This paper is well-written, well-organized and well-illustrated. Its purpose is to obtain a complete study of the existence of a metric of positive scalar curvature on a compact manifold \(X\), with boundary \(\partial X\neq\emptyset\), that is a product near \(\partial X\).
The paper was motivated by Theorems 1.1 and 1.2 which point out the relevance of topological constructions for \((X,\partial X)\) in the study of the existence of a metric of positive scalar curvature on \((X,\partial X)\). This involves the double of \(X\), and a proof of Theorems 1.1 is given in Section 6.
The authors develop a series of results relating the existence problem of a metric of positive scalar curvature on the manifold \((X,\partial X)\), which is assumed to be connected, to that for the double of \(X\) in the spin case, through invoking the relative topological K-homology of a pair of classifying spaces, and the \(\alpha\)-invariant. They also deal with the totally non-spin case, and the situation in which \(\partial X=\sqcup_i(\partial X)_i\) is disconnected, which includes Sullivan’s \(\mathbb{Z}/k\mathbb{Z}\)-manifolds.