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-Fnβ 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-Fnβ 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 FPnβ, cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
M-equivariant collapsing schemes (that is, M-equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-FPββ.Comment: 59 pages, 1 figur