ON GROUPS WHOSE SUBGROUPS ARE EITHER MODULAR OR CONTRANORMAL

A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.

TOPOLOGICAL NOETHERIANITY FOR ALGEBRAIC REPRESENTATIONS OF INFINITE RANK CLASSICAL GROUPS

Draisma recently proved that polynomial representations of GL∞ are topologically noetherian. We generalize this result to algebraic representations of infinite rank classical groups.

Primary decomposition in the smooth concordance group of topologically slice knots

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts, each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$ -signatures, Ozsváth-Szabó d-invariants and Némethi’s result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.

Rational cobordisms and integral homology

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

Congruences on infinite semigroups of transformations preserving a zig-zag order

The purpose of this paper is the study of congruences on semigroups of transformations on a countably infinite fence. We consider the monoid [Formula: see text] of all full transformations on the set [Formula: see text] of all natural numbers preserving the zig-zag order on [Formula: see text], as well as the monoid [Formula: see text] of all idempotent transformations in [Formula: see text] additionally preserving the usual linear order on [Formula: see text] We show that there are uncountably many congruences on [Formula: see text] and determine seven maximal congruences on [Formula: see text] which are all the maximal congruences containing a particular congruence on [Formula: see text] Moreover, we characterize all congruences on the monoid of all transformations in [Formula: see text] with infinite rank. For the semigroup of all transformations in [Formula: see text] with finite rank, we determine the Rees congruences.