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Topological finiteness properties of monoids, I: Foundations

Robert D Gray and Benjamin Steinberg

Algebraic & Geometric Topology 22 (2022) 3083–3170

We initiate the study of higher-dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of M–equivariant homotopy theory where M is a discrete monoid. For projective M–CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the M–equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective M–CW complex. We prove that such a space is unique up to M–homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left- F n and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of M–equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi- F n and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including FP n, cohomological dimension, and Hochschild cohomological dimension. We also introduce a theory of M–equivariant collapsing schemes which gives new results giving sufficient conditions for a monoid to be of type F (or bi- F ). We identify some families of monoids to which these theorems apply, and in particular provide topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left- right- and bi- FP . This is the first in a series of three papers proving that all one-relator monoids are of type FP , settling a question of Kobayashi from 2000.

CW complex, monoid, equivariant homotopy theory, homological finiteness property, cohomological dimension, Hochschild cohomology, rewriting system, collapsing scheme, discrete Morse theory
Mathematical Subject Classification 2010
Primary: 20F65, 20J05, 20M05, 20M50, 57M07
Secondary: 20F10
Received: 17 September 2017
Revised: 30 April 2021
Accepted: 20 September 2021
Published: 30 January 2023
Robert D Gray
School of Mathematics
University of East Anglia
United Kingdom
Benjamin Steinberg
Department of Mathematics
The City College of New York
New York, NY
United States