Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T17:31:13.486Z Has data issue: false hasContentIssue false
  • English
  • Français

A topological proof of Stallings' theorem on lower central series of groups

Published online by Cambridge University Press:  24 October 2008

Tim. D. Cochran
Tim. D. Cochran
Affiliation:
Mathematical Sciences Research Institute, Berkeley, CA 94720, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

For a topologist, the fundamental group G of a space is usually the most important non-abelian algebraic object of study. However, under many equivalence relationships G is not invariant, so topologists have been led to examine other algebraic objects. In particular, for questions of concordance the lower central series of G seems to play the crucial role. Recall that the lower central series Gn(n = 1,2,...) of G is defined by G1 = G, Gn = [G, Gn_1] for n > 1, and the lower central sequence of G is the sequence of quotients G/Gn.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 465 - 472
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Cappell, S. E. and Shaneson, J.. Link cobordism. Comment. Math. Helv. 55 (1980), 20–49.CrossRefGoogle Scholar
[2]Casson, A. J.. Link cobordism and Milnor's invariant. Bull. London Math. Soc. 7 (1975), 39–40.CrossRefGoogle Scholar
[3]Cochran, T. D.. Slice links in S4. Trans. Amer. Math. Soc. 285 (1984), 398–401.Google Scholar
[4]Cochran, T. D.. Geometric invariants of link cobordism. Comment. Math. Helv. (in the Press).Google Scholar
[5]Conner, P. E. and Floyd, E.. Differentiable periodic maps. Bull. Amer. Math. Soc. 68 (1962), 76–86.CrossRefGoogle Scholar
[6]Demeo, R.. Link cobordism. Duke Math. J. 48 (1981), 23–33.CrossRefGoogle Scholar
[7]Hain, R. M.. Iterated integrals, intersection theory and link groups. Topology (in the Press).Google Scholar
[8]Hillman, J. A.. Alexander Ideals of Links (Springer-Verlag, 1981).CrossRefGoogle Scholar
[9]Kojima, S.. Milnor's µ̅-invariants, Massey products and Whitney's trick in 4-dimensions. Topology Appl. 16 (1983), 43–60.CrossRefGoogle Scholar
[10]Milnor, J.. Isotopy of links. Algebraic Geometry and Topology (Princeton University Press, 1957), pp. 280–306.CrossRefGoogle Scholar
[11]Murasugi, K.. On Milnor's invariant for links. Trans. Amer. Math. Soc. 124 (1966), 94–110.CrossRefGoogle Scholar
[12]Porter, R.. Milnor's µ̅-invariants and Massey products. Trans. Amer. Math. Soc. 257 (1979), 39–71.Google Scholar
[13]Stallings, J.. Homology and central series of groups. J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
[14]Traldi, L.. Milnor's invariants and the completions of link modules. Trans. Amer. Math. Soc. 284 (1984), 401–424.Google Scholar