is said to be of type Fn if there exists an aspherical CW-complex whose fundamental group is isomorphic to
is an example of a torsion-free group which is of type F∞ but not of type F.[1] A reformulation of the Fn property is that a group has it if and only if it acts properly discontinuously, freely and cocompactly on a CW-complex whose homotopy groups
Another finiteness property can be formulated by replacing homotopy with homology: a group is said to be of type FHn if it acts as above on a CW-complex whose n first homology groups vanish.
is said to be of type FPn if there exists a resolution of the trivial
such that the n first terms are finitely generated projective
[2] The types FP∞ and FP are defined in the obvious way.
The same statement with projective modules replaced by free modules defines the classes FLn for n ≥ 1, FL∞ and FL.
Either of the conditions Fn or FHn imply FPn and FLn (over any commutative ring).
If it is of type FP then it is of finite cohomological dimension.
Thus finiteness properties play an important role in the cohomology theory of groups.
acts freely on the unit sphere in
, preserving a CW-complex structure with finitely many cells in each dimension.
[4] Since this unit sphere is contractible, every finite cyclic group is of type F∞.
gives rise to a contractible CW-complex with a free
This shows that every finite group is of type F∞.
A non-trivial finite group is never of type F because it has infinite cohomological dimension.
This also implies that a group with a non-trivial torsion subgroup is never of type F. If
The Borel–Serre compactification shows that this is also the case for non-cocompact arithmetic groups.
Arithmetic groups over function fields have very different finiteness properties: if