CW Complex Research Articles

On totally free crossed modules

In [10] we associate to a crossed module (T, G, მ) an invariant abelian crossed module H2(T, G, მ). The construction uses presentations by Set-free crossed modules. Now, Set-free crossed modules are special cases of totally free crossed modules, which are algebraic models of 2-dimensional CW complexes used by several authors (see [1] and [6]). The aim of this paper is to show that H2(T, G, მ) can also be constructed from presentations by arbitrary totally free crossed modules.

Combinatorial vector fields and dynamical systems

In this paper we introduce the notion of a combinatorial dynamical system on any CW complex. Earlier in [Fo3] and [Fo4], we presented the idea of a combinatorial vector field (see also [Fo1] for the one-dimensional case), and studied the corresponding Morse Theory. Equivalently, we studied the homological properties of gradient vector fields (these terms were defined precisely in [Fo3], see also Sect. 2 of this paper). In this paper we broaden our investigation and consider general combinatorial vector fields. We first study the homological properties of such vector fields, generalizing the Morse Inequalities of [Fo3]. We then introduce various zeta functions which keep track of the closed orbits of the corresponding flow, and prove that these zeta functions, initially defined only on a half plane, can be analytically continued to meromorphic functions on the entire complex plane. Lastly, we review the notion of Reidemeister Torsion of a CW complex (introduced in [Re], [Fr]) and show that the torsion is equal to the value at $z=0$ of one of the zeta functions introduced earlier. Much of this paper can be viewed as a combinatorial analogue of the work on smooth dynamical systems presented in [P-P], [Fra], [Fri1, 2] and elsewhere.

Gn,3 ( C) is prime if n is odd

A finite CW complex X is said to be prime if for every Hurewicz fibration F → E → B with E homotopy equivalent to X, and B and F homotopically equivalent to finite CW complexes, either B or F is contractible. We show that the 3-plane complex Grassmannian G n,3 ( C) is prime if n is odd.

Axioms for bundle transfers and traces

In topology it is often profitable to characterize the algebraic invariants of spaces or morphisms by axioms. Of course, the most basic examples along these lines are the Eilenberg-Steenrod axioms for homology and cohomology [EiS], and in many cases there are axiomatic characterizations for more refined algebraic structures that exist in homology and cohomology. For example, axioms for the Steenrod cohomology operations appear in [EpS], and axioms for various sorts of transfers known in the nineteen sixties were described in unpublished work of J. M. Boardman [Bo]. In this paper we shall describe axioms for the bundle transfers considered by the first named author and D. H. Gottlieb in [BG2–3] and from a different perspective by A. Dold, M. Clapp and D. Puppe in [Cl, D1–3, DP]. Previous work of F. Roush [Rou] and L. G. Lewis [L] describes axioms for the special case of finite coverings and bundles whose fibers are compact Lie groups that act smoothly on the fibers; another axiomatic characterization of finite covering transfers appears in [BeS, Sect. 13]. The characterization in this paper is valid for topological fiber bundles for which the fibers and the bases are finite-dimensional compact ANR’s and finite CW complexes respectively. We shall also describe axioms for the homomorphisms in generalized homology and cohomology theories that are determined by bundle transfers.

Approximate finiteness properties of infinite groups

A discrete group Γ is said to be of type F(n) if and only if there is a classifying complex BΓ with finite n-skeleton. We show that Γ is of type F(n) if Γ admits a regular cellular action on a CW complex Y such that Y is homotopy equivalent to a complex with finite ( n−1)-skeleton, Y Γ is a complex with finite n-skeleton, and for each p-cell σ of Y(0⩽ p⩽ n) the stabilizer Γ σ is of type F(n−p) .

Classification ofC*-Algebras of Real Rank Zero and UnsuspendedE-Equivalence Types

In this article, examples are given to prove that the graded scaled orderedK-group is not the complete invariant for aC*-algebra in the class of unital separable nuclearC*-algebras of real rank zero and stable rank one, even for aC*-algebra in the subclass which consists of those real rank zero, stable rank oneC*-algebras being expressed as inductive limits of ⊕kni=1M[n, i](C(Xn, i)), whereXn, iare two-dimensional finite CW complexes and [n, i] are positive integers. (In the case of simple suchC*-algebras, it has been proved that the above invariant is the complete invariant by George Elliott and the author.) These examples prove that the classification conjecture of Elliott for the case of non simple real rank zeroC*-algebras should be revised—one needs extra invariants. The obstruction preventing two suchC*-algebras with the same graded scaled orderedK-group from being isomorphic is that they have different unsuspendedE-equivalence types (a refinement ofKK-equivalence type ofC*-algebras due to Connes and Higson). In this article, it is proved that for the above class of inductive limitC*-algebras, the obstruction of unsuspendedE-equivalence type is the only obstruction (i.e., if twoC*-algebras in the class are unsuspendedE-equivalence, then they are isomorphic). It is a surprise that in the case of simple suchC*-algebras, or even the case ofC*algebras with finitely many ideals, the obstruction will disappear (see Section 4).

Extensions of C ( X ) by Simple C *-algebras of real rank zero

Let Ext ( C ( X ), A ) be the set of unitarily equivalence classes of essential C *-algebra extensions of the following form: 0 → A → E → C ( X ) → 0, where A is a nonunital separable simple C *-algebra of real rank zero, stable rank one with unique normalized trace and X is a finite CW complex. We show that there is a bijection [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: Ext ( C ( X ), A ) → KK ( C ( X ), M ( A )/ A ), where M ( A ) is the multiplier algebra of A . In particular, we determine when an extension is actually splitting. We also, in a more general setting, give a condition when an essential extension is quasidiagonal.

A Note on the Nilpotency of Subgroups of Self-Homotopy Equivalences

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by E(X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup En#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i⩽n, and the subgroup E#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows. 1991 Mathematics Subject Classification 55P62, 55P10.

The fibre of an iterative adjunction of cells

Let ( Y, X) be a relative CW complex with X and Y simply-connected and suppose that the relative homology H ∗(Y, X; k) is nonzero. Denote by F the homotopy fibre of the inclusion X → Y. We show that the grade of H ∗(F;k) as a module over H ∗(ΩY;k) is less than the relative cone length cl( Y, X). This result appears as a corollary of a deeper result concerning differential modules over differential graded algebras.

Alexander stratifications of character varieties

Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group Γ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if Γ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It follows that for Kähler groups the jumping loci must be defined by binomial ideals. As we will show, this is not the case for general finitely presented groups. Thus, the “binomial condition” can be used as a criterion for proving certain finitely presented groups are not Kähler.

Cellular filtration of K-theory and determinants of $C^*$-algebras

In this note, we will disprove the following conjecture raised by Exel–Loring: Let A be a C∗-algebra with trace τ and let det : U∞ → R/τ∗(K0(A)) be a determinant associated to τ . If φt : C(S3)→ A (0 ≤ t ≤ 1) is a continuous family of homomorphisms and b ∈ C(S3)⊗M2 is the canonical matrix valued function on S3 which represents the Bott element in K1(C(S)), then det(φ0(b)) = det(φ1(b)). It should be noticed that the conjecture has been proved by Exel–Loring for the case that φt is a smooth family of homomorphisms. Let bn, or simply b, denote the element of K∗(C(S)) (n ≥ 1) which has the Chern character equal to the fundamental class (i.e., the canonical generator of H(S)). We call b the Bott element. In [ExL], Exel–Loring tried to extend the concept of cohomology of topological spaces (commutative C∗-algebras) to general C∗-algebras and define a filtration of K∗(A) of any C∗-algebra A as the following: Definition 1. Let A be a unital C∗-algebra. The spherical filtration F∗K∗(A) of K∗(A) is defined by F0K0(A) = K0(A) , F1K1(A) = K1(A) , FnKn(A) = {x ∈ Kn(A); x = φ∗(b) for some φ ∈ Hom(C(S),Mk ⊗A)} . Exel-Loring proved that the cohomology Hn(X)⊗Q with rational coefficient of finite CW complex X can be recovered from the filtration of K∗(C(X)) by Hn(X)⊗Q = (FnK∗(C(X))) ⊗Q (Fn+2K∗(C(X))) ⊗Q . Also, they defined Hn(A) = FnK∗(A) Fn+2K∗(A) . It is easy to see that, for any C∗-algebra A, F2K∗(A) ⊂ ⋂ τ∈TA Ker τ∗, Received by the editors January 22, 1996 and, in revised form, March 18, 1996. 1991 Mathematics Subject Classification. Primary 46L80, 46M20, 19K56. c ©1997 American Mathematical Society

Elliptic spaces with the rational homotopy type of spheres

The Milnor-Moore theorem shows that the Q-elliptic spaces are precisely those spaces which have the rational homotopy type of finite, 1-connected CW complexes and have finite total rational homotopy rank. This class of spaces is important because of the dichotomy theorem (the subject of the book [8]) which states that a finite, 1-connected complex either has finite total rational homotopy rank or the rational homotopy groups have exponential growth when regarded as a graded vector space. Moreover, elliptic spaces are the subject of the Moore conjectures, asserting that the homotopy groups of a finite, 1-connected CW complex have finite exponent at all primes if and only if it is Q-elliptic. The p-elliptic spaces form a sub-class of the class of Q-elliptic spaces. A p-elliptic space Z is known to satisfy the following important properties [10, 11].

Extension Theory: The interface between set-theoretic and algebraic topology

Extension Theory can be defined as studying extensions of maps from topological spaces to metric simplicial complexes or CW complexes. One has a natural notion of an absolute (neighborhood) extensor K of X. It is shown that several concepts of set-theoretic topology can be naturally introduced using ideas of Extension Theory. Also, it is shown that several results of set-theoretic topology have a natural interpretation and simple proofs in Extension Theory. Here are sample results. Theorem. Suppose X is a topological space. Then: (a) X is normal iff every finite partition of unity on a closed subset of X extends to a finite partition of unity on X; (b) X is normal iff every countable partition of unity on a closed subset of X extends to a countable partition of unity on X; (c) X is collectionwise normal iff every partition of unity on a closed subset of X extends to a partition of unity on X; (d) if X is paracompact, then every locally finite partition of unity on a closed subset of X extends to a locally finite partition of unity on X; (e) if X is metrizable, then every point-finite partition of unity on a closed subset of X extends to a point-finite partition of unity on X. Theorem. Suppose X is a topological space. Then: (a) finite simplicial complexes are absolute neighborhood extensors of X iff every finite partition of unity on a closed subset of X extends to a partition of unity on X; (b) complete simplicial complexes are absolute neighborhood extensors of X iff every partition of unity on a closed subset of X extends to a partition of unity on X; (c) simplicial complexes are absolute neighborhood extensors of X iff every point-finite partition of unity on a closed subset of X extends to a point-finite partition of unity on X; (d) CW complexes are absolute neighborhood extensors of a first countable X iff every locally finite partition of unity on a closed subset of X extends to a locally finite partition of unity on X. Theorem. A complete simplicial complex K is an absolute neighborhood extensor of X iff its 0-skeleton K 0 is an absolute neighborhood extensor of X. Theorem. Suppose X is a topological space and A is a subset of X. Then: (a) A is C*-embedded in X iff every finite partition of unity on A extends to a finite partition of unity on X; (b) A is C-embedded in X iff every countable partition of unity on A extends to a countable partition of unity on X; (c) A is P-embedded in X iff every partition of unity on A extends to a partition of unity on X; (d) A is M-embedded in X iff every partition of unity α on A extends to a partition of unity β on X so that β(B) = α(A) for some zero-set B of X which contains A.

Category, growth and grade in a fibration

Let F → E → B be a fibration with B a simply connected finite type CW complex. In this paper we consider the relations between the category of E and the growth and the grade of the homology of F as a module over the loop space homology of B.

Indecomposability ofQX

LetX be a finite CW complex localized or complete at a prime greater than 3. Then, under certain technical conditions, it is shown that ifX is stably atomic, thenQX is atomic.

Addendum to “Linear topological classification of certain function spaces on manifolds and CW complexes”, Topology and its Applications 69 (1996) 265–282

We point out that a result of the above paper [3] can be applied to generalize a theorem due to Gul'ko and Khmyleva [2] as follows. As before, C p ( X) denotes the space of all continuous real valued functions on a Tychonoff space X with the pointwise convergent topology.

Actions of symmetry groups

This paper studies maps which are invariant under the action of the symmetry group S k . The problem originates in social choice theory: there are k individuals each with a space of preferences X, and a social choice map ϕ:X k → X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map ψ:X k → X exists which is invariant under the action of S k only if, for all i≥1, the elements of the homotopy group Π i (X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group S k . 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.

Substitutional lemma for G-spaces of 1-dimensional groups

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of Xwhere EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extensionwhere G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order

Linear topological classification of certain function spaces on manifolds and CW complexes

For a Tychonoff space X, C p ( X) denotes the space of all continuous real-valued functions on X with the pointwise convergent topology. Two Tychonoff spaces X and Y are said to be ℓ-equivalent if C p ( X) and C p ( Y) are linearly homeomorphic. We prove that any compact topological n-manifold is ℓ-equivalent to the n-disk. This complements a result of V. Valov on noncompact manifolds. Further the classification of CW complexes up to ℓ-equivalence, originated by D. Pavlovskiĭ, is completed by classifying noncompact CW complexes up to the above equivalence.

Attaching cells to finite complexes, with an application to elliptic spaces

Suppose that f; Sn → E is a continuous map from the n-sphere to a 1-connected CW complex E, with n ≥ 2. One may suppose that f is a cofibration, so that there is a cofibration sequence , with f the attaching map of the cell en+1. Consider the homotopy fibre F of the inclusion E ↪ B, so that there is a homotopy fibration let δ; ΩB → F be the connectant of this fibration. The following definition is given by Félix and Lemaire in [11]: Definition 1·1. Suppose that k is a field of characteristic p ≥ 0. The attaching map f:Sn → E is: 1. p-inert if is surjective; 2. p-lazy if is zero; where H˜ denotes reduced homology and coefficients are taken in the field k.