Geodesic
The $\pi$-$\pi$-Theorem for Manifold Pairs
Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 405-416.

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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.
Keywords: surgery obstruction groups, surgery on manifold pairs, normal maps, splitting obstruction groups, $\pi$-$\pi$-theorem.
Mots-clés : homotopy triangulation 
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Yu. V. Muranov; D. Repovš; M. Cencelj. The $\pi$-$\pi$-Theorem for Manifold Pairs. Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 405-416. http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a8/

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