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In this paper we define a variant of Roe algebras for spaces with cylindricalends and use this to study questions regarding existence and classification ofmetrics of positive scalar curvature on such manifolds which are collared onthe cylindrical end. We discuss how our constructions are related to relativehigher index theory as developed by Chang, Weinberger, and Yu and use thisrelationship to define higher rho-invariants for positive scalar curvaturemetrics on manifolds with boundary. This paves the way for classification ofthese metrics. Finally, we use the machinery developed here to give a conciseproof of a result of Schick and the author, which relates the relative higherindex with indices defined in the presence of positive scalar curvature on theboundary.

A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory