One-relator groups and algebras related to polyhedral products

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

Towards a homological generalization of the direct summand theorem

We present a more general homological characterization of the direct summand theorem (DST). Specifically, we state two new conjectures: the socle-parameter conjecture (SPC) in its weak and strong forms. We give a proof for the weak form by showing that it is equivalent to the DST. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller than or equal to two. Finally, we present a new proof of the DST in the equicharacteristic case, based on the techniques thus developed.

Down closed injectivity and essentialness

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for [Formula: see text]-posets, posets with an action of a pomonoid [Formula: see text] on them. They gave a criterion for dc-injectivity and studied such injectivity for [Formula: see text] itself, and for its poideals. In this paper, we give results about dc-injectivity of [Formula: see text]-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of [Formula: see text]-injectivity for [Formula: see text]-posets, where [Formula: see text] and [Formula: see text] is externally adjoined to the posemigroup [Formula: see text], we find some classes of pomonoids such that for [Formula: see text]-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of [Formula: see text]-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of [Formula: see text]-posets for some classes of pomonoids [Formula: see text].

A characterization of 3D steady Euler flows using commuting zero-flux homologies

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

Toward homological characterization of semirings by e-injective semimodules

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.

On semi-projective modules and their endomorphism rings

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

A Note on Fine Graphs and Homological Isoperimetric Inequalities

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected 2-complex X with a linear homological isoperimetric inequality, a bound on the length of attachingmaps of 2-cells, and finitely many 2-cells adjacent to any edge must have a fine 1-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity and show that a group G is hyperbolic relative to a collection of subgroups P if and only if G acts cocompactly with ûnite edge stabilizers on a connected 2-dimensional cell complex with a linear homological isoperimetric inequality and P is a collection of representatives of conjugacy classes of vertex stabilizers.