  Surgery obstruction of a normal map to a simple Poincare 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 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
$LP_*$ and splitting obstruction groups $LS_*$. 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.
