The Chase radical and reduced products

Let κ be a regular uncountable cardinal. We shall give a criterion for certain reduced products of torsion-free abelian groups to be ℵ₁-free. As an application we shall show that the norm of the Chase radical is ℵ₁ in ZFC, a result which was previously known only under the assumption of the continuum hypothesis 2^ℵ₀=ℵ₁.     

A measure-theoretic proof of Turing incomparability

We prove that if S is an ω-model of weak weak König’s lemma and , is incomputable, then there exists , such that A and B are Turing incomparable. This extends a recent result of Ku?era and Slaman who proved that if S₀ is a Scott set (i.e. an ω-model of weak König’s lemma) and A∈S₀, A⊆ω, is incomputable, then there exists B∈S₀, B⊆ω, such that A and B are Turing incomparable.     

More on perfectly normal non-realcompact spaces

We use the space associated with a guessing sequence on ω₁ to show that it is consistent with CH that there exists a locally countable, first-countable, locally compact, perfectly normal, non-realcompact space of size ℵ₁ which does not contain any sub-Ostaszewski spaces. By a similar technique, it is shown to be consistent with that there exists a locally countable, first-countable, perfectly normal, non-realcompact space of size ℵ₁.     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).     

Club-guessing, stationary reflection, and coloring theorems

We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. In particular, the simplest of our results establishes that if μ is singular and , then there is a regular cardinal θ<μ such that any fewer than cf(μ) stationary subsets of must reflect simultaneously.     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .     

Cardinal characteristics and projective wellorders

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: d<c, b<a=s and b<g.     

Generic embeddings associated to an indestructibly weakly compact cardinal

I use generic embeddings induced by generic normal measures on P_κ(λ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower Q_<κ works much like it does when κ is a Woodin limit of Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness.     

On the Kurosh rank of the intersection of subgroups in free products of groups

The Kurosh rank rK(H) of a subgroup H of a free product of groups Gα, α∈I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H₁, H₂ are subgroups of and H₁, H₂ have finite Kurosh rank, then , where , q^∗ is the minimum of orders >2 of finite subgroups of groups Gα, α∈I, q^∗:=∞ if there are no such subgroups, and if q^∗=∞. In particular, if the factors Gα, α∈I, are torsion-free groups, then .     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Borel subgroups of Polish groups

We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In particular, answering a question of Mauldin, we establish the existence of Borel subgroups which are -complete, α?3, and -complete, α?2, in each uncountable Polish group. Also, for every α<ω₁ we construct an Abelian, locally compact, second countable group which is densely divisible and of Ulm length α+1. All previously known such groups had Ulm length 0 or 1.