The $\pi$-$\pi$-Theorem for Manifold Pairs

The surgery obstruction of a normal map to a simple Poincaré pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(\pi_1(Y)\to\pi_1(X))$. 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 Poincaré 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 the surgery obstruction groups $LP_*$ for manifold pairs and splitting obstruction groups $LS_*$. In the present paper, we formulate and prove for manifold pairs with boundaries results 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.