Surgery up to homotopy equivalence for nonpositively curved manifolds

Abstract:

Let ${M^n}$ be a smooth closed manifold which admits a metric of nonpositive curvature. We show, using a theorem of Farrell and Hsiang, that if $n + k \geqslant 6$, then the surgery obstruction map $\left [ {M \times {D^k},\partial ;G / {\text {TOP}}} \right ] \to L_{n + k}^h\left ( {{\pi _1}M,{w_1}\left ( M \right )} \right )$ is injective, where $L_ * ^h$ are the obstruction groups for surgery up to homotopy equivalence.