CW Complex Research Articles

Categorical sequences

We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skeleta. The k-th term in the categorical sequence of a CW complex X, \sigma_X(k), is the least integer n for which cat_X(X_n) >= k. We show that \sigma_X is a well-defined homotopy invariant of X. We prove that \sigma_X(k+l) >= \sigma_X(k) + \sigma_X(l), which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)).

An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H ∗ ( Ω X ; Q ) is the universal enveloping algebra of a graded Lie algebra L X isomorphic with π ∗ − 1 ( X ) ⊗ Q . Let Q X ⊂ L X be a minimal generating subspace, and set α = lim sup i log rk π i ( X ) i . Theorem: If dim L X = ∞ and lim sup ( dim ( Q X ) k ) 1 / k < lim sup dim ( L X ) k 1 / k , then ∑ i = 1 n − 1 rk π k + i ( X ) = e ( α + ε k ) k , where ε k → 0 as k → ∞ . In particular ∑ i = 1 n − 1 rk π k + i ( X ) grows exponentially in k.

The ranks of the homotopy groups of a space of finite LS category

Let X be a simply connected CW complex with finitely many cells in each degree. The first part of the paper is a report on the different conjectures for the behavior of the sequence rk π i ( X ) . In the second part, we give conditions on the Lusternik–Schnirelmann category of X and on the depth of the Lie algebra π * ( Ω X ) ⊗ Q that imply the exponential growth in k of the sequence ∑ i = 1 d rk π k + i ( X ) , for some d.

A rational splitting of a based mapping space

Let F_*(X, Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form X cup_{alpha}e^{k+1} and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen-Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map alpha: S^k --> X is greater than the Whitehead length WL(Y) of Y, then F_*(X cup_{alpha}e^{k+1}, Y) has the rational homotopy type of the product space F_*(X, Y) times Omega^{k+1}Y. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dim X, then the mapping space F_*(X, Y) can be decomposed rationally as the product of iterated loop spaces.

A Rational Model for the Evaluation Map

Abstract Let 𝑋 be an 𝑙-connected space and 𝑈 a connected CW complex with dim 𝑈 ≤ 𝑙. Let 𝐹(𝑈,𝑋) be the space of continuous maps from 𝑈 to 𝑋. In this paper, an algebraic model for the evaluation map 𝐹(𝑈,𝑋) × 𝑈 → 𝑋 is considered in terms of the model for the function space due to Brown and Szczarba [Trans. Amer. Math. Soc. 349: 4931-4951, 1997]. It turns out that the Brown and Szczarba model for the function space coincides with Haefliger's model.

Universal absolute extensors in extension theory

Let $L$ be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension $\le [L]$ contains a universal element which is an absolute extensor in dimension $[L]$. Our main result shows that $L$ is quasi-finite.

On C. T. C. Wall's suspension theorem

Almost forty years ago, C.T.C. Wall systematically analyzed the set of of a finite CW complex. Of the results he obtained, probably the most computationally important is the suspension theorem, which is an exact sequence relating the n-dimensional thickenings of a finite complex to its (n+1)-dimensional ones. The object of this note is to fill in what we believe is a missing argument in the proof of that theorem.

Phantom Maps from a Strong Homology Viewpoint

Let X and Y be connected nilpotent CW–complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.

Discrete Morse theory and graph braid groups

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

On optimizing discrete Morse functions

In 1998, Forman introduced discrete Morse theory as a tool for studying CW complexes by producing smaller, simpler-to-understand complexes of critical cells with the same homotopy types as the original complexes. This paper addresses two questions: (1) under what conditions may several gradient paths in a discrete Morse function simultaneously be reversed to cancel several pairs of critical cells, to further collapse the complex, and (2) which gradient paths are individually reversible in lexicographic discrete Morse functions on poset order complexes. The latter follows from a correspondence between gradient paths and lexicographically first reduced expressions for permutations. As an application, a new partial order on the symmetric group recently introduced by Remmel is proven to be Cohen–Macaulay.

Algebra of dimension theory

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes dim(A) of graded groups A. There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes K and L, dim(H * (K)) = dim(H * (L)) if and only if the infinite symmetric products SP(K) and SP(L) are of the same extension type (i.e., SP(K) ∈ AE(X) iff SP(L) ∈ AE(X) for all compact X). 2) For pointed compact spaces X and Y, dim(H - *(X)) = dim(Η - *(Y)) if and only if X and Y are of the same dimension type (i.e., dim G (X) = dim G (Y) for all Abelian groups G). Dranishnikov's version of the Hurewicz Theorem in extension theory becomes dim(π*(K)) = dim(H * (K)) for all simply connected K. The concept of cohomological dimension dim A (X) of a pointed compact space X with respect to a graded group A is introduced. It turns out dim A (X) ≤ 0 iff dim A(n) (X) ≤ n for all n ∈ Z. If A and B are two positive graded groups, then dim(A) = dim(B) if and only if dim A (X) = dim B (X) for all compact X.

Subalgebras of group cohomology defined by infinite loop spaces

We study natural subalgebras Ch E ( BG ; R ) of group cohomology H * ( BG ; R ) defined in terms of the infinite loop spaces in spectra E and give representation theoretic descriptions of those based on QS 0 and the Johnson–Wilson theories E ( n ) . We describe the subalgebras arising from the Brown–Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP * ( BG ) in H * ( BG ; F p ) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a non-trivial map of ring spectra E → H F p . We also extend our constructions to define subalgebras of H * ( X ; R ) for any space X; when X is a finite CW complex we show that the subalgebras Ch E ( n ) ( X ; R ) give a natural unstable chromatic filtration of H * ( X ; R ) .

Decomposition of $p$ -complete $H$ -spaces and related problems

We work on based CW complexes of finite types. In this paper, we show that each -completion of the wedge of atomic spaces. We then discuss some related problems on atomic spaces.

Finiteness of Total Cofibres

The suspension of the total cofibre (Y ) of a diagram Y of infinite CW complexes, indexed by the face lattice of a polytope, is in general not homotopy equivalent to a finite CW complex. However, if Y satisfies additional conditions (making it into a locally finite non-linear sheaf on a projective toric variety) we can show that (Y ) is a homotopy finite space. The main tools used in the proof are a Bousfield–Kan spectral sequence for total cofibres, and a modification of the Cech construction to calculate sheaf cohomology groups. Mathematics Subject Classifications (2000): Primary 55P99, Secondary 55N30.

Isomorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In (Dydak and Ruiz del Portal (Monomorphisms and epimorphisms in pro-categories, preprint)) we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro- C, where C has direct sums (resp. weak push-outs). In this paper, we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K. Borsuk regarding a descending chain of retracts of ANRs. If f : X→Y is a bimorphism in the pointed shape category of topological spaces, we prove that f is a weak isomorphism and f is an isomorphism provided Y is sequentially movable and X or Y is the suspension of a topological space. If f : X→Y is a bimorphism in the pro-category pro- H 0 (consisting of inverse systems in H 0, the homotopy category of pointed connected CW complexes) we show that f is an isomorphism provided Y is sequentially movable.

Minimal atomic complexes

We define minimal atomic complexes and irreducible complexes, and we prove that they are the same. The irreducible complexes admit homological characterizations that make them easy to recognize. These concepts apply both to spaces and to spectra. On the spectrum level, our characterizations allow us to show that such familiar spectra as ko, eo 2, and BoP at the prime 2, all BP〈 n〉 at any prime p, and the indecomposable wedge summands of Σ ∞ C P ∞ and Σ ∞ H P ∞ at any prime p are irreducible and therefore minimal atomic. Up to equivalence, the minimal atomic complexes admit descriptions as CW complexes with restricted attaching maps, called nuclear complexes, and this description can be refined further to nuclear minimal complexes, which are nuclear and have zero differential on their mod p chains. As an illustrative example, we construct BoP as a nuclear complex.

Extension dimension for paracompact spaces

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: Theorem. Suppose X is a paracompact space. There is a CW complex K such that (a) K is an absolute extensor of X up to homotopy, (b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. The proof is based on the following simple result (see Theorem 1.2). Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family { Y s } s∈ S of its subspaces with the following properties : (a) Each Y s is an absolute extensor of X, (b) For any two elements s and t of S there is u∈ S such that Y s ∪ Y t ⊂ Y u . If f :A→Y is a map from a closed subset A to Y such that A=⋃ s∈ S Int A ( f −1( Y s )), then f extends over X. That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.

On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs

Using techniques from Robin Forman's discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex Δ n 3 of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of ( n−3)·( n−2)!/2 spheres of dimension 2 n−4, thereby verifying a conjecture by Babson, Björner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex Γ n of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex Δ n 2 of not 2-connected graphs on n vertices. The homotopy type of Δ n 2 has been determined, independently, by the five authors listed above and by Turchin. While Γ n and Δ n 2 are homotopy equivalent for small values on n, they are nonequivalent for n=10.

On the topological social choice model

The main goal of this paper is to show that if a finite connected CW complex admits a continuous, symmetric, and unanimous choice function for some number n>1 of agents, then the choice space is contractible. On the other hand, if one removes the finiteness, we give a complete characterization of the possible spaces; in particular, noncontractible spaces are indeed possible. These results extend earlier well-known results of Chichilnisky and Heal.

Torsion primes in loop space homology

We construct a finite 1-connected CW complex X such that H ∗(ΩX; Z) has p-torsion for the infinitely many primes satisfying p≡5,7,17,19 mod 24 , but no p-torsion for the infinitely many primes satisfying p≡13 or 23 mod 24 .