Relative plus constructions
Guille Carrion Santiago, Jerome Scherer

TL;DR
This paper introduces a functorial relative plus construction in algebraic topology, generalizing Quillen's plus construction using homology theories and Bousfield localization, with applications to fundamental groups and cell attachment.
Contribution
It constructs a functorial relative plus construction for spaces using homology theories, extending previous plus constructions with new functorial properties and clarifying conditions on groups involved.
Findings
Defines a functorial relative plus construction as a Bousfield localization.
Provides a functorial and well-defined counterpart to cell attachment plus construction.
Clarifies the role of strongly R-perfect groups in characteristic zero.
Abstract
Let be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair consisting of a connected space and an -perfect normal subgroup of the fundamental group an -acyclic map inducing the quotient by on the fundamental group. When is an ordinary homology theory with coefficients in a commutative ring with unit , this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen's plus construction. We also clarify the necessity to use a strongly -perfect group in characteristic zero.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Intracranial Aneurysms: Treatment and Complications · Algebraic structures and combinatorial models
