#### Volume 20, issue 3 (2020)

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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 $3\le N\le \infty$. If $3\le N<\infty$, this states that a pair of spaces homotopy equivalent to CW–complexes $\left({X}_{c},{X}_{o}\right)$ is homotopy equivalent to $\left({Y}^{{\mathbb{𝕊}}^{N}},HFib{\left(\iota \right)}^{{\mathbb{𝕊}}^{N-1}}\right)$ for a functorially determined relative space $\iota :B\to Y$ if and only if $\left({X}_{c},{X}_{o}\right)$ is a grouplike ${\overline{\mathsc{𝒮}\mathsc{𝒞}}}_{N}$–space, where ${\overline{\mathsc{𝒮}\mathsc{𝒞}}}_{N}$ is any cofibrant resolution of the Swiss-cheese relative operad ${\mathsc{𝒮}\mathsc{𝒞}}_{N}$. The relative recognition principle for relative $\infty$–loop pairs of spaces states that a pair of spaces $\left({X}_{c},{X}_{o}\right)$ homotopy equivalent to CW–complexes is homotopy equivalent to $\left({Y}_{0},HFib\left({\iota }_{0}\right)\right)$ for a functorially determined relative spectrum ${\iota }_{\bullet }:{B}_{\bullet }↗{Y}_{\bullet +1}$ if and only if $\left({X}_{c},{X}_{o}\right)$ is a grouplike ${\mathsc{ℰ}}^{\to }$–algebra, where ${\mathsc{ℰ}}^{\to }$ 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.

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##### 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