zbMATH — the first resource for mathematics

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

55Q05 Homotopy groups, general; sets of homotopy classes
20F12 Commutator calculus
18G55 Nonabelian homotopical algebra (MSC2010)
Full Text: DOI
[1] Barratt, M.G.; Whitehead, J.H.C., The first non-vanishing group of an (n+1)-ad, (), 417-439, 6 · Zbl 0072.18002
[2] Brown, R.; Loday, J.-L., Excision homotopique en basse dimension, C.R. acad. sci. Paris Sér. I. math., 298, 353-356, (1984) · Zbl 0573.55011
[3] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology, to appear. · Zbl 0622.55009
[4] Brown, R.; Loday, J.-L., Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, (), 176-192, (3) 54 · Zbl 0584.55012
[5] Ellis, G.J., Crossed modules and their higher dimensional analogues, ()
[6] End, W., Groups with projections and applications to homotopy theory, J. pure appl. algebra, 18, 111-123, (1980) · Zbl 0443.55008
[7] Guin-Walery, D.; Loday, J.-L., Obstruction à l’excision en K-théorie algébrique, (), 179-216 · Zbl 0461.18007
[8] Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. pure appl. algebra, 24, 179-202, (1982) · Zbl 0491.55004
[9] Witt, E., Treue darstellung liescher ringe, J. reine angew. math., 177, 152-160, (1937) · JFM 63.0089.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.