Volume 11, Issue 4

We deal with complex points of two-dimensional surfaces. A short review of basic results about complex points of smooth surfaces in is presented at the beginning. For algebraic surfaces, a formula is proved which expresses the number of complex points as the local degree of an explicitly constructible polynomial endomorphism.

Border homology and cohomology groups of pairs of uniform spaces are defined and studied. These groups give an intrinsic characterization of Čech type homology and cohomology groups of the remainder of a uniform space.

Let 𝔎 be a category of pairs of spaces, 𝔏 ⊂ 𝔎 the category of pairs of ANRs or CW-spaces, 𝐴∗ a chain functor (e.g., one associated with a spectrum). Then the derived homology 𝑠 ℎ∗ of the 𝔏-localization of 𝐴∗ is the strong homology theory on 𝔎 which is up to an isomorphism uniquely determined by the fact that 𝑠 ℎ∗ | 𝔏 agrees with the derived homology of 𝐴∗ on 𝔏. This establishes a relationship between localization theory and strong homology theory (e.g., Steenrod–Sitnikov homology theory, whenever all pairs are compact metric).

As observed by J. Beck, and as we know from M. Barr's and his joint work on triple cohomology, the classical isomorphism Opext ≅ 𝐻 2 that describes group extensions with abelian kernels, can be deduced from the equivalence between such extensions and torsors (in an appropriate sense). The same is known for many other “group-like” algebraic structures, and now we present a purely-categorical version of that equivalence, essentially by showing that all torsors are extensions with abelian kernels in any pointed protomodular category, and by giving a necessary and sufficient condition for the converse.

This is an overview of results from our experiment of merging two seemingly unrelated disciplines – higher algebraic 𝐾-theory of rings and the theory of lattice polytopes. The usual 𝐾-theory is the “theory of a unit simplex”. A conjecture is proposed on the structure of higher polyhedral 𝐾-groups for certain class of polytopes for which the coincidence of Quillen's and Volodin's theories is known.

A mapping is said to be confluent over locally connected continua if for each locally connected subcontinuum 𝑄 of the range each component of its preimage is mapped onto 𝑄. For mappings of compact spaces this class is a very natural generalization of locally confluent mappings. Various properties of these mappings are systematically studied in the paper.

For topological spaces 𝑋, 𝑌 with a fixed compatible quasi-uniformity 𝑄 in 𝑌 and for a family (𝑓 𝑖 ) 𝑖∈𝐼 of mappings from 𝑋 to 𝑌, the notions of even continuity in the sense of Kelley, topological equicontinuity in the sense of Royden and 𝑄-equicontinuity (i.e., equicontinuity with respect to the topology of 𝑋 and 𝑄) are compared. It is shown that 𝑄-equicontinuity implies even continuity, and if 𝑄 is locally symmetric, it implies topological equicontinuity too. It turns out that these notions are equivalent provided 𝑄 is a uniformity compatible with a compact topology, but the equivalence may fail even for a locally symmetric quasi-uniformity 𝑄 compatible with a compact metrizable topology.

A subcategory of the category of groups with action is determined and it is proved that the functor defined in [Datuashvili, Georgian Math. J. 9: 671–681, 2002] takes free objects from this category to free Leibniz algebras. This result gives a solution to a problem stated by J.-L. Loday [Enseign. Math. 39: 269–293, 1993], [K-Theory, 30: 105–127, 2003] and is an analogue of Witt's theorem for groups and Lie algebras [Witt, J. Reine Angew. Math. 1: 152–160, 1937].

The paper, consisting of four sections, of which Section 0 is auxiliary, is devoted to some principal questions of the theory of bitoplogical spaces. In Section 1, the 𝑝 -extremally disconnected, (𝑖, 𝑗)-strongly extremally disconnected and (𝑖, 𝑗)-nodec spaces are studied by means of the localization at a point. In Section 2, the (𝑖, 𝑗)-pseudoscattered, (𝑖, 𝑗)-ndscattered, 𝑝 -ultradisconnected and (𝑖, 𝑗)-Moscow spaces are introduced, their interrelations and their relations with the 𝑝 -extremally disconnected spaces are investigated, in particular, when one of the topologies is finer than the other. Section 3 is concerned with the problem of existence of a finer connected topology on a connected topological space and so, it is related to the concept of a maximal connected space.

Let 𝑅 be a local ring and 𝐴 a connected differential graded algebra over 𝑅 which is free as a graded 𝑅-module. Using homological perturbation theory techniques, we construct a minimal free multi-model for 𝐴 having properties similar to those of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi’ refers to the category of multicomplexes.

For a given σ -ideal of sets, the notion of a generalized stepfunction is introduced and investigated in connection with the problem of sup-measurability of certain functions of two variables, regarded as superposition operators.

Some geometric results are presented which can be derived from algebraic formulae for topological degree and Euler characteristic. In particular, it is shown that Euler characteristics of configuration spaces and work spaces of mechanical linkages can be computed in an algorithmic way. We also find the expected gradient degree of rotation invariant Gaussian random polynomial on an even-dimensional space.

It is proved that every regularly movable metric compactum has the shape of the inverse limit of an inverse sequence consisting of compact polyhedra and retractions.

Let 𝐻(𝑋) := (ℝ × 𝑋) ⋋ 𝑋* be the generalized Heisenberg group induced by a normed space 𝑋. We prove that 𝑋 and 𝑋* are relatively minimal subgroups of 𝐻(𝑋). We show that the group 𝐺 := 𝐻(𝐿 4 [0, 1]) is reflexively representable but weakly continuous unitary representations of 𝐺 in Hilbert spaces do not separate points of 𝐺. This answers the question of A. Shtern.

In this paper we continue the investigation of some aspects of descent theory for schemes that was begun in [Mesablishvili, Appl. Categ. Structures]. Let 𝐒𝐂𝐇 be a category of schemes. We show that quasi-compact pure morphisms of schemes are effective descent morphisms with respect to 𝐒𝐂𝐇-indexed categories given by (i) quasi-coherent modules of finite type, (ii) flat quasi-coherent modules, (iii) flat quasi-coherent modules of finite type, (iv) locally projective quasicoherent modules of finite type. Moreover, we prove that a quasi-compact morphism of schemes is pure precisely when it is a stable regular epimorphism in 𝐒𝐂𝐇. Finally, we present an alternative characterization of pure morphisms of schemes.