One resolution that always works, and several that are smaller
The bar resolution has free generators indexed by tuples of group elements, so it exists for every group and yields explicit cocycle formulas — the derivation condition in degree 1 and the factor set condition in degree 2 both read off it directly. It is also far too large for computation. Resolutions built from a presentation, or from a contractible space on which the group acts freely, are the practical alternatives.
Learning objectives
- Write the bar resolution and its differential.
- Read the cocycle and coboundary conditions in low degrees.
- Explain the normalisation and why it is harmless.
- Construct the start of a resolution from a presentation.
Section 01The bar resolution
Bn is free over ℤ[G] on symbols [g1|…|gn], with differential
Exactness is proved by an explicit contracting homotopy over ℤ, which is why the construction works for every group with no hypotheses at all.
Bn has |G|n free generators. For a group of order 60 the degree-4 term already has nearly 13 million generators. The bar resolution is for deriving formulas, not for computing.
Section 02Cocycles in low degrees
| Degree | Cochain | Cocycle condition | Coboundary |
|---|---|---|---|
| 1 | f: G → A | f(gh) = f(g) + g·f(h) | f(g) = g·a − a |
| 2 | f: G² → A | g·f(h,k) − f(gh,k) + f(g,hk) − f(g,h) = 0 | f(g,h) = g·c(h) − c(gh) + c(g) |
| 3 | f: G³ → A | The four-term alternating identity | From a 2-cochain |
It is exactly the associativity requirement for a multiplication on A × G twisted by f. So H2 classifying extensions is not an analogy — the cocycle condition is associativity, read off the resolution.
Section 03Smaller resolutions
Normalised bar resolution
Quotient by degenerate generators, those with some gi = 1. Chain homotopy equivalent to the full bar resolution, and appreciably smaller.
From a presentation
A presentation with generators and relations gives the first two terms of a free resolution, with the differential expressed by Fox derivatives. Enough for H1 and H2.
Geometric
A contractible complex with free G-action gives a resolution by its cellular chains. For a free group, a tree; for a surface group, the hyperbolic plane.
Periodic
For cyclic groups, period 2 as described separately. For groups with periodic cohomology in general, a periodic resolution exists.
Koszul-type
For elementary abelian groups in characteristic p, the resolution has polynomial and exterior structure, and the cohomology ring is computable.
Minimal
Over a local ring such as a group algebra in modular characteristic, minimal resolutions give the Betti numbers directly.
ReferenceFrequently asked questions
Is the normalised bar resolution equivalent to the full one?
Yes — the degenerate subcomplex is acyclic, so the quotient map is a chain homotopy equivalence. Cohomology can therefore be computed with normalised cochains, which is the usual convention.
How much of a resolution does a presentation give?
The first two steps: free on the generators, then the relation module. That determines H1 and H2, which is why Hopf's formula for H2 can be read off a presentation.
Why is the bar resolution still taught?
Because it is the source of the explicit cocycle formulas that give the low-degree interpretations, and because its existence for arbitrary groups makes the general theory unconditional. Its impracticality is separate from its theoretical role.
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