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Relative recognition principle

Renato Vasconcellos Vieira

Algebraic & Geometric Topology 20 (2020) 1431–1486
Abstract

We prove the recognition principle for relative N–loop pairs of spaces for 3 N . If 3 N < , this states that a pair of spaces homotopy equivalent to CW–complexes (Xc,Xo) is homotopy equivalent to (Y 𝕊N ,HFib(ι)𝕊N1 ) for a functorially determined relative space ι: B Y if and only if (Xc,Xo) is a grouplike 𝒮𝒞¯N–space, where 𝒮𝒞¯N is any cofibrant resolution of the Swiss-cheese relative operad 𝒮𝒞N. The relative recognition principle for relative –loop pairs of spaces states that a pair of spaces (Xc,Xo) homotopy equivalent to CW–complexes is homotopy equivalent to (Y 0,HFib(ι0)) for a functorially determined relative spectrum ι: B Y +1 if and only if (Xc,Xo) is a grouplike –algebra, where is a contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal relative operad Com 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