Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Ring-theoretic (In)finiteness in reduced products of Banach algebras

We study ring-theoretic (in)finiteness properties—such as Dedekind-finiteness and proper infiniteness—of ultraproducts (and more generally, reduced products) of Banach algebras. While we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$ -algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if “ultrafilter many” of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś’s Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of fixed bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$ -algebras. Finally, the related notion of having stable rank one is also studied for ultraproducts.

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

After recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.

K-homological finiteness and hyperbolic groups

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a \mathrm{C}^{*} -algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of \mathrm{C}^{*} -algebras associated to hyperbolic groups – the \mathrm{C}^{*} -crossed product for the boundary action, and the reduced group \mathrm{C}^{*} -algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

Polygraphs of finite derivation type

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: Being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squier's results in the contemporary language of polygraphs and higher dimensional categories, with new proofs and relations between them.

Characterization algorithms for shift radix systems with finiteness property

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

CANONICAL HEIGHT FUNCTIONS FOR MONOMIAL MAPS

The height function measures the arithmetic complexity of a point on a variety over [Formula: see text]. The canonical height function measures the asymptotic height growth (relative to the degree growth) of a point under a dominant rational map. One property desired for the canonical height function is the Northcott finiteness property, which states that there are only finitely many points for a bounded degree and a bounded height. We show that the canonical height function for dominant rational maps does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has the desired nice properties.