CW Complex Research Articles

A Hasse diagram for rational toral ranks

Let $X$ be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of $X$ obtained by almost free toral actions, ${\mathcal T}_0(X)=\{[Y_i] \}$, induced by an equivalence relation based on rational toral ranks, we order as $[Y_i] 0$. It presents a variation of almost free toral actions on $X$. We consider about the Hasse diagram ${\mathcal H}(X)$ of the poset ${\mathcal T}_0(X)$, which makes a based graph $G{\mathcal H}(X)$, with some examples. Finally we will try to regard $G{\mathcal H}(X)$ as the 1-skeleton of a finite CW complex ${\mathcal T}(X)$ with base point $X_{\Q}$.

CARDINALITY OF RATIONAL HOMOTOPY TYPES OF SIMPLY CONNECTED CW COMPLEXES

The aim of this paper is to consider the following problem: given a simply connected CW complex X of finite type, find the cardinal of the set of rational homotopy types of CW complexes Y such that H*(Y, ℚ) is isomorphic, as a commutative graded algebra, to H*(X, ℚ). For this purpose we introduce a long exact sequence, called the Whitehead exact sequence, which we associated with the minimal Sullivan model of X and which allows us to define the notion of the adapted couple associated with X. Thus we show that the rational homotopy type of X is completely determined by the equivalence class of its adapted couple.

Discrete Morse theory and Hopf bundles

We use Hopf bundles to give an example of a regular CW complex X and an acyclic matching M on the face poset of X, such that there are no critical cells in neighboring dimensions but the complex X is not homotopy equivalent to the corresponding wedge of spheres. The key fact here is that the higher homotopy groups of spheres are nontrivial. We also give a sufficient condition on an acyclic matching M for concluding that X is homotopy equivalent to a wedge of spheres indexed by the critical cells.

A cell complex in number theory

Let Δn be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system.In this paper we study the asymptotic behavior of the individual Betti numbers βk(Δn) and of their sum. We show that Δn has the homotopy type of a wedge of spheres, and that as n→∞∑βk(Δn)=2nπ2+O(nθ),for all θ>1754. Furthermore, for fixed k,βk(Δn)∼n2logn(loglogn)kk!. As a number-theoretic byproduct we obtain inequalities∂k(σk+1odd(n))⩽σkodd(n/2), where σkodd(n) denotes the number of odd squarefree integers⩽n with k prime factors, and ∂k is a certain combinatorial shadow function.We also study a CW complex Δ˜n that extends the previous simplicial complex. In Δ˜nall numbers ⩽n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex Δ˜n is shown to be homotopy equivalent to a wedge of spheres, and as n→∞∑βk(Δ˜n)=n3+O(nθ),for all θ>2227.

ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS

AbstractIn this paper, we study a homotopy invariant cat(X, B, [ω]) on a pair (X, B) of finite CW complexes with respect to the cohomology class of a continuous closed 1-form ω. This is a generalisation of a Lusternik–Schnirelmann-category-type cat(X, [ω]), developed by Farber in [3, 4], studying the topology of a closed 1-form. This paper establishes the connection with the original notion cat(X, [ω]) and obtains analogous results on critical points and homoclinic cycles. We also provide a similar ‘cuplength’ lower bound for cat(X, B, [ω]).

Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

In this paper, we study the growth with respect to dimension of quite general homotopy invariants Q applied to the CW skeleta of spaces. This leads to upper estimates analogous to the classical “dimension divided by connectivity” bound for Lusternik–Schnirelmann category. Our estimates apply, in particular, to the Clapp–Puppe theory of A–category. We use cat1(X) (which is A–category with A the collection of 1–dimensional CW complexes), to reinterpret in homotopy-theoretical terms some recent work of Dranishnikov on the Lusternik–Schnirelmann category of spaces with fundamental groups of finite cohomological dimension. Our main result is the inequality cat(X)≤ dim(Bπ1(X))+ cat1(X), which implies and strengthens the main theorem of Dranishnikov [Algebr. Geom. Topol. 10 (2010) 917–924].

The Lusternik–Schnirelmann category and the fundamental group

We prove that cat LS X ≤ cd ( π 1 ( X ) ) + dim X − 1 2 for every CW–complex X where cd(π1(X)) denotes the cohomological dimension of the fundamental group of X.

Windmills and extreme $2$-cells

In this article we prove new results about the existence of 2-cells in disc diagrams which are extreme in the sense that they are attached to the rest of the diagram along a small connected portion of their boundary cycle. In particular, we establish conditions on a 2-complex X which imply that all minimal area disc diagrams over X with reduced boundary cycles have extreme 2-cells in this sense. The existence of extreme 2-cells in disc diagrams over these complexes leads to new results on coherence using the perimeter-reduction techniques we developed in an earlier article. Recall that a group is called coherent if all of its finitely generated subgroups are finitely presented. We illustrate this approach by showing that several classes of one-relator groups, small cancellation groups and groups with staggered presentations are collections of coherent groups. In this article we prove some new results about the existence of extreme 2-cells in disc diagrams which lead to new results on coherence. In particular, we combine the diagram results shown here with the theorems from [3] to establish the coherence of various classes of one-relator groups, small cancellation groups, and groups with relatively staggered presentations. The article is organized as follows: § 1 contains background definitions, § 2 recalls how extreme 2-cells lead to perimeter reductions and to coherent fundamental groups, § 3 introduces the concept of a windmill, § 4 uses windmills to prove that extreme 2-cells exist, and finally § 5 uses extreme 2-cells to prove that various groups are coherent. For instance, we obtain the following special case of Corollary 5.12: Corollary 0.1. Let G = 〈a1, . . . , ar, t |W 〉 where W has the form t W1t W2 . . . t Wk, N is arbitrary, and for each i, i is a nonzero integer and Wi is a reduced word in the ai. Suppose that {W1,W2, . . . ,Wk} freely generate a subgroup of the free group 〈a1, . . . , ar | −〉. Then G is coherent. 1. Basic Definitions In this section we review some basic definitions about 2-complexes and diagrams. Definition 1.1 (Combinatorial maps and complexes). A map Y → X between CW complexes is combinatorial if its restriction to each open cell of Y is a homeomorphism onto an open cell of X. A CW complex X is combinatorial provided Date: September 7, 2009. 2000 Mathematics Subject Classification. 20F06,20F67,57M07.

CW Complexes for Complex Algebraic Surfaces

We describe CW complexes for complex projective algebraic surfaces in the context of practical computation of topological invariants.

Noncommutative Koszul algebras from combinatorial topology

Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X). Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski.

Homotopy type of mapping spaces and existence of geometric exponents

Let Y be a simply connected finite complex and let p be a prime. Let Sm[p–1] denote the complex obtained from the m-sphere by inverting p. It is shown in this paper that Y has an eventual H-space exponent at p if and only if the space map*(Sm[p–1], Y) of pointed maps Sm[p–1] → Y has the homotopy type of a CW complex for some (and hence all big enough) m. This makes it possible to interpret the question of eventual H-space exponents in terms of phantom phenomena of mapping spaces.

Twisted K-Theory and Finite-Dimensional Approximation

We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators.

Associahedra, cyclohedra and a topological solution to the [formula omitted] Deligne conjecture

We give a topological solution to the A ∞ Deligne conjecture using associahedra and cyclohedra. To this end, we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally act on the Hochschild cochains of an A ∞ algebra yielding an explicit, topological and minimal solution to the A ∞ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow-up construction.

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part ( G / P ) ⩾ 0 of an arbitrary (generalized, partial) flag variety G / P . He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that ( G / P ) ⩾ 0 – the closure of the top-dimensional cell – is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell ( G / P ) > 0 is homotopic to a sphere, is new for all G / P other than projective space.

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.

Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex

Let X be an n-dimensional, finite, simply connected CW complex and set alpha(X) = lim sup(i) (log rank pi(i) (X))/i. We prove that either rank pi(i) (X) = 0, i >= 2n or else that 0 0 there is a K = K(epsilon) such that e((alpha X - epsilon)k) = K. In particular, this sum grows exponentially in k.

On the connections between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms

The Hilton-Hopf quadratic form is defined for spaces of the homotopy type of a CW complex with one cell each in dimensions 0 and 4n, K cells in dimension 2n and no other cells. If two such spaces are of the same topological genus, then their Hilton-Hopf quadratic forms are of the same weak algebraic genus. For large classes of spaces, such as simply connected differentiable 4-manifolds, the converse is also true, as long as the suspensions of the spaces are also of the same topological genus. This note allays the conjecture that the converse is true in general by offering two techniques for generating infinite families of counterexamples.

Dominant K-theory and integrable highest weight representations of Kac–Moody groups

We give a topological interpretation of the highest weight representations of Kac–Moody groups. Given the unitary form G of a Kac–Moody group (over C ), we define a version of equivariant K-theory, K G on the category of proper G -CW complexes. We then study Kac–Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable highest weight representations of a Kac–Moody group G of compact type, maps isomorphically onto K ˜ G ∗ ( E G ) , where E G is the classifying space of proper G -actions. For the affine case, this agrees very well with recent results of Freed–Hopkins–Teleman. We also explicitly compute K G ∗ ( E G ) for Kac–Moody groups of extended compact type, which includes the Kac–Moody group E 10.

A relative version of the finiteness obstruction theory of C. T. C. Wall

In his 1965 paper C. T. C. Wall demonstrated that if a CW complex Y is finitely dominated, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite CW complex. Wall also demonstrated that such an obstruction is invariant under homotopy equivalences. Subsequently Sum and Product Theorems for this obstruction were proved by L. C. Siebenmann. In his second paper on the subject Wall gives an algebraic definition of the relative finiteness obstruction. If a CW complex Y is finitely dominated rel. a subcomplex X, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite complex rel. X. In this paper we will use a geometric construction to reduce the relative finiteness obstruction to the non-relative version. We will demonstrate that the relative finiteness obstruction is invariant under certain types of homotopy equivalences. We will also prove the relative versions of the Sum and the Product Theorems.

Minimal Nielsen Root Classes and Roots of Liftings

Given a continuous map from a 2-dimensional CW complex into a closed surface, the Nielsen root number and the minimal number of roots of satisfy . But, there is a number associated to each Nielsen root class of and an important problem is to know when . In addition to investigate this problem, we determine a relationship between and , when is a lifting of through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.