Topological finiteness properties of monoids, I: Foundations

We initiate the study of higher dimensional topological finiteness properties monoids. This is done by developing theory monoids acting on CW complexes. For this we establish foundations $M$-equivariant homotopy where $M$ a discrete monoid. projective $M$-CW complexes prove several fundamental results such as extension and lifting property, which use to Whitehead theorems. define left equivariant classifying space contractible complex. that unique up $M$-homotopy equivalence give canonical model for via nerve right Cayley graph category The conditions left-$\mathrm{F}_n$ geometric dimension are then defined in terms existence satisfying appropriate properties. also introduce bilateral notion space, proving uniqueness giving two-sided category, associated bi-$\mathrm{F}_n$ dimension. explore connections between all these well-studied homological important string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, Hochschild develop corresponding collapsing schemes (that is, Morse theory), among other things apply it proofs Anick, Squier Kobayashi admit presentations complete systems left-, right- bi-$\mathrm{FP}_\infty$.