An operadic model for a mapping space and its associated spectral sequence. (English) Zbl 1123.55004
M. Mandel has proved that the homotopy category of nilpotent, \(p\)-complete spaces of finite type is equivalent to a full subcategory of the homotopy category of algebras over an \(\overline{\mathbb{F}}_p\)-operad \({\mathcal E}\). Here \(\overline{\mathbb{F}}_p\) denotes the closure of the finite field \(\mathbb{F}_p\). In this paper the authors construct an \({\mathcal E}\)-algebra model for a mapping space \({\mathcal F}(X, Y)\) when \(X\) is a finite simplicial set, \(Y\) is a connected nilpotent simplicial set of finite type and the connectivity of \(Y\) is greater than or equal to the dimension of \(X\). In this case, they also construct a left-half plane spectral sequence with \(E_2\)-term expressed in terms of the left derived functor of the Lannes’ division functor and converging to the cohomology of \({\mathcal F}(X, Y)\).
Reviewer: Jean Claude Thomas (Angers)
MSC:
| 55P48 | Loop space machines and operads in algebraic topology |
| 18D50 | Operads (MSC2010) |
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