On $\pi - \pi$ theorem for manifold pairs with boundaries

TL;DR

This paper extends the classical $$-$$ theorem to manifold pairs with boundaries, providing geometric proofs and applications to surgery on filtered manifolds, enhancing understanding of surgery obstructions in these contexts.

Contribution

It formulates and proves a $$-$$ type theorem for manifold pairs with boundaries, with direct geometric proofs and applications to filtered manifolds.

Findings

Established a $$-$$ theorem for manifold pairs with boundaries.

Provided geometric proofs based on original Wall's results.

Applied results to surgery on filtered manifolds.

Abstract

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.