%0 Journal Article
%T On $¦Ğ- ¦Ğ$ theorem for manifold pairs with boundaries
%A M. Cencelj
%A Yu. V. Muranov
%A D. Repov£¿
%J Mathematics
%D 2007
%I arXiv
%X 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.
%U http://arxiv.org/abs/0705.4155v1