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Higher-dimensional crossed modules and the homotopy groups of \((n+1)\)- ads. (English) Zbl 0622.55010
The higher dimensional crossed modules of the title are the crossed n- cubes of groups which are defined in section 1. They are essentially contravariant functors M from the set of subsets of \(<n>=\{1,2,...,n\}\) to the category of groups together with functions \(h: M_ A\times M_ B\to M_{A\cup B}\) \((A,B\subseteq <n>)\) satisfying eleven axioms closely related to commutator identities. In particular, crossed n-cubes of groups are groups for \(n=0\); crossed modules for \(n=1\); and are for \(n=2\) the crossed squares of D. Guin-Waléry and J.-L. Loday [Lect. Notes Math. 854, 179-216 (1981; Zbl 0461.18007)].
J.-L. Loday [J. Pure Appl. Algebra 24, 179-202 (1982; Zbl 0491.55004)] defines the notion of cat\({}^ n\)-group. Succintly, these are groups with n compatible category structures (or, equivalently, \((n+1)\)-fold groupoids such that the last structure is a group). Their importance is Loday’s result (loc.cit.) that \(cat^ n\)-groups model all homotopy \((n+1)\)-types (see also a correction of one lemma in the second author’s paper [J. Lond. Math. Soc., II. Ser. 34, 169-176 (1986; Zbl 0576.55007)].
Loday (loc.cit.) exposes also the joint result with Guin-Waléry that \(cat^ 2\)-groups are equivalent to crossed squares. The present paper proves the important result that \(cat^ n\)-groups are equivalent to crossed n-cubes of groups. Part of the proof involves a generalization of a theorem of W. End [J. Pure Appl. Algebra 18, 111-123 (1980; Zbl 0443.55008)] on groups with n commuting projections.
The generalized Van Kampen Theorem (GVKT) proved by the reviewer and J.-L. Loday [Topology 26, 311-335 (1987; see the preceding review)] involves Loday’s fundamental \(cat^ n\)-group functor from n-cubes of spaces to \(cat^ n\)-groups, and makes colimits of \(cat^ n\)-groups useful for applications in homotopy theory. The reviewer and J.-L. Loday [Proc. Lond. Math. Soc., III. Ser. 54, 176-192 (1987; Zbl 0584.55012)] define universal \(cat^ n\)-groups and apply them with the GVKT to generalize the Blakers-Massey result on the first nonvanishing group of a triad. The current paper defines r-universal crossed n-cubes of groups, and gives some computations of the top group \(M_{<n>}\) of such a crossed n-cube. Further, the GVKT is applied to generalize a result of M. G. Barratt and J. H. C. Whitehead [ibid. 6, 417- 439 (1956; Zbl 0072.180)] on the first nonvanishing homotopy group of an \((n+1)\)-ad. In particular the formalism allows for a presentation of this group in the cases when it may be non-abelian, and when some of the spaces are not simply connected. The main point is that the GVKT specifies an appropriate universal property, and that the axioms for a crossed n-cube dictate further simplifications of the presentation.
Reviewer: R.Brown

MSC:
55Q05 Homotopy groups, general; sets of homotopy classes
20F12 Commutator calculus
18G55 Nonabelian homotopical algebra (MSC2010)
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