This is a joint Algebraic Topology / Topology seminar.  We use recent results on the moduli spaces of manifolds, relevant index and surgery theory to study the index-difference map from the space  ${\mathcal R}iem^+(W^d)$ of psc-metrics to the space $\Omega^{d+1}KO$ representing the real $K$-theory.  In particular, we show that the index map induces nontrivial homomorphism in homotopy groups $\pi_k {\mathcal R}iem^+(W^d) \to \pi_k \Omega^{d+1}KO$  once the target groups $\pi_k \Omega^{d+1}KO= KO_{k+d+1}$ are not trivial. This work is joint with J. Ebert and O. Randall-Williams. In this talk, I also plan to discuss some recent applications of those results.