Relative recognition principle

Vasconcellos Vieira, Renato¹

¹IME-USP, São Paulo, Brazil

Algebraic and Geometric Topology, Tome 20 (2020) no. 3, pp. 1431-1486

Résumé

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)𝕊N−1 ) 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 ℰ→–algebra, where ℰ→ 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.

Détail

DOI : 10.2140/agt.2020.20.1431

Classification : 55P35, 55P48, 55R15, 55P42
Keywords: infinite loop spaces, recognition principle, stable homotopy theory, relative loop spaces, spectra, relative operads, model category theory, operads

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