Relative recognition principle

Vasconcellos Vieira, Renato

doi:10.2140/agt.2020.20.1431

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Relative recognition principle

Renato Vasconcellos Vieira

Algebraic & Geometric Topology 20 (2020) 1431–1486

DOI: 10.2140/agt.2020.20.1431

Abstract

We prove the recognition principle for relative $N$ –loop pairs of spaces for $3 \leq N \leq \infty$ . If $3 \leq N < \infty$ , this states that a pair of spaces homotopy equivalent to CW–complexes $(X_{c}, X_{o})$ is homotopy equivalent to $(Y^{𝕊^{N}}, HFib {(ι)}^{𝕊^{N - 1}})$ for a functorially determined relative space $ι : B \to Y$ if and only if $(X_{c}, X_{o})$ is a grouplike ${\bar{𝒮 𝒞}}_{N}$ –space, where ${\bar{𝒮 𝒞}}_{N}$ is any cofibrant resolution of the Swiss-cheese relative operad ${𝒮 𝒞}_{N}$ . The relative recognition principle for relative $\infty$ –loop pairs of spaces states that a pair of spaces $(X_{c}, X_{o})$ homotopy equivalent to CW–complexes is homotopy equivalent to $(Y_{0}, HFib (ι_{0}))$ for a functorially determined relative spectrum $ι_{∙} : B_{∙} ↗ Y_{∙ + 1}$ if and only if $(X_{c}, X_{o})$ is a grouplike $ℰ^{\to}$ –algebra, where $ℰ^{\to}$ is a contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal relative operad ${Com}^{\to}$ of continuous homomorphisms of commutative monoids. These principles are proved as equivalences of homotopy categories.

Keywords

infinite loop spaces, recognition principle, stable homotopy theory, relative loop spaces, spectra, relative operads, model category theory, operads

Mathematical Subject Classification 2010

Primary: 55P35, 55P48, 55R15

Secondary: 55P42

References

Publication

Received: 26 September 2018

Revised: 21 February 2019

Accepted: 6 March 2019

Published: 27 May 2020

Authors

Renato Vasconcellos Vieira
IME-USP São Paulo Brazil

Volume 20, issue 3 (2020)