Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

Topology of branched surfaces which admit expanding immersions

We investigate the topology of branched surfaces K which have the disjoint union of embedded circles as their branch sets SK, and which admit expanding immersions f with injective induced homomorphisms . If every connected component L of K?SK is orientable, then L is homeomorphic to a surface of genus ?1 with holes. In particular if there is a component homeomorphic to a 2-torus with holes, then K is the union of immersed tori. If every L is a 2-sphere with holes, under an additional assumption K is the union of immersed annuli.     

Right projective semigroups with 0

We describe the semigroups S with 0 that are projective in the category of right S-objects, i.e. of centered right S-operands. Among them are the free semigroups with an adjoined zero, and several characterizations of this subclass are given. Two other classes, one of them close to free semigroups with 0, the other rather far apart, are discussed in detail. The first class consists of semigroups of finite and infinite sequences, with multiplication based on concatenation. The second is formed by the projectives in the category of (S,S)-biobjects.     

Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⇒β(X)τK for all paracompact spaces X. Here are the main results of the paper:

Theorem 0.1. If{Ks}_s∈Sis a family of pointed quasi-finite complexes, then their wedge_?s∈SKsis quasi-finite.     

Ideal semigroups of noetherian domains and Ponizovski decompositions

Let R be an integral domain with quotient field K and L⊃K a finite extension field. By an R-lattice in L we mean a finitely generated R-module containing a basis of L over K. The set of all R-lattices is a commutative multiplicative semigroup. If R is one-dimensional and noetherian, we determine the structure of this semigroup and of the corresponding class semigroup by means of its partial Ponizovski factors. If moreover R is a Dedekind domain and L⊃K is separable, we give criteria for the partial Ponizovski factors to be groups in terms of the different and the conductor of their endomorphism rings.     

Vines and partial transformations

We introduce the partial vine monoid PVn. This monoid is related to the partial transformation semigroup PTn in the same way as the braid group Bn is related to the symmetric group Sn, and contains both the vine monoid [T.G. Lavers, The theory of vines, Comm. Algebra 25 (4) (1997) 1257-1284] and the inverse braid monoid [D. Easdown, T.G. Lavers, The inverse braid monoid, Adv. Math. 186 (2) (2004) 438-455]. We give a presentation for PVn in terms of generators and relations, as well as a faithful representation in a monoid of endomorphisms of a free group. We also derive a new presentation for PTn.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G  . We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group _A5$A_5$, and give various other examples.     

On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis-Meakin expansion M(H) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin-Zelmanov varieties K_n of all locally finite groups of exponent dividing n.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

How strange can an attractor for a dynamical system in a 3-manifold look?

The aim of this paper is to characterise those compact subsets K of 3-manifolds M that are (stable and not necessarily global) attractors for some flow on M. We will show that it is the topology of M−K, rather than that of K, the one that plays a relevant role in this problem.A necessary and sufficient condition for a set K to be an attractor is that it must be an “almost tame” subset of M in a sense made precise under the equivalent notions of “weakly tame” and “tamely embedded up to shape”, defined in the paper. These are complemented by a further equivalent condition, “algebraic tameness”, which has the advantage of being checkable by explicit computation.A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.     

Two, more readily computable, equivariant Nielsen numbers II

In this paper, we present and make computations of two equivariant Nielsen type numbers NG(f(H),k(H)) and NG(f(H),k(H)). The second one is new, while the first one extends and clarifies one given earlier by the author and Jianhan Guo. Both numbers were defined here in terms of Nielsen theory of M-ads introduced in the prequel to this work. The theory of M-ads is also used to give both upper and lower bounds on our numbers, and to make specific computations. Our numbers moreover, fit together in the same way that the two Nielsen type periodic point numbers NPn(f) and NΦn(f) fit together. In particular, we show that NG(f(H),k(H)) is greater than or equal to a sum of numbers of the form NG(f(K),k(K)), and give conditions for equality and Möbius inversion. The periodic point theory results are then seen to follow from what are actually generalizations of them.We work with both fixed point, and coincidence point classes, in the context of a category with essentiality which we introduced in the prequel on M-ads. It is intended that this paper be read in tandem with said prequel.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Standard P3-bundles of exponent two on algebraic surfaces

Let V_K be a twisted form of P³ over the function field of an algebraic surface with Pic V_K generated by the half of the canonical line bundle. We construct an algebraic fibre space V → X projective flat over a smooth projective surface X with the generic fibre V _K → Spac K satisfying some properties.     

Braids and partial permutations

We study the inverse braid monoid IBn introduced by Easdown and Lavers in 2004. We completely describe the factorizable structure of IBn and use this to give a new proof of the Easdown-Lavers presentation; we also derive several new presentations, each of which gives rise to a new presentation of the symmetric inverse monoid. We then define and study the pure inverse braid monoid IPn which is related to IBn in the same way that the pure braid group is related to the braid group.