Some Topological Aspects of Proper Group Actions; Non-compact Dimension of Groups

Let G be a Lie group and let X be a space. It is believed that G and X are strongly related if there is a proper action of G on X, especially if the orbit space G\X is compact. The paper makes explicit one such relation, namely non-compact dimension. For a field k and a locally compact space X we call nc(X ; k) = inf {/: H'C(X ;k) ^ 0} the non-compact dimension of X (see §2). Let us fix a field k of positive characteristic. Suppose that the Lie group G acts properly on X with compact orbit space. Suppose further that the cohomology algebra H*(X ; k) has finite dimension. Finally we need the technical assumption that the action is /c-free (see Definition 1.5) or admits a freeing (§5), which is satisfied in several cases of interest (Theorem 5.3). Under these hypotheses define the non-compact dimension NC0(G ; k) of the group G by NC0(G ; k) := nc(X ; k). The claim is of course that this number is well defined, that is, it depends only on G and k and not on the particular space X. More generally we obtain the following.