Twisted K-theory and obstructions against positive scalar curvature metrics

We decompose the twisted index obstruction $\theta(M)$ against positive scalar curvature metrics for oriented manifolds with spin universal cover into a pairing of a twisted $K$-homology with a twisted $K$-theory class and prove that $\theta(M)$ does not vanish if $M$ is an orientable enlargeable manifold with spin universal cover.