Cohomological dimension

In abstract algebra, cohomological dimension is an invariant which measures the homological complexity of representations of a group. It has important applications in geometric group theory, topology, and algebraic number theory.

Cohomological dimension of a group

As most (co)homological invariants, the cohomological dimension involves a choice of a "ring of coefficients" R, with a prominent special case given by R = Z, the ring of integers. Let G be a discrete group, R a non-zero ring with a unit, and RG the group ring. The group G has cohomological dimension less than or equal to n, denoted cd_R(G) ≤ n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules P₀, …, P_n and RG-module homomorphisms d_k: P_k$\to$P_k − 1 (k = 1, …, n) and d₀: P₀$\to$R, such that the image of d_k coincides with the kernel of d_k − 1 for k = 1, …, n and the kernel of d_n is trivial.

Equivalently, the cohomological dimension is less than or equal to n if for an arbitrary RG-module M, the cohomology of G with coeffients in M vanishes in degrees k > n, that is, H^k(G,M) = 0 whenever k > n.

The smallest n such that the cohomological dimension of G is less than or equal to n is the cohomological dimension of G(with coefficients R), which is denoted n = cd_R(G).

A free resolution of Z can be obtained from a free action of the group G on a contractible topological space X. In particular, if X is a contractible CW complex of dimension n with a free action of a discrete group G that permutes the cells, then cd_Z(G) ≤ n.

Examples

In the first group of examples, let the ring R of coefficients be Z.

• A free group has cohomological dimension one. As shown by John Stallings (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups.
• The fundamental group of a compact, connected, orientable Riemann surface other than the sphere has cohomological dimension two.
• More generally, the fundamental group of a compact, connected, orientable aspherical manifold of dimension n has cohomological dimension n. In particular, the fundamental group of a closed orientable hyperbolic n-manifold has cohomological dimension n.
• Nontrivial finite groups have infinite cohomological dimension over Z. More generally, the same is true for groups with nontrivial torsion.

Now let us consider the case of a general ring R.

• A group G has cohomological dimension 0 if and only if its group ring RG is semisimple. Thus a finite group has cohomological dimension 0 if and only if its order (or, equivalently, the orders of its elements) is invertible in R.
• Generalizing the Stallings–Swan theorem for R = Z, Dunwoody proved that a group has cohomological dimension at most one over an arbitrary ring R if and only if it is the fundamental group of a connected graph of finite groups whose orders are invertible in R.

See also

• Eilenberg−Ganea conjecture
• Group cohomology
• Global dimension

References

• Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. MR1324339. ISBN 0-387-90688-6
• Warren Dicks, Groups, Trees, and Projective Modules, Lecture Notes in Mathematics, vol 790, Springer-Verlag, Berlin, 1980 MR0584790 ISBN 3-540-09974-3
• Jerzy Dydak, Cohomological dimension theory, in: Handbook of geometric topology, 423–470, North-Holland, Amsterdam, 2002. MR1886675. ISBN 0-444-82432-4
• John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334. MR0228573
• Richard G. Swan, "Groups of cohomological dimension one", Journal of Algebra 12 (1969), 585–610. MR0240177