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Geometric realizations for free quotients

Published online by Cambridge University Press:  09 April 2009

William Jaco
William Jaco
Affiliation:
Department of MathematicsRice UniversityHouston, Texas U.S.A.
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In [7] Lyndon introduced the concept of inner rank for groups. He defined the inner rank of an arbitrary group G to be the upper bound of the ranks of free homomorphic images of G. Both Lyndon and Jaco have shown that the inner rank of the fundamental group of a closed 2-manifold with Euler characteristic 2 − p, p ≧ 0, is [p/2] where [p/2] is the greatest integer ≦ p/2. The proof given by lyndon [8] uses algebraic techniques; whereas, the proof by Jaco [4] is geometrical.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 411 - 418
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Hilton, P. J. and Wylie, S., Homology Theory (Camb. Univ. Press, Cambridge, 1962).Google Scholar
[2]Hocking, J. G. and Young, G. S., Topology, (Addison-Wesley, Reading, Mass., 1961).Google Scholar
[3]Hudson, J. E. P., Piecewise Linear Topology (W. A. Benjamin, New York, 1969).Google Scholar
[4]Jaco, W., ‘Heegaard splitting and splitting homomorphisms’, Trans. A. M. S., Vol. 146 (1969), 365–375.CrossRefGoogle Scholar
[5]Jaco, W., ‘Non-retractible cubes-with-holes’, Michigan Math. J., Vol. 18 (1971), 193–201.CrossRefGoogle Scholar
[6]Kurosh, A. G., The Theory of Groups, Vol. 1, II (Chelsea, New York, N.Y., 1960).Google Scholar
[7]Lyndon, R. C., ‘The equation a2 b2 = c2 in free groups’, Mich. Math. J. 6 (1959), 89–95.CrossRefGoogle Scholar
[8]Lyndon, R. C., ‘Dependence in groups,’ Colloq, Mathe. (Warsaw) XIV (1966), 275–283.CrossRefGoogle Scholar
[9]Markov, A., ‘The insolubility of the problem of homeomorphy’, Dokl. Akad. Nauk SSSR 121 (1958), 218–220.Google Scholar