Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T08:08:39.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Free Crossed Resolutions of Groups and Presentations of Modules of Identities among Relations

Published online by Cambridge University Press:  01 February 2010

Ronald Brown  and
Abdul Razak Salleh
Ronald Brown
Affiliation:
School of Mathematics, University of Wales, Dean Street, Bangor, Gwynedd LL57 1UT, U.K., [email protected] http://www.bangor.ac.uk/~mas010
Abdul Razak Salleh
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D.E., Malaysia, [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The paper gives formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. These formulae are seen as a special case of formulae for extending a partial free crossed resolution of a group, given a partial contracting homotopy of the universal cover of the partial resolution.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 2 , 1999 , pp. 28 - 61
Copyright
Copyright © London Mathematical Society 1999

References

1.Baik, Y. G., Harlander, J. and Pride, S. J., ’The geometry of group extensions’, J. Group Theory 1 (1998) 395–16.Google Scholar
2.Blakers, A. L., ‘Some relations between homology and homotopy groups’, Ann. of Math. 49 (1948) 428–461.CrossRefGoogle Scholar
3.Brown, R., ‘Fibrations of groupoids’, J. Algebra 15 (1970) 103–132.CrossRefGoogle Scholar
4.Brown, R., Topology: a geometric account of general topology, homotopy types and the fundamental groupoid (Horwood, Ellis, Chichester, 1988).Google Scholar
5.Brown, R., ‘Higher order symmetry of graphs’, Bull. Irish Math. Soc. 32 (1994) 46–59.CrossRefGoogle Scholar
6.Brown, R., ‘Homotopy theory, and change of base for groupoids and multiple groupoids’, Applied Categorical Structures 4 (1996) 175–193.CrossRefGoogle Scholar
7.Brown, R., ‘Groupoids and crossed objects in algebraic topology’, Homotopy, Homology and Applications 1 (1999) 1–78.Google Scholar
8.Brown, R., ’Higher dimensional group theory’, 1997. http://www.bangor.ac.uk/~mas010/hdaweb2.htmlGoogle Scholar
9.Brown, R., Heath, P. R. and Kamps, H., ‘Coverings of groupoids and Mayer-Vietoris type sequences, Categorical topology, Proc. Conf. Toledo, Ohio, 1983 (Verlag, Heldermann, Berlin, 1984) 147–162.Google Scholar
10.Brown, R. and Higgins, P. J., ‘On the connection between the second relative homotopy groups of some related spaces’, Proc. London Math. Soc. (3) 36 (1978) 193–212.CrossRefGoogle Scholar
11.Brown, R. and Higgins, P. J., ‘On the algebra of cubes’, J. Pure Appl. Algebra 21 (1981) 233–260.CrossRefGoogle Scholar
12.Brown, R. and Higgins, P. J., ‘Colimit theorems for relative homotopy groups’, J. Pure Appl. Algebra 22 (1981) 11–41.CrossRefGoogle Scholar
13.Brown, R. and Higgins, P. J., ‘Crossed complexes and non-abelian extensions’, Proc. International Conference on Category Theory, Gummersbach, 1981, Lecture Notes in Math. 962 (Springer–Verlag, Berlin/New York, 1982) 39–50.CrossRefGoogle Scholar
14.Brown, R. and Higgins, P. J., ‘Tensor products and homotopies for ω-groupoids and crossed complexes’, J. Pure Appl. Algebra 47 (1987) 1–33.CrossRefGoogle Scholar
15.Brown, R. and Higgins, P. J., ‘Crossed complexes and chain complexes with operators’, Math. Proc. Camb. Phil. Soc. 107 (1990) 33–57).CrossRefGoogle Scholar
16.Brown, R. and Higgins, P. J., ‘The classifying space of a crossed complex’, Math. Proc. Cambridge Phil. Soc. 110 (1991) 95–120.CrossRefGoogle Scholar
17.Brown, R. and Huebschmann, J., ’Identities among relations’, Low dimensional topology, London Math. Soc. Lecture Notes 46 (ed. Brown, R. and Thickstun, T. L., Cambridge University Press, Cambridge, 1982) 153–202.Google Scholar
18.Brown, R. and Porter, T., On the Schreier theory of non-abelian extensions: generalisations and computations’, Proc. Roy. Irish Acad. 96 (1996) 213–227.Google Scholar
19.Brown, R. and Wensley, C. D., ‘On finite induced crossed modules, and the homotopy 2-type of mapping cones’, Theory Appl. Categ. 1 (1995) 54–71.Google Scholar
20.Brown, R. and Wensley., C. D., ‘Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types’, Theory Appl. Categ. 2(1996) 3–16.Google Scholar
21.Carloson, J. F., ‘The cohomology of 2-groups mod 2’. http://www.math.uga.edu/~jfc/groups/cohomology.htmlGoogle Scholar
22.Crowell, R. H., ‘The derived module of a homomorphism’, Adv. Math. 6 (1971) 210–238.CrossRefGoogle Scholar
23.Fox, R. H., ‘Free differential calculus I. Derivations in the free group ring’, Ann. of Math. 57 (1953) 547–560.CrossRefGoogle Scholar
24.Groves, J. R. J., ‘An algorithm for computing homology groups’, J. Pure Appl. Algebra 194(1997)331–361.CrossRefGoogle Scholar
25.Gruenberg, K. W., ‘Resolution by relations’, J. London Math. Soc. 35 (1960) 481–494.CrossRefGoogle Scholar
26.Heyworth, Anne, ‘Applications of rewriting systems and Gröbner bases to computing Kan extensions and identities among relations’, Ph.D. thesis, University of Wales, 1998. http://www.soton.ac.uk/abs/math.CT/9812097Google Scholar
27.Heyworth, Anne and Reinert, Birgit, ‘Applications of Gröbner bases to group rings and identities among relations’, University of Wales Bangor Maths Preprint 99.09. http://www.bangor.ac.uk/ma/research/preprints/99/99_09.html/Google Scholar
28.Heyworth, Anne and Wensley, C. D., ‘Logged Knuth-Bendix procedures and identities among relations’, University of Wales Bangor Maths Preprint 99.07. http://www.bangor.ac.uk/ma/research/preprints/99/99_07.html/Google Scholar
29.Higgins, P. J., ‘Presentations of groupoids, with applications to groups’, Proc. Camb. Phil. Soc. 60 (1964) 7–20.CrossRefGoogle Scholar
30.Higgins, P. J., Categories and groupoids (Van Nostrand, London, 1971).Google Scholar
31.Hog-Angelont, C., Metzler, W. and Sieradski, A. J. (Eds), Two-dimensional homotopy and combinatorial group theory, London Math. Soc. Lecture Note Series 197 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
32.Howie, J., ‘Pullback functors and crossed complexes’, Cahiers Topologie Géom. Différentielle Catég. 20 (1979) 281–295.Google Scholar
33.Hu, S. T., Homotopy theory (Academic Press, New York, 1959).Google Scholar
34.Huebschmann, J., ‘Crossed n-fold extensions and cohomology’, Comment. Math. Helv. 55 (1980)302–314.CrossRefGoogle Scholar
35.Johansson, L., Lambe, L. and Sköldberg, E., ‘Normal forms and iterative methods for constructing resolutions’, Preprint, Stockholm, 1998.Google Scholar
36.Pride, S. J., ‘Identities among relations’, Proc. Workshop on Group Theory from a Geometrical Viewpoint, International Centre of Theoretical Physics, Trieste, 1990 (ed. Ghys, E., Haefliger, A. and Verjodsky, A., World Scientific, Singapore, 1991) 687–716.Google Scholar
37.Schönert, M. et al. , ‘GAP: groups, algorithms, and programming’, 4th edn (Lehrstuhl D für Mathematik, Rheinisch Westfalische Technische Hochschule, Aachen, Germany, 1997).Google Scholar
38.Smith, J., ‘Equivariant Moore spaces’, Algebraic and geometric topology, Proceedings Rutgers 1983, Springer Lecture Notes in Math. 1126 (ed. Ranicki, A., Levitt, N. and Quinn, F., Springer-Verlag, Berlin/New York, 1985) 238–270.CrossRefGoogle Scholar
39.Tonks, A. P., ’Theory and applications of crossed complexes’, PhD thesis, University of Wales, Bangor, 1993. http://www.bangor.ac.uk/ma/research/tonksGoogle Scholar
40.Whitehead, J. H. C., ‘Combinatorial homotopy II’, Bull. Amer. Math. Soc. 55 (1949) 453–496.CrossRefGoogle Scholar