dimension of infinite covers

We prove the following theorem and some generalizations. Theorem. Let X X be a connected CW complex which satisfies Poincaré duality of dimension n ≥ 4 n\ge 4 . For any subgroup H H of π 1 ( X ) \pi _1(X) , let X H X_H denote the cover of X X corresponding to H H . If H H has infinite index in π 1 ( X ) \pi _1(X) , then X H X_H is homotopy equivalent to an ( n − 1 ) (n-1) -dimensional CW complex.

Finite CW complexes with maximal torsion gaps

We investigate some properties of finite CW complexes with maximal homotopy torsion gaps and prove a revision of the Halperin conjecture under an additional condition.

Action of Symmetry Groups

This paper studies maps which are invariant under the action of the symmetry group S,. The problem originates in social choice theory: there arc k individuals each with a space of preferences X, and a social choice map phi:Xk - X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map Psi: Xk -> X exists which is invariant under the action of S, only if, for all i>1, the elements of the homotopy group IL(X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.

Realizing dimension functions via homology

The following theorem is the main result of the paper: Theorem. Let G be an Abelian group and m > 0. Let G be a countable family of countable Abelian groups and let D : G → Z + be a function. The following conditions are equivalent: 1. (1) For any CW complex P and any a ϵ H m ( P; G) − {0} there is a compactum X and a map π : X → P such that dim X = m, dim H X ⩽ D( H) for each H ϵ G and a ϵ im( H ̌ m(X ; G) → H ̌ m(P;G)) . 2. (2) H ̃ k(K(H,D(H));G) = 0 for all k < m and all H ϵ G . As an application, we prove the existence of compacta realizing dimension functions, a result due to A.N. Dranishnikov.

Normal Elements of C ∗-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C∗-algebras. One example, which has real-rank zero, is the middle term of an exact sequence 0 → ℐ → 𝒜 → π ℬ → 0 , where ℬ is a Bunce-Deddens algebra and ℐ is AF. Using Berg's technique, we produce a normal element N ∈ 𝒜 that is not the limit of finite-spectrum normals. Moreover, this is an example of a normal element in an inductive limit that is not the limit of normal elements of the approximating subalgebras. A second example is an embedding of C(ⅅ) (ⅅ the closed disk) into 𝒜 ⊗ 𝒥, where 𝒜 is a simple AF algebra and 𝒥 is the Toeplitz algebra. Let ⅅn, for n ⩾ 2, be the CW complex obtained as the quotient of ⅅ by an n-fold identification of the boundary. (So ⅅ2 = RP2.) Regarding C(ⅅn) as a subalgebra of C(ⅅ), we find nontrivial embeddings of C(ⅅn) into type I inductive limits. From this, we produce a ∗-homomorphism, for n odd, C0(ⅅn\{pt}) → 𝒪n + 1, that induces an isomorphism on K-theory. More generally, for X a connected CW complex minus a point, and for n odd, we show that the map [ C 0 ( X ) , 𝒪 n + 1 ] → K K ( C 0 ( X ) , 𝒪 n + 1 ) is a split surjection.

Universal separable metrizable spaces of given cohomological dimension

For every countable CW complex K, we construct a universal separable metrizable space X with K ϵ AE( X). From this theorem we derive several corollaries devoted to the question of existence of universal spaces in cohomological dimension theory.

K-Homology, Asymptotic Representations, and Unsuspended E-Theory

Connes and Higson defined a bivariant homology theory E( A, B) for separable C*-algebras. The elements of E( A, B) are taken to be homotopy classes of asymptotic momorphisms from SA ⊗ K to SB ⊗ K . In symbols E( A, B) ≅ [[ SA, SB ⊗ K ]]. If A is K-nuclear then E-theory agrees with Kasparov′s bivariant K-theory. We show that, in many cases, one need not take suspensions to calculate the E-theory group E( A, B). For many A, we show E(A, B) ≅ [[A, B ⊗ K]] for all B. Among the A for which this is true are C 0( X\\{ pt}) for X with the homotopy type of a finite, connected CW complex. This gives a concrete realization of K-homology, related to the Brown-Douglas-Fillmore description. For example, K 0( X) arises as asymptotic representations, K 0(X) ≅ [[C 0(X), K]]. Other A for which our isomorphic holds include the nonunital dimension-drop intervals. In this case, there is no distinction between ∗-homomorphisms and asymptotic morphisms so we have succeeded in classifying all ∗-homomorphisms from a dimension-drop interval to a stable C*-algebra. This was subsequently used by Elliott ( J. Reine Angew. Math. 443 (1993), 179-219) in the classification of certain C*-algebras. The dimension-drop interval may also be used to describe K *( X; Z/ n) in terms of paths of asymptotic representations of C 0( X).

A cell structure for the set of autoregressive systems

The set of autoregressive systems generalizes the set of transfer functionsin a natural way. We describe a topology for the set of all autoregressive systems of fixed size and bounded McMillan degree. We show that this topological space has the structure of a finite CW complex.

Algebraic K-theory with continuous control at infinity

Let ( E ̄ , Σ ) be a pair of spaces consisting of a compact Hausdorff space Ē and a closed subspace Σ. Let U be an additive category. This paper introduces the category B( E ̄ , Σ; U of geometric modules over E with coefficients in U and with continuous control at infinity. One of the main results is to show that the functor that sends a CW complex X to the algebraic K-theory of B(cX, X; U) is a homology theory. Here cX is the closed cone on X and X is its base. The categories B( E ̄ , Σ; U) are generalizations of the categories C(Z; U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here ( Z, ϱ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C( O(X); U)→ B(cX, X; U) . A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B( E ̄ , Σ; U) depends only on the topology of ( E ̄ , Σ ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

A relative generalized Lefschetz number

The generalized Lefschetz number of a selfmap on a finite CW complex is a trace-like quantity that captures both the Lefschetz number and the Nielsen number of the map. In this paper, we will define a relative generalized Lefschetz number for a selfmap f:( X, A) →( X, A) of a finite CW pair, in terms of the generalized Lefschetz numbers of the maps f on X and f A = f∣ A on A. From this number one can extract the relative Nielsen number, thereby allowing us to compute it as a trace.

Smooth extensions for finite CW complexes

In this paper, we have completely classified the C n {C_n} -smooth elements of Ext ⁡ ( X ) \operatorname {Ext} (X) modulo torsion for X being an arbitrary finite CW complex.

Postnikov sections of formal and hyperformal spaces

Let X be a simply connected CW complex and X n {X_n} its nth Postnikov section. We prove that X is formal provided H q ( X n , Q ) {H^q}({X_n},\mathbb {Q}) is additively generated by decomposables for all q and n with q &gt; n q &gt; n . Recall from [4] that a space X is said to be hyperformal if its rational cohomology algebra is the quotient of a free graded algebra by an ideal generated by a regular sequence. Using the main result of Felix and Halperin’s paper (Trans. Amer. Math. Soc. 270 (1982), 575-588) we show our sufficient condition for formality is actually equivalent to hyperformality.

Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups

For simply connected CW complexes X with finitely many, finitely generated homotopy groups, 1 ^{1} the path components of the function space M ( X , X ) M(X,X) of free self-maps of X are all of the same rational homotopy type if and only if all the k-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space M 0 ( X , X ) {M_0}(X,X) of inessential self-maps of X has the structure of rational H-space if and only if the k-invariants of X are of finite order.

Mild and tame homotopy theory

Extending recent results on R-local homotopy theory, we demonstrate that ‘mild’ r-reduced Hopf algebras up to homotopy over Z can be modeled by ‘mild’ r-reduced differential graded Lie algebras (DGL's) over Z . This equivalence can be applied to the Adams–Hilton model of a 1-connected CW complex, yielding a mild DGL model. We then show that this model induces an equivalence of the tame homotopy categories of spaces and DGL's, generalizing slightly the equivalence proved by Dwyer. The advantage of the new proof lies in the computability of the DGL model.

The boundedly controlled Whitehead theorem

This note contains a version of the Whitehead Theorem for boundedly controlled maps of CW complexes that is often useful in applications and complements the Whitehead Theorem in our book Boundedly controlled topology (Lecture Notes in Math., vol. 1323, Springer-Verlag, 1988). We also include a version of the Whitehead Theorem valid for simply connected boundedly controlled CW complexes.

The bounded and thin Whitehead theorems

This paper deals with finite-dimensional CW complexes equipped with reference maps to a fixed metric space and maps between such complexes that respect the reference maps up to a bounded distortion. We prove two Whitehead Theorems for such maps f f . The Bounded Whitehead Theorem allows one to decide whether f f is a bounded homotopy equivalence. The Thin Whitehead Theorem allows one to decide when a map of bound zero admits homotopy inverses of arbitrarily small bound (also on the homotopies). Both theorems come in two versions: One that deals with homotopy in all dimensions; one where homotopy in dimensions at least two is replaced by homology of "universal covers".

On the homology of Postnikov fibres

Let k k be a field of positive characteristic and X X be a simply connected space of the homotopy type of a finite type CW complex. The Postnikov fibre X [ n ] {X_{[n]}} of X X is defined as the homotopy fibre of the n n -equivalence f n : X → X n {f_n}:X \to {X_n} coming from the Postnikov tower { X n } \{ {X_n}\} of X X . We prove that if the Lusternik-Schnirelmann category of X X is finite, then H ∗ ( X [ n ] ; k ) {H_{\ast }}({X_{[n]}};k) contains a free module on a subalgebra K K of H ∗ ( Ω X n ; k ) {H_{\ast }}(\Omega {X_n};k) such that H ∗ ( Ω X n ; k ) {H_{\ast }}(\Omega {X_n};k) is a finite-dimensional free K K -module.

Generalized Group Presentation and Formal Deformations of CW Complexes

A Peiffer-Whitehead word system $\mathcal {W}$, or generalized group presentation, consists of generators, relators (words of order $2$), and words of higher order $n$ that represent elements of a free crossed module $(n = 3)$ or a free module $(n > 3)$. The ${P_n}$-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected ${\text {CW}}$ complex $K$ by removing a maximal tree and selecting one generator or word per cell, via relative homotopy. Given homotopy readings ${\mathcal {W}_1}$ and ${\mathcal {W}_2}$ of finite ${\text {CW}}$ complexes ${K_1}$ and ${K_2}$ respectively, we show that ${\mathcal {W}_1}$ is ${P_n}$-equivalent to ${\mathcal {W}_2}$ if and only if ${K_1}$ formally $(n + 1)$-deforms to ${K_2}$. This extends results of P. Wright (1975) and W. Metzler (1982) for the case $n = 2$. For $n \geq 3$, it follows that ${\mathcal {W}_1}$ is ${P_n}$-equivalent to ${\mathcal {W}_2}$ if and only if ${K_1}$ and ${K_2}$ have the same simple homotopy type.