Let \(\pi\) be a finitely presented group, and \({\omega}: {\pi}\to {\mathbb Z}/2\) be an orientation character. Given a closed topological \(n\)-manifold \(X\), \(n\geq 5\), with the fundamental group \(\pi\) and orientation character \(\omega\), let \(C_n(X,{\omega})\) denote the subgroup of the Wall surgery obstruction group \(L_n({\mathbb Z}{\pi},{\omega})\) which is the image of the map \({\sigma}_n(X): [X,G/TOP]\to L_n({\mathbb Z}{\pi},{\omega})\) in the surgery exact sequence. In other words, \(C_n(X, {\omega})\) is given by the surgery obstructions of all degree \(1\) normal maps \((f,b): M\to X\) for closed \(n\)-manifolds \(M\). The closed manifold subgroup \(C_n({\pi}, {\omega})\) is defined as the subgroup of \(L_n({\mathbb Z}{\pi},{\omega})\) generated by all of the closed manifold subgroups \(C_n(X,{\omega})\).
Similarly, for a fixed \((X,{\omega})\), let \(I_{n+1}(X, {\omega})\) denote the image of the map \({\sigma}_{n+1}(X)\) in the surgery exact sequence, \({\sigma}_{n+1}(X): [{\Sigma}(X),G/TOP]\to L_{n+1}({\mathbb Z}{\pi}, {\omega})\). The elements of \(I_{n+1}(X, {\omega})\) may be interpreted as surgery obstructions of relative degree \(1\) normal maps \((f,b): (W,\partial W)\to (X\times I, X\times \partial I)\) where restricts to a homeomorphism on the boundary \(\partial W\). The inertia subgroup \(I_{n+1}({\pi}, {\omega})\subset L_{n+1}({\mathbb Z}{\pi}, {\omega})\) is generated by all of the subgroups \(I_{n+1}(X, {\omega})\) for various choices of \(X\) with the orientation character \(\omega\).
The main result of the paper is that the inertia subgroup \(I_{n+1}({\pi}, {\omega})\) equals the closed manifold subgroup \(C_{m+1}({\pi}, {\omega})\) for all \(n\geq 5\). This result holds for the simple surgery obstructions in \(L^s_{n+1}({\mathbb Z}{\pi}, {\omega})\), as well as for the \(L^h\) and \(L^p\) versions. It was previously shown by the author in [Lect. Notes Math. 967, 101–131 (1982; Zbl 0503.57018)] that the images of these two subgroups are equal in the projective surgery obstruction groups \(L^p_{n+1}({\mathbb Z}{\pi}, {\omega})\) for finite groups \(\pi\). The main result stated above answers the question posed in this reference in 1980.