On Finite Groups Acting on Acyclic Complexes of Dimension Two

We conjecture that every finite group G acting on a contractible CW-complex X of dimension 2 has at least one fixed point. We prove this in the case where G is solvable, and under this additional hypothesis, the result holds for X acyclic.


Introduction
Let G be a group and A an abelian group. Dicks and Dunwoody ([4, Chapter IV]) proved that for each element ζ of H 1 (G; AG) there exists a G-tree T with finite edge stabilizers, with the property that for each subgroup H of G, the restriction of ζ to H is zero if and only if H fixes a point of T . It is natural to look for analogous geometric explanations of elements of higher cohomology groups; thus, for example, one can ask if for each element ζ of H 2 (G; AG) there exists a contractible 2-dimensional CW -complex X admitting an action of G with finite stabilizers for 2-cells, with the property that for each subgroup H of G, the restriction of ζ to H is zero if and only if H acts trivially on X in some sense, perhaps leaving invariant a subtree of the 1-skeleton of X. The restriction of ζ to any finite subgroup of G is zero, but if a finite group leaves a subtree invariant then it fixes a point. With this motivation, we optimistically conjecture that every finite group G acting on a contractible 2-dimensional CW -complex X has at least one fixed point.
In this note we prove this conjecture in the case where G is solvable. Our argument is based on a classical result of P.A. Smith ([8], [9]), stating that every action of a finite p-group on a finite dimensional Z/p-acyclic CW -complex has a Z/p-acyclic fixed-point set (see [2,ter III] and further developments e.g. in [1], [3], [7]).
In our context, the hypothesis that X has no cells above dimension 2 is essential. It is known that any finite nilpotent group whose order is not a prime power acts on some contractible 3-dimensional CW -complex without fixed points ( [1]).
On the other hand, we shall prove that for a finite solvable group G acting on a 2-dimensional CW -complex X, in order to ensure the existence of a fixed point it suffices to assume that X is acyclic. For X acyclic, however, the condition that G be solvable cannot be removed, because the alternating group A 5 acts on the 2-skeleton of the Poincaré sphere -which is acyclic-without fixed points ( [6]). Recall that the 1-skeleton of the Poincaré sphere is the complete graph on 5 vertices, and the 2-skeleton is obtained by adding 6 pentagonal faces so as to extend the natural action of A 5 on the set of vertices. The fundamental domain of the action is a triangle with angles π/2, π/5, 3π/10, and the 60 copies of this fundamental domain triangulate the 2-skeleton, from which it follows that there are no fixed points. The fundamental group of this space is isomorphic to SL 2 (F 5 ).
Since X being contractible is equivalent to X being simply-connected and acyclic, the question that remains open is: If we add the condition that X be simply-connected, can we delete the condition that G be solvable?

Statement and proof of the result
Let G be a finite group acting on a CW -complex X of dimension 2, and denote by X G the set of fixed points under the action of G. We shall assume that the action is cellular ( [5]); that is, each translation of an open cell is an open cell, and, if a cell is invariant, then it is pointwise fixed. Thus X G is a subcomplex of X. For a subcomplex Y ⊆ X, we denote by C n (X, Y ) the group of relative cellular n-chains of the pair (X, Y ).
Given a nonzero abelian group A, a space X is said to be A-acyclic if H k (X; A) = 0 for all k, where H denotes reduced homology. Recall that the condition H −1 (X; A) = 0 is equivalent to the augmentation homomorphism C 0 (X) ⊗ A → A being surjective, and hence equivalent to X being nonempty.
We prove Theorem 1.1. Let G be a finite solvable group acting on a CWcomplex X of dimension 2. If H * (X; Z) is finite, and the orders of G, H 1 (X; Z) are coprime, then the natural map H * (X G ; Z) → H * (X; Z) is injective.

C. Casacuberta, W. Dicks
Proof: Under our assumptions, the graded group H * (X; Z) is necessarily concentrated in degree 1, since it is free abelian in all other degrees. Moreover, H 1 (X; Z/p) = 0 (and hence X is Z/p-acyclic) for every prime p dividing the order of G.
Since G is solvable, we can find a series of subgroups Then the action of G on X induces an action of G i /G i−1 on We prove inductively that the map H * (X G i ; Z) → H * (X; Z) is a monomorphism for all i = 0, . . . , k. This is trivial for G 0 . Thus suppose that it has been established for G i−1 . Then X G i−1 is Z/p-acyclic for every prime p dividing the order of G. Since the order of G i /G i−1 is a prime p i , applying Smith's Theorem ( [9]) to the action of G i /G i−1 on X G i−1 we obtain, by (1.2), that X G i is Z/p i -acyclic. This tells us in particular that X G i is nonempty and connected. Further, for every abelian group A we have an exact sequence from which we infer that H 2 (X, X G i ; Z/p i ) = 0. But, since X has no cells above dimension 2, the group H 2 (X, X G i ; Z) embeds in the free abelian group C 2 (X, X G i ) and hence it is free abelian itself. This forces H 2 (X, X G i ; Z) = 0, showing that H 1 (X G i ; Z) embeds in H 1 (X; Z). Corollary 1.2. Every action of a finite solvable group G on a Z-acyclic CW -complex X of dimension 2 has at least one fixed point.
Proof: It follows from Theorem 1.1 that the fixed-point set X G is Z-acyclic, so in particular it is nonempty.
Note. Robert Oliver has kindly informed us that an, as yet unpublished, paper by Yoav Segev contains a different proof of Corollary 1.2, with the additional assumption that X be finite.
Acknowledgements. The authors are supported by the DGICYT through grants PB91-0467 and PB90-0719. We are indebted to Enric Ventura for several useful observations in connection with this note.