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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.
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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

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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