On $π- π$ theorem for manifold pairs with boundaries

Cencelj, M.; Muranov, Yu. V.; Repovš, D.

On $π- π$ theorem for manifold pairs with boundaries

Authors

Cencelj, M.

Muranov, Yu. V.

Repovš, D.

Description

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ 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 $π$-$π$ 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.