On the group of equivariant self equivalences of free actions (Q1090968)

Let \(q: P\to B\) be a numerable principal G-bundle, and \(F_ G(P)\) (resp. \(E_ G(P))\) the group of unbased (resp., based) G-equivariant self- homotopy equivalence classes under the free G-action on P. In the case where \(B=SZ\) is a suspended 1-connected CW complex and q is classified by \(k\in [Z,G]\), a commutative diagram of exact sequences (of groups except at \(\theta)\) is obtained (to complicated to state here). This extends earlier results of Tsukiyama in the case where P is simply connected. Note the special case where G is 1-connected. Calculations are made in several cases. A further result given here is that \(F_ G(P)\) and \(E_ G(P)\) are finitely presented in the case where B is a 1-connected, and G a path-connected, finite CW complex, and E(B).k is a finite set. The proof relies on the known result for E(X) in case X is a 1-connected finite CW complex.