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
–equivariant homotopy
theory where
is a discrete
monoid. For projective
–CW
complexes we prove several fundamental results such as the
homotopy extension and lifting property, which we use to prove the
–equivariant
Whitehead theorems. We define a left equivariant classifying space as a contractible projective
–CW
complex. We prove that such a space is unique up to
–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-
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
–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- 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
, cohomological
dimension, and Hochschild cohomological dimension. We also introduce a theory of
–equivariant
collapsing schemes which gives new results giving sufficient conditions for a monoid to be
of type
(or bi-).
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-.
This is the first in a series of three papers proving that all one-relator monoids are of
type
,
settling a question of Kobayashi from 2000.
