Higher-dimensional crossed modules and the homotopy groups of \((n+1)\)- ads.

*(English)*Zbl 0622.55010The 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.

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) |

##### Keywords:

higher dimensional crossed modules; crossed n-cubes of groups; commutator identities; \(cat^ n\)-group; groups with n commuting projections; colimits of \(cat^ n\)-groups
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\textit{G. Ellis} and \textit{R. Steiner}, J. Pure Appl. Algebra 46, 117--136 (1987; Zbl 0622.55010)

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##### References:

[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 |

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