CW-complex Of Dimension Research Articles

The homology groups $H_{n+1} \left( \mathbb{C}\Omega_n \right)$

The topic of the paper is the investigation of the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and are the analogue of the spaces $\Omega_n$, widely known in the approximation theory. The spaces $\mathbb{C}\Omega_n$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $\mathbb{C}\Omega_n$ and using this CW-structure established that the spaces $\mathbb{C}\Omega_n$ are simply connected. Note that the mentioned CW-structure of the spaces $\mathbb{C}\Omega_n$ is the analogue of the CW-structure of the spaces $\Omega_n$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $\mathbb{C}\Omega_n$ in the dimensionalities $0, 1, \ldots, n, 2n-1, 2n, 2n+1$. The goal of the present paper is to find the homology group $H_{n+1}\left ( \mathbb{C}\Omega_n \right )$. It is proved that $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+1}{2}$ if $n$ is odd and $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+2}{2}$ if $n$ is even.

BFT2: a general class of 2d mathcal{N} = (0, 2) theories, 3-manifolds and toric geometry

We introduce and initiate the study of a general class of 2d mathcal{N} = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of theories as BFT2’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their master and moduli spaces. Invariance of the moduli space is indeed a powerful criterion for identifying theories in the same triality class. We also investigate the reducibility of these theories.

On the homology groups $H_k(\mathbb{C}\Omega_n)$, $k=1, ..., n$

In the paper the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$ are investigated. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and generalize the widely known in the approximation theory spaces $\Omega_n$. The research of the homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban who in 1985 found the n-dimensional homology group of the space $\Omega_n$ and in 1999 found all the cohomology groups of this space. The spaces $\mathbb{C}\Omega_n$ have been introduced by A.M. Pasko who in 2015 has built the structure of CW-complex on these spaces. This CW-structure is analogue of the CW-structure of the space $\Omega_n$ introduced by V.I. Ruban. In present paper in order to investigate the homology groups of the spaces $\mathbb{C}\Omega_n$ we calculate the relative homology groups $H_k(\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1})$, it turned out that the groups $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )$ are trivial if $1\leq k < n$ and $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}^{C^{k-n}_{n+1}}$ if $n \leq k \leq 2n+1$, in particular $H_n \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}$. Further we consider the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ and prove that its inclusion operator $i_*: H_k(\mathbb{C}\Omega_n) \rightarrow H_k(\mathbb{C}\Omega_{n+1})$ is zero. Taking into account that the relative homology groups $H_k \left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ are zero if $1\leq k \leq n$ and the inclusion operator $i_*=0$ we have derived from the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ that the homology groups $H_k \left ( \mathbb{C}\Omega_n \right ), 1\leq k<n,$ are trivial. The similar considerations made it possible to calculate the group $H_n(\mathbb{C}\Omega_n)$. So the homology groups $H_k(\mathbb{C}\Omega_n), n \geq 2, k=1,...,n,$ have been found.

Stabilized convex symplectic manifolds are Weinstein

We show that a stabilized convex symplectic (also called Liouville) manifold with the homotopy type of a half dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.

A revised augmented Cuntz semigroup

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

Linear systems over [formula omitted] and roots of maps of some 3-complexes into [formula omitted

Let Z[Q32] be the group ring where Q32=〈x,y|x8=y2,xyx=y〉 is the quaternion group of order 32 and ε the augmentation map. We show that, if PX=K(x−1) and PX=K(−xy+1) has solution over Z[Q32] and all m×m minors of ε(P) are relatively prime, then the linear system PX=K has a solution over Z[Q32]. As a consequence of such results, we show that there is no map f:W→MQ32 that is strongly surjective, i.e., such that MR[f,a]=min⁡{#(g−1(a))|g∈[f]}≠0. Here, MQ32 is the orbit space of the 3-sphere S3 with respect to the action of Q32 determined by the inclusion Q32⊆S3 and W is a CW-complex of dimension 3 with H3(W;Z)=0.

The topological period-index problem over $8$-complexes, II

We complete the study of the topological period-index problem over 8 dimensional finite CW complexes started in a preceding paper. More precisely, we determine the sharp upper bound of the index of a topological Brauer class $\alpha\in H^3(X;\mathbb{Z})$, where $X$ is of the homotopy type of an 8 dimensional finite CW complex and the period of $\alpha$ is divisible by 4.

A topological equivalence relation for finitely presented groups

In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups G and H are “proper 2-equivalent” if there exist (equivalently, for all) finite 2-dimensional CW-complexes X and Y, with π1(X)≅G and π1(Y)≅H, so that their universal covers X˜ and Y˜ are proper 2-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are 1-ended and semistable at infinity are classified, up to proper 2-equivalence, by their fundamental pro-group, and we study the behavior of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type Fn,n≥3, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups G which admit a finite 2-dimensional CW-complex X with π1(X)≅G and whose universal cover X˜ has the proper homotopy type of a 3-manifold. We show that if such a group G is 1-ended and semistable at infinity then it is proper 2-equivalent to either Z×Z×Z, Z×Z or F2×Z (here, F2 is the free group on two generators). As it turns out, this applies in particular to any group G fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper 2-equivalence.

Chern rank of complex bundle

We introduce notions of {\it upper chernrank} and {\it even cup length} of a finite connected CW-complex and prove that {\it upper chernrank} is a homotopy invariant. It turns out that determination of {\it upper chernrank} of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$-dimensional complex manifold we obtain an upper bound of its even cup length. For a finite connected even dimensional CW-complex with its {\it upper chernrank} equal to its dimension, we provide a method of computing its even cup length. Finally, we compute {\it upper chernrank} of many interesting spaces.

A note on Gorenstein spaces

Associated with an augmented differential graded algebra R=R≥0 is a homotopy invariant T(R). This is a graded vector space, and if H0(R) is the ground field and H>N(R)=0 then dimT(R)=1 if and only if H(R) is a Poincaré duality algebra. In the case of Sullivan extensions ∧W→∧W⊗∧Z→∧Z in which dimH(∧Z)<∞ we show thatT(∧W⊗∧Z)=T(∧W)⊗T(∧Z). This is applied to finite dimensional CW complexes X where the fundamental group G acts nilpotently in the cohomology H(X˜;Q) of the universal covering space. If H(X;Q) is a Poincaré duality algebra and H(X˜;Q) and H(BG;Q) are finite dimensional then they are also Poincaré duality algebras.

The capacity of some classes of polyhedra

K. Borsuk in 1979, in Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of the product of two spheres of the same or different dimensions and the capacity of lense spaces. Also, we present an upper bound for the capacity of a $\mathbb{Z}_n$-complex, i.e., a connected finite 2-dimensional CW-complex with finite cyclic fundamental group $\mathbb{Z}_n$.

Regularity of Villadsen algebras and characters on their central sequence algebras

We show that if $A$ is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of $A$ does not admit characters.Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast $-algebra with a unique tracial state which fails to have this property.

Free and Properly Discontinuous Actions of Groups on Homotopy 2n-spheres

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

Partial Euler characteristic, normal generations and the stable $D(2)$ problem

We obtain relations among normal generation of perfect groups, Swan’s inequality involving partial Euler characteristic, and deficiency of finite groups. The proof is based on the study of a stable version of Wall’s D(2) problem. Moreover, we prove that a finite 3-dimensional CW complex of cohomological dimension at most 2 with fundamental group G is homotopy equivalent to a 2-dimensional CW complex after wedging n copies of the 2-sphere S 2 , where n depends only on G.

Additive posets, CW-complexes, and graphs

We introduce and study additive posets. We show that the top homology group (with coefficients in Z∕2Z) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.

On the group of self-homotopy equivalences of (n + 1)-connected and (3n + 2)-dimensional CW-complex

Let X be a CW complex, E(X) the group of homotopy classes of homotopy self-equivalences of X and Ψ:E(X)→aut(H⁎(X,Z)) the map sending [α] to H⁎(α). This paper deals with the following question: Characterize f⁎∈aut(H⁎(X,Z)) such that f⁎∈ImΨ. For the R-localized XR of an (n+1)-connected and (3n+2)-dimensional CW-complex X; n≥2, where R is a certain subring of Q we define the notion of strong automorphism in aut(H⁎(X,Z)), in term of the Whitehead exact sequence of the Anick model of XR and we show that f⁎∈ImΦ if and only if f⁎ is a strong automorphism. Consequently we prove that E(XR)(E⁎(X))R≅B(XR), where E⁎(X) is the subgroup of the elements that induce the identity on homology and B(XR) is the subgroup of the strong automorphisms.

Genera and minors of multibranched surfaces

We say that a 2-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to the 2-dimensional Euclidean space R2, then we obtain a 1-dimensional complex which is homeomorphic to a disjoint union of some S1's. We define the genus of a multibranched surface X as the minimum number of genera of 3-dimensional manifold into which X can be embedded. We prove some inequalities which give upper bounds for the genus of a multibranched surface. A multibranched surface is a generalization of graphs. Therefore, we can define “minors” of multibranched surfaces analogously. We study various properties of the minors of multibranched surfaces.

Geometric dimension of lattices in classical simple Lie groups

We prove that if $\Gamma$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $\Gamma$-equivariantly homotopy equivalent to a proper cocompact $\Gamma$-CW complex of dimension the virtual cohomological dimension of $\Gamma$.

A Note on Group Extensions and Proper 3-Realizability

The interaction between the study of three-dimensional manifolds and a particular stream of group theory has often been fruitful. In the realm of this, we recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with \({\pi_1(X) \cong G}\), the universal cover of X has the proper homotopy type of a 3-manifold. In this paper, we generalize a previous result on the direct products of groups; more precisely, we show that if \({N \rightarrow G \rightarrow Q}\) is a short exact sequence of infinite finitely presented groups, then G is properly 3-realizable. In particular, any semidirect product of two infinite finitely presented groups is properly 3-realizable. As an application, we show proper 3-realizability for certain classes of groups.

An obstruction to embedding 2-dimensional complexes into the 3-sphere

We consider an embedding of a 2-dimensional CW complex into the 3-sphere, and construct its dual graph. Then we obtain a homogeneous system of linear equations from the 2-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equations does not have an integral solution, we show that some 2-dimensional CW complexes cannot be embedded into the 3-sphere.