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Computation and homotopical applications of induced crossed modules. (English) Zbl 1026.55012
Working on the generalization of the fundamental group to higher dimensions, the main structures are crossed modules, particularly through explicit computations. The present paper explains how the computation of induced crossed modules allows the computation of certain homotopy 2-types (like second homotopy groups) and gives many examples and applications.

##### MSC:
 55P15 Classification of homotopy type 18D35 Structured objects in a category (MSC2010) 20J15 Category of groups 55Q99 Homotopy groups 18G55 Nonabelian homotopical algebra (MSC2010)
##### Keywords:
higher homotopy; crossed modules; induced crossed modules
Gpd; GAP; XMod
Full Text:
##### References:
 [1] Alp, M., 1997. GAP, Crossed Modules, Cat^{1}-groups: Applications of Computational Group Theory, Ph.D. Thesis, University of Wales, Bangor. Available from http://www.informatics.bangor.ac.uk/public/math/research/ftp/theses/alp.ps.gz [2] Alp, M.; Wensley, C.D., Enumeration of cat^{1}-groups of low order, Int. J. algebra comput., 10, 407-424, (2000) · Zbl 1006.18002 [3] Ashley, N.K., 1978. Simplicial T-complexes, Ph.D. Thesis, University of Wales, Bangor. Published as: Simplicial T-complexes: a non-abelian version of a theorem of Dold and Kan, Diss. Math. (Rozprawy Mat.) (1988), 265, 1-61 [4] Atiyah, M., Mathematics in the 20th century, Bull. London math. soc., 34, 1-15, (2002) · Zbl 1022.01007 [5] Breen, L., Théorie de Schreier supérieure, Ann. sci. écol. norm. sup., 25, 465-514, (1992) · Zbl 0795.18009 [6] Brown, K.S., () [7] Brown, R., Homotopy theory, and change of base for groupoids and multiple groupoids, Appl. categ. struct., 4, 175-193, (1996) · Zbl 0859.55001 [8] Brown, R.; Higgins, P.J., On the connection between the second relative homotopy groups of some related spaces, Proc. London math. soc., 36, 193-212, (1978) · Zbl 0405.55015 [9] Brown, R.; Higgins, P.J., The classifying space of a crossed complex, Math. proc. camb. phil. soc., 110, 95-120, (1991) · Zbl 0732.55007 [10] Brown, R.; Huebschmann, J., Identities among relations, (), 153-202 [11] Brown, R.; Spencer, C.B., Double groupoids and crossed modules, Cah. top. Géom. diff., 17, 343-362, (1976) · Zbl 0344.18004 [12] Brown, R.; Spencer, C.B., $$G$$-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. K. ned. akad. wet., 79, 296-302, (1976) · Zbl 0333.55011 [13] Brown, R.; Wensley, C.D., On finite induced crossed modules and the homotopy 2-type of mapping cones, Theory appl. categ., 1, 51-74, (1995) [14] Brown, R.; Wensley, C.D., Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy theory, Theory appl. categories, 2, 3-16, (1996) · Zbl 0856.18011 [15] Dakin, K., 1977. Multiple Compositions for Higher Dimensional Groupoids, Ph.D. Thesis, University of Wales, Bangor [16] The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.3 (2002). Available from http://www.gap-system.org [17] Heyworth, A., Wensley, C.D., Logged rewriting and identities among relators. In: Proceedings of Groups St Andrews 2001 in Oxford (to appear) · Zbl 1049.20020 [18] Higher-dimensional discrete algebra, Available from http://www.informatics.bangor.ac.uk/public/math/research/hdda/ [19] () [20] Jones, D.W., 1984. Polyhedral T-complexes, Ph.D. Thesis, University of Wales, Bangor. Published as: A general theory of polyhedral sets and the corresponding T-complexes, Diss. Math. (Rozprawy Mat.) (1988), 266, 1-110 [21] Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. pure appl. algebra, 24, 179-202, (1982) · Zbl 0491.55004 [22] MacLane, S.; Whitehead, J.H.C., On the 3-type of a complex, Proc. nat. acad. sci., 36, 41-48, (1950) · Zbl 0035.39001 [23] Moore, E.J., 2001. Graphs of Groups: Word Computations and Free Crossed Resolutions, Ph.D. Thesis, University of Wales, Bangor. Available from http://www.informatics.bangor.ac.uk/public/math/research/ftp/theses/moore.ps.gz [24] Nan Tie, G., A Dold-kan theorem for crossed complexes, J. pure appl. algebra, 56, 177-194, (1989) · Zbl 0679.18005 [25] Pride, S.J., 1991. Identities among relations. In: Ghys, E., Haefliger, A., Verjodsky, A. (Eds.), Proc. Workshop on Group Theory from a Geometrical Viewpoint, International Centre of Theoretical Physics, Trieste, 1990, World Scientific, pp. 687-716 · Zbl 0843.20026 [26] Schönert, M. et al., 1997. GAP-Groups, Algorithms, and Programming, 6th edn. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany [27] Whitehead, J.H.C., On adding relations to homotopy groups, Ann. math., 41, 806-810, (1941) · Zbl 0060.41104 [28] Whitehead, J.H.C., Note on a previous paper entitled on adding relations to homotopy groups, Ann. math., 47, 806-810, (1946) · Zbl 0060.41104 [29] Whitehead, J.H.C., Combinatorial homotopy II, Bull. am. math. soc., 55, 453-496, (1949) · Zbl 0040.38801
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