Nonlinearity, proper actions and equivariant stable cohomotopy (Q2437996)

In this paper the author extends the notion of equivariant cohomotopy for compact Lie groups to the case of non compact Lie group actions. The author first presents an example of a discrete group such that all its finite dimensional representations are trivial, which explains why there is a need to do this. The equivariant cohomotopy proposed here is constructed by introducing a cocyle consisting of a four-tuple \((E, F, \ell, c)\) where \(E\) and \(F\) are real \(G\)-Hilbert bundles over \(X\), \(\ell : E \to F\) is a fiberwise real Fredholm operator and \(c : E \to F\) is a (possibly nonlinear) fibrewise map. Besides the maps \(\ell\) and \(c\) above are assumed to satisfy a further condition for localization such that \(\ell +c\) extends to a map \(S^E \to S^F\) between one-point compactification bundles. This allows us to define the \(G\)-equivariant cohomotopy \(\Pi^{[\ell]}_G(X)\) of \(X\) to be the set of perturbations of \(\ell +c\). We then find that this new theory is graded by the group \(KO^0_G(X)\) since it is represented by cocycles \((E, F, \ell)\). The main result of this paper states that if \([\ell]\) represents an equivariant \(KO\)-theoretical class of a Fredholm morphism whose fibrewise index is a trivial virtual vector bundle of dimension \(p\), then there is an isomorphism \(\Pi^{[\ell]}_G(X)\cong \pi^p_G(X)\) which is given by an index.