Finite CW-complex Research Articles

𝐾𝑈-local zeta-functions of finite CW-complexes

Begin with the Hasse-Weil zeta-function of a smooth projective variety over Q \mathbb {Q} . Replace the variety with a finite CW-complex, replace étale cohomology with complex K K -theory K U ∗ KU^* , and replace the p p -Frobenius operator with the p p th Adams operation on K K -theory. This simple idea yields a kind of “ K U KU -local zeta-function” of a finite CW-complex. For a wide range of finite CW-complexes X X with torsion-free K K -theory, we show that this zeta-function admits analytic continuation to a meromorphic function on the complex plane, with a nice functional equation, and whose special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 . We then consider a more general and sophisticated version of the K U KU -local zeta-function, one which is suited to finite CW-complexes X X with nontrivial torsion in their K K -theory. This more sophisticated K U KU -local zeta-function involves a product of L L -functions of complex representations of the torsion subgroup of K U 0 ( X ) KU^0(X) , similar to how the Dedekind zeta-function of a number field factors as a product of Artin L L -functions of complex representations of the Galois group. For a wide range of such finite CW-complexes X X , we prove analytic continuation of the zeta-function, and we show that the special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 if and only if the skeletal filtration on the torsion subgroup of K U 0 ( X ) KU^0(X) splits completely.

Algorithmic reconstruction of the fiber of persistent homology on cell complexes

Let K be a finite simplicial, cubical, delta or CW complex. The persistence map PH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}$$\\end{document} takes a filter f:K→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:K\\rightarrow \\mathbb {R}$$\\end{document} as input and returns the barcodes of the sublevel set persistent homology of f in each dimension. We address the inverse problem: given target barcodes D, computing the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. For this, we use the fact that PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document} decomposes as a polyhedral complex when K is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth-first search that recovers the polytopes forming the fiber PH-1(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PH}^{-1}(D)$$\\end{document}. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.

ON TOPOLOGICAL COMPLEXITY OF GORENSTEIN SPACES

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q}, C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then compare these invariants and their respective ordinary ones for Gorenstein spaces.
 We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

Computing cohomology groups that classify bundles of strongly self-absorbing C∗-algebras

Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory.

Fixed point sets and orbit spaces of wedge of three spheres

Let X be a finite CW-complex having mod p cohomology isomorphic to a wedge of three spheres Sn∨Sm∨Sl,1≤n≤m≤l. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on X. We also classify the orbit spaces of free actions of G=Zp,p a prime or G=Sd,d=1 or 3, on X and derive the Borsuk-Ulam type results.

Tomiyama’s K-commutative diagrams of minimal dynamical systems

Let [Formula: see text] be the Cantor space and [Formula: see text] be an even-dimensional sphere. By applying a result of the existence of minimal skew products, we show that, associated with any Cantor minimal system [Formula: see text], there is a class [Formula: see text] of minimal skew products on [Formula: see text], such that for any two rigid homeomorphisms [Formula: see text] and [Formula: see text], the notions of approximate [Formula: see text]-conjugacy and [Formula: see text]-strongly approximate conjugacy coincide, which are also equivalent to a [Formula: see text]-version of Tomiyama’s commutative diagram. In fact, this is also the case if [Formula: see text] is replaced by any (infinite) connected finite CW-complex with torsion free [Formula: see text]-group, vanished [Formula: see text]-group and the so-called Lipschitz-minimal-property.

Posets for which Verdier duality holds

We discuss two known sheaf-cosheaf duality theorems: Curry’s for the face posets of finite regular CW complexes and Lurie’s for compact Hausdorff spaces, i.e., covariant Verdier duality. We provide a uniform formulation for them and prove their generalizations. Our version of the former works over the sphere spectrum and for more general finite posets, which we characterize in terms of the Gorenstein* condition. Our version of the latter says that the stabilization of a proper separated infty -topos is rigid in the sense of Gaitsgory. As an application, for stratified topological spaces, we clarify the relation between these two duality equivalences.

Fixed point sets and the fundamental group I: semi-free actions on G-CW-complexes

Smith theory says that the fixed point set of a semi-free action of a group $G$ on a contractible space is ${\mathbb {Z}}_p$ -acyclic for any prime factor $p$ of the order of $G$ . Jones proved the converse of Smith theory for the case $G$ is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain $K$ -theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of $K$ -theoretical obstructions.

Polyhedra for which every homotopy domination over itself is a homotopy equivalence

In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? Since every ANR has the homotopy type of a polyhedron, this question is related in an obvious way to the longstanding Borsuk's problem (1967) Is it true that two ANR's homotopy dominating each other have the same homotopy type? The answer is positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearly for 1-dimensional polyhedra. In some previous paper, we proved that this also holds for all 2-dimensional polyhedra with weakly Hopfian (hence Hopfian and co-Hopfian) fundamental groups (a group is weakly Hopfian if it is not isomorphic to a proper retract of itself). In this paper we extend this result to all (G,n)-complexes (where n≥2). A (G,n)-complex, is an n-dimensional finite CW-complex with fundamental group G and all the homotopy groups πr in dimensions 1<r<n trivial.We also discuss fundamental groups of potential counterexamples.As a corollary we obtain that for the same classes of spaces, the Borsuk's question: Is it true that the homotopy types of two quasi-homeomorphic ANR's are equal?, has positive answer (this question was also asked by S. Ferry in the recent edition of Scottish Book (1981)).

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n\\ge 6$$\\end{document} with finite fundamental group, we show that the classifying space BDiff(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\ extrm{Diff}(M)$$\\end{document} of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\subset M$$\\end{document} of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

Harmonic representatives in homology over arbitrary fields

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets.

Elementary abelian group actions on a product of spaces of cohomology type (a, b)

Let Xn be a finite CW complex with cohomology type (a, b), characterized by an integer n &gt; 1 [20]. In this paper, we show that if G = (Z2)q acts freely on the product Y = Qmi=1 Xin, where Xin are finite CW complexes with cohomology type (a, b), a and b are even for every i, then q ? m. Moreover, for n even and a = b = 0, we prove that G = (Z2)q (q ? m) is the only finite group which can act freely on Y. These are generalizations of the results which says that the rank of a group acting freely on a space with cohomology type (a, b) where a and b are even, is one and for n even, G = Z2 is the only finite group which acts freely on spaces of cohomology type (0, 0) [17].

Free Cyclic Actions on Surfaces and the Borsuk—Ulam Theorem

Let M and N be topological spaces, let G be a group, and let τ: G × M → M be a proper free action of G. In this paper, we define a Borsuk—Ulam-type property for homotopy classes of maps from M to N with respect to the pair (G, τ) that generalises the classical antipodal Borsuk—Ulam theorem of maps from the n-sphere $${\mathbb{S}^n}$$ to ℝn. In the cases where M is a finite pathwise-connected CW-complex, G is a finite, non-trivial Abelian group, τ is a proper free cellular action, and N is either ℝ2 or a compact surface without boundary different from $${\mathbb{S}^2}$$ and ℝℙ2, we give an algebraic criterion involving braid groups to decide whether a free homotopy class β ∈ [M, N] has the Borsuk—Ulam property. As an application of this criterion, we consider the case where M is a compact surface without boundary equipped with a free action τ of the finite cyclic group ℤn. In terms of the orientability of the orbit space Mτ of M by the action τ, the value of n modulo 4 and a certain algebraic condition involving the first homology group of Mτ, we are able to determine if the single homotopy class of maps from M to ℝ2 possesses the Borsuk—Ulam property with respect to (ℤn, τ). Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk—Ulam property for maps whose target is ℝ2.

Small non-Leighton two-complexes

AbstractHow many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.

On the Homotopy Type of Intersections of Two Real Bruhat Cells

Abstract Real Bruhat cells give an important and well-studied stratification of such spaces as $\operatorname {GL}_{n+1}$, $\operatorname {Flag}_{n+1} = \operatorname {SL}_{n+1}/B$, $\operatorname {SO}_{n+1}$, and $\operatorname {Spin}_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group $\operatorname {Lo}_{n+1}^{1}$. We are particularly interested in the homotopy type of such intersections. In this paper, we define a stratification of such intersections. As a consequence, we obtain a finite CW complex that is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for $n \le 4$ all connected components of such subsets of $\operatorname {Lo}_{n+1}^1$ are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for $n \ge 5$ and the top permutation, there is always a connected component with even Euler characteristic and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation $S_6$, there exists a connected component with Euler characteristic equal to $2$. We also give an example of a permutation in $S_6$ for which there exists a connected component that is homotopically equivalent to the circle ${{\mathbb {S}}}^1$.

Fluctuations of cycles in a finite CW complex

We use algebraic topology to study the stochastic motion of cellular cycles in a finite CW complex. Inspired by statistical mechanics, we introduce a homological observable called the average current. The latter measures the average flux of the probability in the process. In the low temperature, adiabatic limit, we prove that the average current fractionally quantizes, in which the denominators are combinatorial invariants of the CW complex.

On the homotopy of closed manifolds and finite CW-complexes

We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when X X is a finite CW-complex of dimension n n and π 1 ( X ) \pi _1(X) is virtually a Poincaré duality group of dimension ≥ n − 1 \geq n-1 , then π i ( X ) \pi _i(X) is not finitely generated for some i i unless X X is homotopy equivalent to the Eilenberg–MacLane space K ( π 1 ( X ) , 1 ) K(\pi _1(X),1) ; when M M is an n n -dimensional closed manifold and π 1 ( M ) \pi _1(M) is virtually a Poincaré duality group of dimension ≥ n − 1 \ge n-1 , then for some i ≤ [ n / 2 ] i\leq [n/2] , π i ( M ) \pi _i(M) is not finitely generated, unless M M itself is an aspherical manifold. These generalize theorems of M. Damian [Trans. Amer. Math. Soc. 361 (2009), pp. 1791–1809] from polycyclic groups to any virtually Poincaré duality groups.

On K0 of locally finite categories

We calculate the Grothendieck group K0(A), where A is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category SW of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category A and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group K0(SW).

Principal bundles on two-dimensional CW-complexes with disconnected structure group

AbstractGiven any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

The homotopy type of elliptic arrangements

We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.