Twisted homotopy theory and the geometric equivariant 1-stem

We develop methods for computing the equivariant homotopy set [M,SV]G, where M is a manifold on which the group G acts freely, and V is a real linear representation of G. Our approach is based on the idea that an equivariant invariant of M should correspond to a twisted invariant of the orbit space M/G. We use this method to make certain explicit calculations in the case dimM=dimV+dimG+1.