On $\pi -\pi$ theorem for manifold pairs with boundaries

Abstract

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_{\ast }({\pi }_1(Y)\to {\pi }_1(X))$. A well known result of Wall, the so called $\pi$-$\pi$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with ${\pi }_1(X)\cong {\pi }_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $L{P}_{\ast }$ and splitting obstruction groups $L{S}_{\ast }$. In the present paper we formulate and prove for manifold pairs with boundaries the results which are similar to the $\pi$-$\pi$ theorem. We give direct geometric proofs, which are based on the original statements of Wall's results and apply obtained results to investigate surgery on filtered manifolds.