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We investigate the mod $p$ Buchstaber invariant of the skeleta of simplices, for a prime number $p$, and compare them for different values of $p$. For $p=2$, the invariant is the real Buchstaber invariant. Our findings reveal that these values are generally distinct. Additionally, we determine or estimate the mod $p$ Buchstaber invariants of certain universal simplicial complexes $X(\F_p^n)$.

In [1], the author considered the problem of the optimal approximation of symmetric surfaces by biquadratic B\'ezier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal approximation of spherical squares. A detailed analysis and a numerical algorithm are given, providing the best approximant according to the (simplified) radial error, which differs from the one obtained in [1]. The sphere is then approximated by the continuous spline of two and six tensor product quadratic B\'ezier patches. It is further shown that the $G^1$ smooth spline of six patches approximating the sphere exists, but it is not a good approximation. The problem of an approximation of spherical rectangles is also addressed and numerical examples indicate that several optimal approximants might exist in some cases, making the problem extremely difficult to handle. Finally, numerical examples are provided that confirm theoretical results.

We study combinatorial and topological properties of the universal complexes $X(\mathbb{F}_p^n)$ and $K(\mathbb{F}_p^n)$ whose simplices are certain unimodular subsets of $\mathbb{F}_p^n$. We calculate their $\mathbf f$-vectors and their Tor-algebras, show that they are shellable but not shifted, and find their applications in toric topology and number theory. We showed that the Lusternick-Schnirelmann category of the moment angle complex of $X(\mathbb{F}_p^n)$ is $n$, provided $p$ is an odd prime, and the Lusternick-Schnirelmann category of the moment angle complex of $K(\mathbb{F}_p^n)$ is $[\frac n 2]$. Based on the universal complexes, we introduce the Buchstaber invariant $s_p$ for a prime number $p$.

In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $\sigma$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.

A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by parametric polynomial patches if the measure of quality is the (simplified) radial error. As a consequence, optimal approximations of the unit sphere by parametric polynomial spline patches underlying on particular regular spherical triangulations arising from a tetrahedron, an octahedron and an icosahedron inscribed in the unit sphere are provided. Some low total degree spline patches with corresponding geometric smoothness are analyzed in detail and several numerical examples are shown confirming the quality of approximants.

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic $G^0$, cubic $G^1$, quartic $G^2$ and quintic $G^3$ interpolation. Several numerical examples are presented which confirm theoretical results.

The aim of this paper is a construction of quartic parametric polynomial interpolants of a circular arc, where two boundary points of a circular arc are interpolated. For every unit circular arc of inner angle not greater than $\pi$ we find the best interpolant, where the optimality is measured by the simplified radial error.

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.

Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)$ are $(n-2)$-connected simplicial complexes. We discuss applications of these universal simplicial complexes to toric topology and number theory., Comment: In the previous preprint, there were gaps in the proofs that $K(\mathbb{Z}^n)$ and $X(\mathbb{Z}^n)$ and their links of its simplices have homotopy type of a wedge of countable infinite number of spheres $S^{n-1}$. The fact was pointed to the authors by unanimous referee who read the previous version carefully. The result is proved using direct approach instead of using discrete Morse functions

We study the Lie ring $\mathfrak{nil}_n$ of all strictly upper-triangular $n\!\times\!n$ matrices with entries in $\mathbb{Z}$. Its complete homology for $n\!\leq\!8$ is computed. We prove that every $p^m$-torsion appears in $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ for $p^m\!\leq\!n\!-\!2$. For $m\!=\!1$, Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ when prime $p\!>\!n\!-\!2$. In general, for $m\!>\!1$ the bound is not sharp, as we show that there is $8$-torsion in $H_\ast(\mathfrak{nil}_8;\mathbb{Z})$. As a sideproduct, we derive the known result, that the ranks of the free part of $H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ are the Mahonian numbers (=number of permutations of $[n]$ with $k$ inversions), using a different approach than Kostant. Furthermore, we determine the algebra structure (cup products) of $H^\ast(\mathfrak{nil}_n;\mathbb{Q})$., Comment: 10 pages, 1 table

In this paper we find sufficient and necessary conditions under which vector lattice $C(X)$ and its sublattices $C_b(X)$, $C_0(X)$ and $C_c(X)$ have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space $X$. We also consider the countable sup property of $C(X\times Y)$. Even when both $C(X)$ and $C(Y)$ have the countable sup property it is possible that $C(X\times Y)$ fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when $C(\prod_{\lambda\in\Lambda} X_\lambda)$ has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest.

In this paper we consider the relationship between order and topology in the vector lattice $C_b(X)$ of all bounded continuous functions on a Hausdorff space $X$. We prove that the restriction of $f\in C_b(X)$ to a closed set $A$ induces an order continuous operator iff $A=\overline{\mathrm{Int} A}.$ This result enables us to easily characterize bands and projection bands in $C_0(X)$ and $C_b(X)$ through the one-point compactification and the Stone-\v{C}ech compactification of $X$, respectively. With these characterizations we describe order complete $C_0(X)$ and $C_b(X)$-spaces in terms of extremally disconnected spaces. Our results serve us to solve an open question on lifting un-convergence in the case of $C_0(X)$ and $C_b(X)$.

In 'Hochschild (co)homology of exterior algebras' (Han, Xu, 2007), the authors computed the additive and multiplicative structure of $H\!H^\ast\!(A;\!A)$, where $A$ is the $n$-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT), as well as calculate the additive structure of $H\!H_k\!(A;\!A)$ and $H\!H^k\!(A;\!A)$ over $\mathbb{Z}$. We provide concise presentations of algebras $H\!H_\ast\!(A;\!A)$ and $H\!H^\ast\!(A;\!A)$, as well as determine their generators in the Hochschild complex. Lastly, we compute an explicit free resolution (spanned by multisets) of the $A^e$-module $A$ and describe the homotopy equivalence to its bar resolution., Comment: 7 pages

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces., Comment: 9 pages

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also contains $p'$-torsion for every prime $p'\!<\!p$. \par We obtain new explicit formulas for the (co)homology of some families over arbitrary fields. Among them are the solvable non-nilpotent analogs of the Heisenberg Lie algebras from the Cairns & Jambor article, the 2-step Lie algebras from Armstrong & Cairns & Jessup article, strictly block-triangular Lie algebras, etc. The resulting generating functions and the combinatorics of how they are obtained are interesting in their own right. \par All this is done by using AMT (algebraic Morse theory). This article serves as a source of examples of how to construct useful acyclic matchings, each of which in turn induces compelling combinatorial problems and solutions. It also enables graph theory to be used in homological algebra., Comment: 17 pages, 2 homology tables, 5 figures, 3 distribution plots

E. Sk\"oldberg's Morse Theory from an Algebraic Viewpoint and M. J\"ollenbeck's Algebraic Discrete Morse Theory and Applications to Commutative Algebra, which is the algebraic generalization of R. Forman's discrete Morse Theory for Cell Complexes, is discussed in the context of general chain complexes of free modules. Using this, we compute the Chevalley-Eilenberg (co)homology of the Lie algebra of all triangular matrices $\frak{sol}_n$ over $\mathbb{Q}$ or $\mathbb{Z}_p$ for large enough prime $p$. We determine the column and row in the table of $H_k(\frak{sol}_n;\mathbb{Z})$ where the $p$-torsion first appears. Every $\mathbb{Z}_{p^k}$ appears as a direct summand of some $H_k(\frak{sol}_n;\mathbb{Z})$. Module $H_k(\frak{sol}_n;\mathbb{Z}_p)$ is expressed by the homology of a chain subcomplex for the Lie algebra of all strictly triangular matrices $\frak{nil}_n$, using the K\"unneth formula. All conclusions are accompanied by computer experiments. Then we generalize some results to the more general Lie algebras of (strictly) triangular matrices $\frak{gl}_n^\prec$ and $\frak{gl}_n^\preceq$ with respect to any partial ordering $\preceq$ on $[n]$. Furthermore, the matchings used can be analogously defined for other Lie algebra families and are useful for theoretical as well as computational purposes., Comment: 19 pages, 2 homology tables, 5 figures

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modelled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite non-trivial group., Comment: 24 pages, version 3 as a result of comments by a great referee

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension., Comment: 10 pages

The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\leq n$, [b.] For each $R,\epsilon > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \epsilon$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$, [c.] For each $R > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$., Comment: 3 pages

We extend the definition of Bockstein basis $\sigma(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) = \sup\{\dim_H(X) | H\in \sigma(G)\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\dim_G(X) \leq 1$ if and only if $\sup\{\dim_H(X) | H\in\sigma(G)\}\leq 1$. $X$ is a Bockstein space if and only if $\dim_{\Z_{(l)}} (X) = \dim_{\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers., Comment: 10 pages

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $X\tau_h K\implies \beta(X)\tau K$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite. Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite. Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$. Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of $X\tau {\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept., Comment: 20 pages

The purpose of the paper is to characterize the dimension of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{m+1}$. One should compare it to the result of Dranishnikov \cite{Dr1} who characterized the dimension of the Higson corona $\nu(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(\nu_L(X))\leq n$ if and only if $\R^{n+1}$ is an absolute extensor of $X$ in the category $\tilde\AAA$. \par As an application we reprove the following result of Dranishnikov and Smith \cite{DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension $\asdim_{AN}(X)$. If $X$ is cocompact and connected, then $\asdim_{AN}(X)$ equals the dimension of the sublinear Higson corona $\nu_L(X)$ of $X$., Comment: 13 pages

We give a systematic definition of the fundamental groups of gropes, which we call grope groups. We show that there exists a nontrivial homomorphism from the minimal grope group M to another grope group G only if G is the free product of M with another grope group.