Mathematics – Geometric Topology
Scientific paper
2007-05-29
Mathematical Notes 81: 3-4 (2007), 356-364
Mathematics
Geometric Topology
Scientific paper
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.
Cencelj Matija
Muranov Yu. V.
Repovš Dušan
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