An equivariant Novikov conjecture (Q2640951)

The authors discuss and formulate the correct equivariant generalization of the strong Novikov conjecture. This is a statement that certain G- equivariant signatures (living in suitable equivariant K-groups) are invariant under G-maps of manifolds which, nonequivariantly, are homotopy equivalences (such maps are called pseudoequivalences) preserving orientation. They prove this conjecture for manifolds modeled on a complete Riemannian manifold of nonpositive curvature on which G (a compact Lie group) acts by isometries. They also use the theory of harmonic maps to construct (in some cases) G-maps into such model spaces. Let D be the signature operator in the sense of Atiyah-Singer computed with respect to some G-invariant Riemannian metric. Let \(B\pi\) (M) be the classifying space of the fundamental groupoid of M, as defined by P. May in an appendix to the paper. Conjecture. Let h: \(M\to M'\) be an orientation preserving pseudoequivalence of connected, closed, oriented G-manifolds, and consider the associated commutative diagram. If \(K^ G_*(B\pi (M'))\) is finitely generated over R(G), then the higher G-signatures agree, i.e. \[ h_*\circ (f_ M)_*([D_ M])=(f_{M'})_*([D_{M'}]). \] More generally, suppose X is a G CW complex and \(\psi: M\to X\) and \(\phi\) : M\({}'\to X\) are G maps such that \(\psi\) is G homotopic to \(\phi\circ h\). Then if \(K^ G_*(B\pi (X))\) is finitely generated over R(G) \[ (f_ X)_*\circ \psi ([D_ M])=(f_ X)_*\circ \phi_*([D_{M'}]) \] in \(K^ G_*(B\pi (X))\). There are also localized versions. Theorem. Given data as in the general conjecture and suppose that X is a complete Riemannian manifold of nonpositive curvature on which G acts via isometries. Assume that \(K^*_ G(X)\) is finitely generated. Then \(\psi_*([D_ M])=\phi_*([D_{M'}])\) in \(K^ G_*(X)\).