Old and new categorical invariants of manifolds

Abstract

The Lusternik-Schnirelmann category of a space X is the smallest number of open and (in X ) contractible sets that cover X . More generally, a categorical invariant of X is defined to be the smallest number of open sets that cover X and that have certain properties. For example, the sets may be required to be contractible in themselves; or the inclusion of each of the sets may be required to factor homotopically through some fixed given space K ; or for each component of the sets the inclusion into X may be required to be π 1 -injective; or for each component of the sets the image of the fundamental group under the homomorphism induced by inclusion into X may be required to belong to a given class of groups (e.g. the class of amenable or the class of solvable groups). The aim of this paper is to describe several such categorical Dedicated to Fico González-Acuña on his 70th birthday.

Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

“…An invariant of a space X is usually defined as the smallest number of open sets that cover X and that satisfy certain properties, such as being elementary in a certain sense, for instance acyclic or contractible (see [23,26] for more examples). More generally, and analogously, a categorical invariant of an object X (such as a simplicial complex) can be defined as the smallest number of subobjects needed to cover X and that verify certain properties (see for example [14,15,31,39]).…”

Expanders and right-angled Artin groups

“…An invariant of a space X is usually defined as the smallest number of open sets that cover X and that satisfy certain properties, such as being elementary in a certain sense, for instance acyclic or contractible (see [23,26] for more examples). More generally, and analogously, a categorical invariant of an object X (such as a simplicial complex) can be defined as the smallest number of subobjects needed to cover X and that verify certain properties (see for example [14,15,31,39]).…”

Morse-Bott Theory on posets and an homological Lusternik-Schnirelmann Theorem

“…An invariant of a space X is the smallest number of open sets that cover X and that satisfy certain properties, such as: being elementary in a certain sense, for instance acyclic or contractible (see [23,26] for more examples). More generally, and analogously, a categorical invariant of an object X (such as a simplicial complex) can be defined as the smallest number of subobjects needed to cover X and that verify certain properties (see for example [30,15,14,38]).…”