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Convergence of the homology spectral sequence of a cosimplicial space. (English) Zbl 0864.55017
As the formula \(\pi_0(\Omega X)= \pi_1(X)\) reveals, the components of a mapping space can be interesting. One of the main tools for analyzing the algebraic topology of mapping spaces is the Eilenberg-Moore spectral sequence. However, convergence results for this spectral sequence depend for the most part on assumptions of connectivity, or algebraic properties of the action of the fundamental group. In this paper the Eilenberg-Moore spectral sequence is studied as a special case of the homology spectral sequence of a cosimplicial space (as introduced by D. Rector [Comment. Math. Helv. 45, 540-552 (1970; Zbl 0209.27501)], and greatly generalized in [A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lect. Notes Math. 304 (1972; Zbl 0259.55004)]). Thus convergence results for the more general homology spectral sequence of a cosimplicial space apply to the Eilenberg-Moore spectral sequence of a fibre square. The paper works simplicially and uses the full picture of the closed model category structure on simplicial sets and cosimplicial spaces. The main ideas are the notions of pro-isomorphisms of towers of abelian groups, weak pro-homotopy equivalence, strong convergence in this context, and pro-convergence. Let \(H_nX\) denote \(H_n(X;\mathbb{F}_p)\) where \(p\) is a fixed prime. Let CSA denote the category of cosimplicial simplicial abelian groups. The normalization of a simplicial (cosimplicial) group, \(N_*G_\bullet\) \((N^*B^\bullet)\) is defined by \(N_nG_\bullet= G_n/ \text{im} s_0+ \cdots+ \text{im} s_{n-1}\) \((N^nB^\bullet=B^n \cap\ker s^0\cap \cdots \cap\ker s^{n-1})\). This leads to a double complex \(N^*N_*B^\bullet\), a filtration, and hence, a spectral sequence. If we set \((TB^\bullet)_n= \prod_{m\geq 0} N^mN_{m+n} B^\bullet\), equipped with differential \(\partial_T= \partial_\bullet\pm \delta^\bullet\), then the filtration of \(TB^\bullet\) given by \((F^{m+1} TB^\bullet)_n= \prod_{k\geq m+1} N^kN_{k+n} B^\bullet\) provides a tower \(T_m B^\bullet= TB^\bullet/F^{m+1} TB^\bullet\). To a cosimplicial space \(X^\bullet\) one associates \(R\otimes X^\bullet\) in CSA where \((R\otimes X^\bullet)^m_n\) is the vector space over \(\mathbb{F}_p\) with basis the set \(X_n^m\). The spectral sequence associated to the filtration described has \(E^2_{s,t}\cong \pi^sH_t(X^\bullet)\). There is another tower associated to \(X^\bullet\) given by \(\{\text{Tot}_s(X^\bullet)\}\). This tower maps into the tower \(\{T_s(R\otimes X^\bullet)\}\) and provides a mapping from the constant tower \(\{H_n\text{Tot} (X^\bullet)\}\) to \(\{H_nT_s(R\otimes X^\bullet)\}\). A cosimplicial space \(X^\bullet\) is said to be strongly convergent if this map of towers is a pro-isomorphism for all \(n\), when we replace \(X^\bullet\) by \(\overline X^\bullet\), a fibrant object homotopy equivalent to \(X^\bullet\). A cosimplicial space is called pro-convergent if the map of towers \(\{H_n\text{Tot}_s(\overline X^\bullet)\} \to \{H_nT_s(R\otimes \overline X^\bullet)\}\) is a pro-isomorphism for all \(n\). In §2 of the paper, the author explains how these notions relate to our algebraic characterizations of convergence of spectral sequences. Assume that \(H_*X^s\) is of finite type for all \(s\). The condition that guarantees pro-convergence (Theorem 5.3) is that \(X^s\) be \(p\)-nilpotent for all \(s\) (that is, \(\pi_1X^s\) is nilpotent and \(\pi_iX^s\) is a \(p\)-group with bounded torsion for all \(i)\). The condition that is equivalent to strong convergence (Theorem 6.1) is that \(X^s\) is \(p\)-nilpotent for all \(s\), either \(H_*\text{Tot}(\overline X^\bullet)\) is of finite type or \(\varprojlim H_*\text{Tot}_s (\overline X^\bullet)\) is of finite type and \(\text{Tot}(\overline X^\bullet)\) is \(p\)-good (that is, \(\widetilde H_*(\text{Tot}(\overline X^\bullet))\) is isomorphic to \(\widetilde H_*(R_\infty\text{Tot}(\overline X^\bullet))\) via the completion mapping. With such characterizations of strong convergence and pro-convergence, the author applies these results to the Eilenberg-Moore spectral sequence (§8) and to the \(p\)-resolution of a cosimplicial space (§10). These results go a long way in providing tools for studying the homotopy theory of mapping spaces. Generalizations of results of A. K. Bousfield [Am. J. Math. 109, 361-394 (1987; Zbl 0623.55009)] are obtained.

MSC:
55T20 Eilenberg-Moore spectral sequences
55U10 Simplicial sets and complexes in algebraic topology
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