Volume 20, issue 3 (2020)

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Relative recognition principle

Renato Vasconcellos Vieira

Algebraic & Geometric Topology 20 (2020) 1431–1486

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.

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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
Received: 26 September 2018
Revised: 21 February 2019
Accepted: 6 March 2019
Published: 27 May 2020
Renato Vasconcellos Vieira
São Paulo