Let Gk(PSp(2)) and Gk(PSp(3)) be the gauge groups of principal PSp(2)-bundles over S8 and principal PSp(3)-bundles over S4 classified by kε1′ and kε2′, where ε1′ and ε2′ are generators of π8(BPSp(2))≅Z and π4(BPSp(3))≅Z, respectively. In this article we partially classify the homotopy types for these gauge groups.

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.,

We give estimations of the nilpotency class and the p-primary exponent of the total Cohen groups [Ω(Sr+1),Ω(X)] especially, when X is the projective space over K=R,C, the field of reals or complex numbers and H, the quaternionic skew R-algebra.

The main purpose of this note is to prove that the product of a cellular-Lindelöf space with a space of countable spread need not be cellular-Lindelöf.

We calculate the asymptotic behavior of the Kashaev invariant of a twice-itarated torus knot and obtain topological interpretation of the formula in terms of the Chern–Simons invariant and the twisted Reidemeister torsion.

In this paper we extend the notion of a clopen subset of a topological space to that of a clopen subobject in a topological category. We relate clopen subobjects to the concept of normalized topological category via a theorem which, in turn, shows why certain subsets must be clopen in a topological space. We employ clopen subobjects to extend the topological notion of connectedness to topological categories

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15]. Furthermore, we strengthen a result in the above paper, namely “Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice

2-stratifolds are a generalization of 2-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of 1-connected trivalent 2-stratifolds. In this paper we classify trivalent 2-stratifolds that have fundamental group Z. This classification implicitly gives an efficient algorithm

A simple closed curve α in the boundary of a genus two handlebody H is primitive if adding a 2-handle to H along α yields a solid torus. If adding a 2-handle to H along α yields a Seifert-fibered space and not a solid torus, the curve is called Seifert. If α is disjoint from an essential separating disk in H, does not bound a disk in H, and is not primitive in H, then it is said to be proper power

Let Pk:=F2[x1,x2,…,xk] be the polynomial algebra over the prime field of two elements, F2, in k variables x1,x2,…,xk, each of degree 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra. In this paper, we extend a result in [12] on the hit problem in degree (k−1)(2d−1) with k⩾6, by explicitly computing the hit

We raise the problem of realisability of rings as {X,X} the ring of stable homotopy classes of self-maps of a space X. By focusing on An2-polyhedra, we show that the direct sum of three endomorphism rings of abelian groups, one of which must be free, is realisable as {X,X} modulo the acyclic maps. We also show that Fp3 is not realisable in the setting of finite type An2-polyhedra, for p any prime.

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse–Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations

We use rewriting systems to spell out cup-products in the (twisted) cohomology groups of a product of surface groups. This allows us to detect a non-trivial obstruction bounding from below the effective topological complexity of an orientable surface with respect to its antipodal involution. Our estimates are at most one unit from being optimal, and are closely related to the (regular) topological

In this paper we characterize the Menger property and the selection principles star-Menger and strongly star-Menger in the hyperspaces CL(X), K(X), F(X) and CS(X), endowed with the hit-and-miss topology. To characterize the corresponding principles type star, we introduce a couple of technical selection principles, which we have denoted by SM(ΠΔ(Γ),ΠΔ(Γ)) and SM⁎(ΠΔ(Γ),ΠΔ(Γ)). Also, we give an equivalence

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of R which is Baire (= has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by an element of the set. Here effective proofs are recognized to yield (i) analogous category and Haar-measure metrizable generalizations for Baire groups and locally compact

We introduce new formulas that are Vassiliev invariants of flat vertex isotopy classes of 2-bouquet graphs. Although any Gauss diagram formula of Vassiliev invariants of 2-bouquet graphs in a 3-space has been unknown explicitly, this paper gives the first and simple example.

A topological group G is M-factorizable if for every continuous real-valued function f on G, one can find a continuous homomorphism π:G→H onto a metrizable topological group H and a continuous function h on H such that f=h∘π. We continue the study of M-factorizability in topological groups started in Zhang et al. (2020) [14], with a special emphasis on P-groups, i.e., the groups in which all Gδ-sets

A function f from a metric space (X,d) to another metric space (Y,ρ) is said to be Cauchy-continuous if for every Cauchy sequence (xn) in (X,d), (f(xn)) is Cauchy in (Y,ρ). It is well known that a metric space is complete if and only if every real-valued continuous function defined on it is Cauchy-continuous. Here we consider a well-studied intermediate class of metric spaces which lies between the

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of 3-dimensional flows. We prove that for an embedded template in S3 with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set

An algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let M(P) and M(E) be the cusped and compact hyperbolic real moment-angled manifolds associated with the hyperbolic right-angled 24-cell P and the hyperbolic right-angled 120-cell E, respectively. Jankiewicz, Norin, and Wise recently showed that π1(M(P)) and π1(M(E)) are algebraically

Let K denote the Klein bottle. We compute the algebraic K-theory groups of the ring Z[π1(K)] in terms of the algebraic K-theory groups of the integers. Following the same arguments, we do the same for the ring Z[π1(Ng)], where Ng denotes a nonorientable closed surface of genus g>2.

Given a group G and an automorphism φ of G, two elements x,y∈G are said to be φ-conjugate if x=gyφ(g)−1 for some g∈G. The number of equivalence classes is the Reidemeister number R(φ) of φ, and if R(φ)=∞ for all automorphisms of G, then G is said to have the R∞-property. A finite simple graph Γ gives rise to the right-angled Artin group AΓ, which has as generators the vertices of Γ and as relations

Our main objective is to study g-reversible groups introduced in Chatyrko and Shakhmatov (2020) [3]. We first give a necessary and sufficient condition for a locally compact group with an open compact subgroup to be g-reversible, it can be applied to give complete solutions of Questions 8.5, 9.7, 9.8, and Problem 9.6 in the previous article [3]. We also investigate g-reversibility of some precompact

Given a metric continumm X, let CB(X) be the hyperspace of subcontinua of X with connected boundary. In this paper we present results concerning CB(X), first about continua for which every subcontinuum has connected boundary. Then we study continua that only have one-point sets as connected boundary proper subcontinua. Later we include characterizations of arcs and simple closed curves. Finally we

A.V. Arhangel'skiı̌ introduced the notion of an almost s-mapping. It is known that the open almost s-images of metric spaces coincide with the open boundary s-images of metric spaces. In this paper, we investigate some questions related to the sequence-covering almost s-images and sequence-covering boundary s-images of metric spaces. We establish some new characterizations of the images of metric spaces

The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.

Let C(X) be the hyperspace of all subcontinua of a metric continuum X. An element A∈C(X) makes a hole in C(X) if C(X)−{A} is not unicoherent. In this paper, we characterize the elements A∈C(X) satisfying that A makes a hole in C(X) when X has property (b).

Let Topd, Topw and Sob be the category of all d-spaces, that of all well-filtered spaces and that of all sober spaces respectively. For a full subcategory K of Topd containing Sob, it is proved that the product of an arbitrary family of K-determined sets is a K-determined set and if K is adequate, then the K-reflection preserves arbitrary products of T0 spaces. In particular, the Keimel-Lawson reflection

In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T0 spaces is H-sober iff

Under finite regular covering spaces, each preimage class of the base spaces is covered by some preimage classes of the covering spaces. We find an explicit relationship between those preimage classes in terms of covering transformation groups. Thereby we obtain a relationship of the homomorphism indices between them in terms of covering transformation groups. This is a local averaging formula and

In this paper, we investigate the special Kähler geometry of the base of the Hitchin integrable system in terms of spectral curves. We give explicit formulas of the metric for some special cases by using the affine coordinates introduced by Freed in 1999. We also find that the Taylor expansion of the special Kähler metric about any point in the base may be computed by topological recursion.

In [2], T. Bartsch provided detailed and broad exposition of a numerical cohomological index theory for G-spaces, known as the length, where G is a compact Lie group. We present the length of G-spaces which are cohomology spheres and G is a p-torus or a torus group, where p is a prime. As a consequence, we obtain Borsuk-Ulam and Bourgin-Yang type theorems in this context. A sharper version of the Bourgin-Yang

In this paper we study the following problem: given X↪M→B and Y↪N→B smooth fibre bundles over B and f1,…,fk:M→N fibre-preserving maps, is it possible to deform (f1,…,fk)∼(f1′,…,fk′) via a fibrewise homotopy over B such that the set of coincidence points of f1′,…,fk′:M→N is empty? We study this question making use of obstruction theory.

Let n≥3. In this paper we deal with the conjugacy problem in the Artin braid group quotient Bn/[Pn,Pn]. To solve it we use systems of equations over the integers arising from the action of Bn/[Pn,Pn] over the abelianization of the pure Artin braid group Pn/[Pn,Pn]. Using this technique we also realize explicitly infinite virtually cyclic subgroups in Bn/[Pn,Pn].

We prove that both stated skein algebras and their reduced versions at odd roots of unity are almost-Azumaya and compute the rank of a reduced stated skein algebra over its center, extending a theorem of Frohman, Kania-Bartoszynska and Lê to the case of open punctured surfaces. We deduce that generic irreducible representations of the reduced stated skein algebras are of quantum Teichmüller type and

Given a virtual biquandle multi-switch (S,V) on an algebraic system X (from some category) and a virtual link L, we introduce a general approach to construct an algebraic system XS,V(L) (from the same category) which is an invariant of L. As a corollary we introduce a new quandle invariant for virtual links which generalizes the previously known quandle invariants for virtual links.

This work is devoted to the metrization of probabilistic spaces. More precisely, given such a space (G,D,⋆) and provided that the triangle function ⋆ is continuous, we exhibit an explicit and canonical metric σD on G such that the associated topology is homeomorphic to the so-called strong topology. As applications, we make advantage of this explicit metric to present some fixed point theorems on such

In [1], a deformation of spherical curves called deformation type α was introduced. Then, it was showed that if two spherical curves P and P′ are equivalent under the relation consisting of deformations of type RI and type up to ambient isotopy, and satisfy certain conditions, then P′ is obtained from P by a finite sequence of deformations of type α. In this paper, we introduce a new type of deformations

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms

For a topological space X its reflection in a class T of topological spaces is a pair (TX,iX) consisting of a space TX∈T and a continuous map iX:X→TX such that for any continuous map f:X→Y to a space Y∈T there exists a unique continuous map f¯:TX→Y such that f=f¯∘iX. In this paper for an infinite cardinal κ and a nonempty set M of ultrafilters on κ, we study the reflections of topological spaces in

Given a metric continuum X, we say that X is 12-homogeneous provided that X has exactly two orbits under the action of the group of homeomorphisms of X onto itself. In this paper, we introduce the hyperspace C12(X) consisting of all 12-homogeneous subcontinua of X. We study compactness and connectedness of C12(X) for some classes of continua, mainly on finite graphs and dendroids, and we characterize

We continue the study of M-factorizability in topological groups started in [18], with a special emphasis on feathered groups. It is shown that a feathered group G is M-factorizable if and only if G is either metrizable or R-factorizable. We also prove that an M-factorizable Čech-complete subgroup H of a topological group G is C-embedded in G. We show that the product G=∏n∈ωGn of countably many M-factorizable

In this paper, we prove the equivariant James splitting theorem, and we give the generalizations of EHP sequences in the classical homotopy theory to the C2-equivariant case.

We prove that every compact group with a bi-invariant metric is isometrically isomorphic to the isometry group of some compact metric space and, more generally, that every separable group with a bi-invariant metric is isometrically isomorphic to the isometry group (equipped with the supremum metric) of some separable metric space.

A topological group X is called duoseparable if there exists a countable set S⊆X such that SUS=X for any neighborhood U⊆X of the identity. We construct a functor F assigning to each (abelian) topological group X a duoseparable (abelian-by-cyclic) topological group FX, containing an isomorphic copy of X. In fact, the functor F is defined on the category of unital topologized magmas. Also we prove that

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of the surface braid group Bn(M) by the commutator subgroup Γ2(Pn(M)) of the pure braid group Pn(M). If M is different from the 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, and that Bn(M)/Γ2(Pn(M)) is a crystallographic group if and only if M is orientable. If M is orientable, we prove

We study the Wecken property for coincidences in the setting of boundary-preserving maps between surfaces with nonempty boundary. We consider the situation where the domain surface has non-negative Euler characteristic and obtain boundary Wecken results for almost all target surfaces.

We describe the set [X,Y] of path-components of pointed mapping spaces M⁎(X,Y), where X is chosen to be the reduced kth suspension EkFPm of a projective space FPm and Y is a sphere Sn or FPn for F=R,C, the fields of real or complex numbers, respectively. In particular, the cohomotopy sets πn(EkFPm) are studied for k≥0 and certain m,n≥1.

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e, the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) is known as a generalized Dold manifold. Suppose that a group G≅Z2s acts smoothly on X such that g∘σ=σ∘g for all g∈G. Using the action of the diagonal subgroup D=O(1)m+1⊂O(m+1)

Let n≥3. In this work we study almost-crystallographic subgroups H˜n,3=σ−1(H)/Γ3(Pn) of Bn/Γ3(Pn) with cyclic holonomy group H, where Bn is the Artin braid group, σ:Bn→Sn is the natural projection and Γ3(Pn) is the third-step element of the lower central series of the Artin pure braid group Pn. We study in detail the holonomy representation of H˜n,3 describing it as a matrix representation. As an application

A group G has the R∞-property if for every φ∈Aut(G), there are an infinite number of φ-twisted conjugacy classes of elements in G. In this note, we determine the R∞-property for G=π1(M) for all geometric 3-manifolds M.

We classify oriented ribbon 2-knots of 1-fusion with length up to six. We show the difference by: the Alexander polynomial; the trace set, which is obtained from the representations of the knot group to SL(2,C); the twisted Alexander polynomial.

There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification

We estimate the number of simplices required for triangulations of compact Lie groups. As in the previous work [12], our approach combines the estimation of the number of vertices by means of the covering type via a cohomological argument from [11], and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the exceptional Lie groups, we present a complete calculation

We construct two practical algorithms for twisted conjugacy classes of polycyclic groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that their Reidemeister coincidence number is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister

For a based CW-complex X, A♯n(X) is the submonoid of [X,X] which consists of all homotopy classes of self-maps of X that induce an automorphism on πk(X) for all 0≤k≤n. Since, for m

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot

We show that if f:I→I is piecewise monotone, post-critically finite, and locally eventually onto, then for every point there exists a planar embedding of X such that x is accessible. In particular, every point x in Minc's continuum XM from [11, Question 19 p. 335] can be embedded accessibly. All constructed embeddings are thin, i.e., can be covered by an arbitrary small chain of open sets which are

An Alexander pair (f1,f2) and an (f1,f2)-twisted 2-cocycle can be used to define a generalization of twisted Alexander matrices and twisted Alexander invariants. In this paper, we introduce row relation maps with respect to Alexander pairs, and we show the following two properties: First, a row relation map gives a linear relation among the row vectors of an (f1,f2)-twisted Alexander matrix. Second