Link concordance and algebraic closure of groups

Partially supported by NSF.     

Link concordance invariants and homotopy theory

Summary We show that each of a sequence of concordance invariants for codimension-two links of spheres inS ⁿ⁺² , defined by Kent Orr, is identically zero forn>1. For classical links (n=1), the same proof shows that these invariants vanish if and only if Milnor's vanish (a result obtained independently and earlier by Orr himself). We offer sufficient conditions for the vanishing of Orr's -invariant (not covered by the above). We discuss how this relates to positive results.Supported by a National Science Foundation Postdoctoral Fellowship and the hospitality of Univ. Calif. San Diego Math. Dept.     

Root closure in algebraic orders

We obtain a characterization of root closed algebraic orders by means of their conductor. It provides the root closure of an algebraic order. Actually, non-integrally closed root closed orders are exceptional. In the same way, we study n-root closedness of algebraic orders, for a given integer n.     

New representations of algebraic domains and algebraic L-domains via closure systems

Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.

     

On algebraic semigroups and monoids,II

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.     

On the algebraic closure in rings

The ``algebraic closure" of a subset of a ring is an algebraic analogue of topological closure.

     

The algebraic closure of a function

The algebraic operations on a set A which admit (as a homomorphism) a particular mapf∈A ^A, admit all powers off and certain other maps as well. The algebraic closure off, , is the totality of all maps which admit the algebraic operations thatf does. The purpose of this paper is to characterize in terms off, both forA finite and forA infinite. This research was supported in part by N.R.C. Operating Grants A7213 and A8094 from the National Research Council of Canada. Presented by B. Jónsson     

Hodge genera of algebraic varieties, II

We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic ${\chi^c_y}$ -genus satisfies the “stratified multiplicative property”, which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann–Hurwitz formula. We also study the monodromy contributions to the ${\chi_y}$ -genus of a family of compact complex manifolds, and prove an Atiyah–Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the ${\chi_y}$ -genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah–Meyer type are also obtained by making use of Saito’s theory of mixed Hodge modules.     

The algebraic closure of a semigroup of functions

A semigroup of functionsS υA ^A isalgebraic providedA can be endowed with some collection of (finitary) operations to produce an algebraU withS = EndU, the endomorphisms ofU. Thealgebraic closure ofS is =End,U _s , whereU _s is the algebra of all finitary operations which admit eachf σS as a homomorphism. Here we prove, for A finite, thatg σ iffg is the unique solution to a system ζ of functional equations each of the formfx=h orfx=y with, coefficientsf, h σS. For A infinite a similar local condition holds. Applications to related problems are given. Presented by B. Jónsson This research was supported in part by N.R.C. Operating Grants A 7213 and A8094 from The National Research Council of Canada.     

On many‐sorted algebraic closure operators

A theorem of Birkhoff‐Frink asserts that every algebraic closure operator on an ordinary set arises, from some algebraic structure on the set, as the corresponding generated subalgebra operator. However, for many‐sorted sets, i.e., indexed families of sets, such a theorem is not longer true without qualification. We characterize the corresponding many‐sorted closure operators as precisely the uniform algebraic operators. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bousfield localization as an algebraic closure of groups

We show that the notion ofHZ-local groups due to A. Bousfield which is based on considerations from algebraic topology can be defined and understood in terms of solubility of certain systems of equations overG.     

Finitary algebraic logic II

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.