Complexes with free actions by the group of automorphisms of a free group

We define two finite-dimensional complexes on which the group Aut(Fn) acts freely, and study their homotopy groups. The definitions of these complexes were motivated by lowdimensional pseudo-isotopy theory. So not surprisingly, the complexes are related to a number of spaces arising in algebraic K-theory, and to another in low-dimensional topology. The complexes are denoted by Xn and X alg n . The cells of Xn correspond bijectively to certain pointed graphs which are marked by homotopy equivalences with ∨n i=1 S , the wedge of n circles. Each of these marked graphs is required to have some additional structure which we call a coloring. The action of Aut(Fn) on Xn is defined by precomposing the markings with maps ∨n i=1 S 1 → ∨n i=1 S 1 corresponding to elements of Aut(Fn). The complex X alg n is defined by algebraically mimicing the definition of Xn. There is an equivarient map Xn → X alg n which, after stabalizing, may be a homotopy equivalence. Forgetting the colorings defines an equivarient map from Xn to the sphere complex Sn. This last complex was introduced by Hatcher in [3]. Forgetting basepoints as well gives a map from Xn to Outer space. Outer space was defined by Culer and Vogtmann in [1], and denoted by Xn there. The colorings provide just enough structure to eliminate the fixed points of Sn, which Aut(Fn) acts on, but not freely. From the point of view of pseudo-isotopy theory, the cells of Xn may be thought of as parametrized families of certain handle-body structures of the 4-disk, or equivalently, of smooth functions on D. The colorings of the graphs prevent certain higher singularities from occuring among these functions so that the homotopy groups ofXn resemble a K-theory. There is no such restriction on the singularities with Sn, which may be thought of as a sort of low-dimensional Whitehead space, with a little added structure. The space X = limn→∞Xn (and probably X alg = limn→∞X alg n as well) should perhaps be thought of as a pre-K-theory, with the corresponding K-theory indicated in the remarks at the end of §2.4. The homotopy groups πi(Xn) depend on n, even for i >> n, but seem to be more amenable to computation using elementary techniques than the probably stable homotopy groups of the corresponding K-theory. The homotopy groups of X and X map to the algebraic K-groups of the ring Z, essentially by abelianizing Fn, or, with pseudo-isotopy theory in mind, by suspending the families of smooth functions mentioned above. More specifically, there is a map h : X → V , where V ≡ V (Z) is the space with πn−1(V ) = Kn(Z) defined by Volodin, as in [11]. The complex X n is, in some ways, similar to the unstable Volodin space Vn, but with Aut(Fn) in place of GLn(Z). A difference is that, while Ṽ /St(Z) is acylic, X̃/StN is not. (Here the tildes denote universal covers of a component of V and X. Also, the group StN, introduced by Gersten in [2], and also studied in [7], is a nonabelian version of St(Z),