1,174 research outputs found
Topological finiteness properties of monoids. Part 1: Foundations
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 develop the corresponding theory of
-equivariant collapsing schemes (that is, -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-.Comment: 59 pages, 1 figur
Homological finiteness conditions for groups, monoids and algebras
Recently Alonso and Hermiller introduced a homological finiteness
condition\break (here called {\it weak} ) for monoid
rings, and Kobayashi and Otto introduced a different property, also called
(we adhere to their terminology). From these and other papers we
know that: left and right weak
; the first implication is not reversible in general; the second
implication is reversible for group rings. We show that the second implication
is reversible in general, even for arbitrary associative algebras (Theorem 1'),
and we show that the first implication {\it is} reversible for group rings
(Theorem 2). We also show that the all four properties are equivalent for
connected graded algebras (Theorem 4). A result on retractions (Theorem 3') is
proved, and some questions are raised.Comment: 10 page
- β¦