Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T12:41:00.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Basic Commutators and Minimal Massey Products

Published online by Cambridge University Press:  20 November 2018

Roger Fenn  and
Denis Sjerve
Roger Fenn
Affiliation:
University of Sussex, Sussex, England
Denis Sjerve
Affiliation:
University of British Columbia, Vancouver, British Columbia
Rights & Permissions [Opens in a new window]

Extract

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 purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.

A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1119 - 1146
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Buoncristiano, S., Rourke, C. P. and Anderson, B., A geometric approach to homology theory, Lond. Math. Soc. Lecture notes 18 (Cambridge, 1976).CrossRefGoogle Scholar
2. Fenn, R. A. and Sjerve, D., One relator groups satisfying Poincaré duality, to appear in the Pacific Journal.Google Scholar
3. Fox, R., Free differential calculus I, Ann. of Math. 57 (1953), 547–560.Google Scholar
4. Fox, R., Chen, K. T. and Lyndon, R. C., Free differential calculus IV, Annals of Math. 68 (1958), 81–95.Google Scholar
5. Goresky, M. and MacPherson, R., Intersection homology theory, Topology 19 (1980), 135–162.Google Scholar
6. Hall, P., A contribution to the theory of groups of prime power order, Proc. Lond. Math. Soc. Ser 2, 56 (1933), 29–95.Google Scholar
7. Magnus, W., Beziehungen zwischen Gruppen and Idealen in einem speziellen Ring, Math Ann. 111 (1935), 259–280.Google Scholar
8. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (Dover Publications, New York, 1976).Google Scholar
9. Massey, W., Higher order linking numbers, Conference on Algebraic Topology, University of Illinois at Chicago Circle (June, 1968).Google Scholar
10. May, J. P., Matric Massey products, Jour. Alg. 12 (1969), 533–568.Google Scholar
11. Milnor, J., Isotopy of links in Algebraic geometry and topology; a symposium in honor of S. Lefschetz (Princeton Univ. Press, 1957), 280–306.Google Scholar
12. O'Neill, E. J., On Massey products, Pacific Jour. Math. 76 (1978), 123–127.Google Scholar
13. Porter, R., Milnors fi-invariants and Massey products, Transactions of the Amer. Math. Soc 257 (1980), 39–71.Google Scholar
14. Stallings, J., Homology and central series of groups, J. Algebra 2 (1965), 170–181.Google Scholar
15. Turaev, V. G., The Milnor invariants and Massey products — Studies in topology II, Zap. Nanc Sem. Lenin. Ot. Mat. Kogo in Stet. Acad. Nauk SSSR. 66 (1976), 189–203.Google Scholar
16. Whitney, H., On products in a complex, Annals of Math. 39 (1938), 397–432.Google Scholar
17. Whitney, H., Geometric methods in cohomology theory, Proc. Nat. Acad. Sci. 33 (1947), 7–9.Google Scholar
18. Witt, E., Treue Darstellung Lieschen Ringe, Jour, f.r.a. Math. 177 (1937), 152–160.Google Scholar