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Let n ≥ 1, and let ι n : F n (M) → ∏ 1 n M be the natural inclusion of the nth configuration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the map ιn, the homotopy fibre In of ιn and its homotopy groups, and the induced homomorphisms (ιn)#k on the kth homotopy groups of Fn(M) and ∏ 1 n M for all k ≥ 1, where M is the 2-sphere S 2 or the real projective plane ℝP2.It is well known that the group πk(In) is the homotopy group π k + 1 (∏ 1 n M , F n (M)) for all k ≥ 0. If k ≥ 2, we show that the homomorphism (ιn)#k is injective and diagonal, with the exception of the case n = k = 2 and M = S 2 , where it is anti-diagonal. We then show that In has the homotopy type of K (R n − 1 , 1) × Ω (∏ 1 n − 1 S 2 ) , where Rn−1 is the (n − 1)th Artin pure braid group if M = S 2 , and is the fundamental group Gn−1 of the (n−1)th orbit configuration space of the open cylinder S 2 \ { z ˜ 0 , − z ˜ 0 } with respect to the action of the antipodal map of S 2 if M = ℝP2, where z ˜ 0 ∈ S 2 . This enables us to describe the long exact sequence in homotopy of the homotopy fibration I n → F n (M) → ι n ∏ 1 n M in geometric terms, and notably the image of the boundary homomorphism π k + 1 (∏ 1 n M) → π k (I n) . From this, if M = S 2 and n ≥ 3 (resp. M = ℝP2 and n ≥ 2), we show that Ker((ιn)#1) is isomorphic to the quotient of Rn−1 by the square of its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order 2 generated by the centre of Pn (M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5]. [ABSTRACT FROM AUTHOR]

Let E be the fundamental group of a closed S o l 3 -manifold. We describe the groups A u t (E) and O u t (E). We first consider the case where E is the fundamental group of a torus bundle, and then the case where E is the fundamental group of a closed S o l 3 -manifold which is not a torus bundle. The groups are described in terms of some iterated semidirect products of well-known groups, where one exception is for O u t (E) for some E ′ s , in which case O u t (E) is described as an extension and a presentation is given. [ABSTRACT FROM AUTHOR]

A root of an n-valued map ϕ: X → Dn(Y ) at a ∈ Y is a point x ∈ X such that a ∈ ϕ(x). We lift the map ϕ to a split n-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of ϕ. We describe the relationship between the root classes at a of the lift factors and those of ϕ. We define the Reidemeister root number RR(ϕ) in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number NR(ϕ), the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the Φ-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if Y is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of ϕ. If X and Y are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of ϕ that is invariant under Φ-relation and this implies that if its index is non-zero, then the root class is essential. [ABSTRACT FROM AUTHOR]

We determine a lower bound for the dimension of the Čech cohomology of the root sets of maps from the sphere S^{2n+1} and from the real projective space {\mathbb {R}\mathrm {P}}^{2n+1} into the complex projective space {\mathbb {C}\mathrm {P}}^n, for n\geq 1. For each such a map, we construct a representative of its homotopy class which realize the lower bound and whose root set is minimal in the class. We prove that the circle is a minimal root set for any non-trivial homotopy class. We present analogous results for maps from both S^{4n+3} and {\mathbb {R}\mathrm {P}}^{4n+3} into the orbit space {\mathbb {C}\mathrm {P}}^{2n+1}\!/\tau, for n\geq 0, where \tau is a free involution on {\mathbb {C}\mathrm {P}}^{2n+1}. In this setting, we prove that the disjoint union of two circles is a minimal root set for any non-trivial homotopy class. [ABSTRACT FROM AUTHOR]

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups. [ABSTRACT FROM AUTHOR]

Let X, Y, S be closed connected surfaces and τ × β a diagonal involution on X × Y where τ and β are free involutions on X and Y, respectively. In this work we study when the triple (X ×Y, τ ×β, S) satisfies the Borsuk-Ulam property. The problem is formulated in terms of an algebraic diagram, involving the 2-string braid group B2(S). [ABSTRACT FROM AUTHOR]

We study groups G where the -conjugacy class of the unit element is a subgroup of G for every automorphism of G. If G has n generators, then we prove that the k-th member of the lower central series has a finite verbal width bounded in terms of n, k. Moreover, we prove that if such group G satisfies the descending chain condition for normal subgroups, then G is nilpotent, what generalizes the result from [Bardakov, Nasybullov, and Neshchadim]. Finally, if G is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of G. [ABSTRACT FROM AUTHOR]

Let G be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of ℤ over ℤG and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings H∗(G;A) for A=ℤ and A=ℤp for an odd prime p, and indicate how to compute the groups H∗(G;A) and the multiplicative structure given by the cup product for any system of coefficients A. [ABSTRACT FROM AUTHOR]

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional C W-complex Σ(2 n) which has the homotopy type of the 2 n -sphere 𝕊2 n . Then, that action induces a homomorphism G → Aut(H 2 n (Σ(2 n))). 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(2 n). 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(2 n) with dim Σ(2 n) ≤ 2 n + 1. [ABSTRACT FROM AUTHOR]

In this work, we study the decomposability property of branched coverings of degree odd, over the projective plane, where the covering surface has Euler characteristic . The latter condition is equivalent to say that the defect of the covering is greater than . We show that, given a datum with an even defect greater than , it is realizable by an indecomposable branched covering over the projective plane. The case when is even is known. [ABSTRACT FROM AUTHOR]

Let be a virtually cyclic of the form or . We compute the integral cohomology ring of , and then obtain the periodicity of the Farrell cohomology of these groups. [ABSTRACT FROM AUTHOR]

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d, n ∈ ℕ, and let M̃ be a d-fold covering space of M. We show that the covering map induces an embedding of the nth braid group Bn(M) of M in the dnth braid group Bdn(M̃) of M̃, and give several applications of this result. First, we classify the finite subgroups of the nth braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of Bn due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear. [ABSTRACT FROM PUBLISHER]

Let G be any of the (binary) icosahedral, generalized octahedral (tetrahedral) groups or their quotients by the center. We calculate the automorphism group Aut(G). [ABSTRACT FROM AUTHOR]

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups Pn(ℝ P2) of the projective plane. The maximal finite subgroups of Pn(ℝ P2) are isomorphic to the quaternion group of order 8 if n = 3, and to ℤ4 if n ⩾ 4. Further, for all n ⩾ 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of Pn(ℝ P2): ℤ, ℤ2 × ℤ and the amalgamated product ℤ4 ∗ℤ2 ℤ4. [ABSTRACT FROM AUTHOR]

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(핊2\{x1,...,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type $\widetilde{C}$ (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bm is completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(핊2\{x1,...,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(핊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(핊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2 * ℤ. Further, its lower central series quotients Γi/Γi + 1 are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1 are 2-elementary finitely-generated groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We study the automorphism group Aut (G) to compute the number of distinct homotopy types of orbit spaces X(2dn — 1)/μ with respect to free and cellular G-actions μ on all CW — complexes X(2dn—1). At the end, the groups ϵ(X(2dn—1)/μ) of self homotopy equivalences of orbit spaces X(2dn — 1)/μ associated with free and cellular G-actions μ on X(2dn — 1) are determined. [ABSTRACT FROM AUTHOR]

Let m,n ∈ ℕ. We define $B_{n,m}(\mathbb{S}^{2})$ to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup Sn × Sm of the symmetric group Sn+m on n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence: \[ 1\to B_m(\mathbb{S}^{2}\setminus\{x_1,\ldots,x_n\})\to B_{n,m}(\mathbb{S}^{2})\stackrel{p_{\ast}}{\longrightarrow} B_n(\mathbb{S}^{2})\to 1, \] where ${p_{\ast}}{:}\, B_{n,m}(\mathbb{S}^{2}) \to B_n(\mathbb{S}^{2})$ is the group homomorphism (defined for all n ∈ ℕ) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration $p{:}\, D_{n,m}(\mathbb{S}^{2}) \to D_n(\mathbb{S}^{2})$ of the quotients of the corresponding configuration spaces. Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p* and the fibration p admit (respectively, do not admit) a section. If n = 3, then p* and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p* and p admit a section then m ≡ ε1(n - 1)(n - 2) - ε2n(n - 2) (mod n(n - 1)(n - 2)), where ε1,ε2 ∈ {0,1}. Finally, we show that $B_n(\mathbb{S}^{2})$ is generated by two of its torsion elements. [ABSTRACT FROM AUTHOR]