This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).

Many interesting integral representations of groups arise from a group acting on a simplicial complex that is homotopy equivalent to a wedge of spheres, by applying homology. A classical example is the action of groups of Lie type on spherical buildings. On homology this gives an integral form of the Steinberg representation.

One may ask if there exists a complex of lower dimension than the Tits building that realises the (integral) Steinberg representation in this way. I expect that the answer is No, but how to prove it?

More generally, given an integral $G$-representation that can be realised as the homology of a spherical complex with an action of $G$, is there an effective lower bound on the dimension of such a complex? One obvious lower bound is given by the minimal length of a resolution by permutation representations. Is this something that has been studied?

  • 1
    $\begingroup$ If you do not require the complex to be spherical, then any $G$-representation can be realized as $H_1$ of a 2-complex, or as $H_2$ of a simply-connected 3-complex - I got as far as writing this out in detail as an answer before noticing the condition `spherical'. $\endgroup$
    – IJL
    Jul 19, 2018 at 16:24
  • $\begingroup$ @Ian It sounds interesting, even if it does not quite answer the question. I would encourage you to post your construction in an answer, especially if it is not something that is already in the literature, and you already have it written out :-). $\endgroup$ Jul 19, 2018 at 16:48
  • $\begingroup$ I have put up what I wrote. I'm sure that most books about $G$-CW-complexes will contain the result, but I don't know of a good reference offhand. $\endgroup$
    – IJL
    Jul 20, 2018 at 8:50

1 Answer 1


This does not answer Greg's question, but it is related. You can realize any $\mathbb{Z}G$-module you that like as $H_1$ of a based 2-complex, or as $H_2$ of a 3-complex if you insist that the complex should be simply-connected. Furthermore, you can require $G$ to act freely on the complex except for fixing the base point.

Given a $\mathbb{Z}G$-module $M$, take a presentation for $M$, i.e. an exact sequence $F_1\rightarrow F_0\rightarrow M\rightarrow 0$ in which each $F_i$ is a free $\mathbb{Z}G$-module. Now you can realize $F_0$ as the 2nd homology of a wedge of 2-spheres with $G$ acting freely except on the basepoint.

You can attach a disjoint union of 3-balls permuted freely by $G$ (with $H_0$ isomorphic to $F_1$) in such a way that the cellular chain complex is just $F_1\rightarrow F_0\rightarrow 0\rightarrow \mathbb{Z}$, with the given map from degree 3 to degree 2. $H_3$ of this space is of course the kernel of the map $F_1\rightarrow F_0$, while $H_2$ is isomorphic to $M$.

With a wedge of 1-spheres and attached 2-cells everything works the same way (the Hurewicz theorem tells you that $\pi_1$ surjects onto $H_1$), except that the 2-dimensional complex will probably have fundamental group a lot bigger than the abelian group $M$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.