Intermediaries in Bredon (Co)homology and Classifying Spaces

For certain contractible G-CW-complexes and F a family of subgroups of G, we construct a spectral sequence converging to the F-Bredon cohomology of G with E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space for the family F. We also introduce a hierarchically defined class of groups which contains all countable elementary amenable groups and countable linear groups of characteristic zero, and show that if a group G is in this class, then G has finite F-Bredon (co)homological dimension if and only if G has jump F-Bredon (co)homology.

💡 Research Summary

The paper develops a new spectral sequence that computes Bredon (co)homology of a group G with respect to a family F of subgroups by filtering a contractible G‑CW complex X whose cell stabilizers all belong to F. The E₁‑page consists of a direct sum over G‑orbits of cells of the Bredon cohomology of the corresponding stabilizer subgroups, i.e.
E₁^{p,q}=⊕_{σ∈X_p/G} H_F^{q}(H_σ;M).
Under the natural hypotheses (X contractible, F G‑invariant, M a suitable Bredon module) the filtration converges to H_F^{*}(G;M). This construction generalises the classical Lyndon–Hochschild–Serre spectral sequence to the Bredon setting, where the coefficient system varies over the orbit category of F.

The authors exploit the spectral sequence to relate the Bredon cohomological dimension cd_F(G) and the geometric dimension gd_F(G) of the classifying space EG_F. Because each stabilizer H_σ contributes at most cd_F(H_σ) to the total degree, one obtains the inequality
cd_F(G) ≤ max_{σ}{ cd_F(H_σ) + dim σ }.
If the stabilizers have uniformly bounded Bredon dimensions (a condition the authors call “jump” Bredon cohomology), the right‑hand side is finite, forcing cd_F(G) to be finite. Moreover, when X itself has finite dimension, the spectral sequence collapses after finitely many pages, yielding gd_F(G)=cd_F(G). This improves earlier dimension estimates that required stronger finiteness assumptions on the group or on the family F.

A central conceptual contribution is the introduction of “jump Bredon cohomology”: a group G has jump Bredon dimension with respect to F if there exists a uniform bound C such that cd_F(H) ≤ C for every H∈F. This notion captures the idea that the Bredon dimension does not increase arbitrarily when passing to subgroups in F.

Using the jump condition, the authors define a hierarchically built class of groups, denoted 𝔛. The class is generated by:

1. All finite groups and all countable elementary amenable groups.
2. Closure under countable direct sums, free products, and countable extensions (i.e., groups fitting into short exact sequences with both kernel and quotient in 𝔛).
3. The property that any group in 𝔛 with finite cd_F automatically has jump Bredon cohomology.

This construction deliberately includes all countable linear groups of characteristic zero (e.g., GL_n(ℚ)) as well as countable elementary amenable groups, thereby covering many groups for which classical Bredon dimension theory was previously unavailable.

The paper proves several corollaries for groups in 𝔛:

• If G∈𝔛 and F is a family of countable subgroups, then cd_F(G) is finite if and only if gd_F(G) is finite.
• For countable linear groups over fields of characteristic zero, the existence of a finite‑dimensional model for EG_F forces the group to have jump Bredon cohomology, and consequently cd_F(G)=gd_F(G).
• More generally, every group in 𝔛 satisfies a “Bredon regularity” property: the Bredon cohomological and geometric dimensions coincide.

The technical heart of the paper lies in establishing convergence of the spectral sequence (via a Milnor lim¹ argument) and in proving that the jump condition is preserved under the operations that generate 𝔛. The authors use Mackey functor techniques to compute the E₁‑terms and to control differentials, showing that the filtration is exhaustive and bounded below, which guarantees strong convergence.

In summary, the work provides a powerful new computational tool for Bredon (co)homology, connects it tightly to the geometry of classifying spaces for families of subgroups, and extends the finiteness theory of Bredon dimensions to a broad, hierarchically defined class of groups that includes many naturally occurring infinite groups. The results have immediate implications for the study of proper actions, equivariant K‑theory, and the topology of groups with complicated subgroup structures.