Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-28T19:59:28.801Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on coverings and Kervaire complexes

Published online by Cambridge University Press:  17 April 2009

Stephen G. Brick
Stephen G. Brick
Affiliation:
Department of Mathematics California, State University at Fresno Fresno, CA 93740, United States of America
Rights & Permissions [Opens in a new window]

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.

In the combinatorial category, a two-complex X is said to be Kervaire if any set of equations modelled on X, over any group, has a solution in a larger group. The Kervaire-Laudenbach conjecture speculates that if H2(X) = 0 then X is Kervaire. We show that the validity of this conjecture would imply that all aspherical two-complexes are Kervaire. In particular, any two-complex homotopically equivalent to a two-manifold (≠ S2, RP2) would be Kervaire. We show that this is indeed the case for certain such two-complexes. We generalise this to staggered two-complexes, and, more generally, one-relator extensions of Kervaire complexes. We obtain similar results for diagrammatically reducible two-complexes. Our proofs make use of covering spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 1 - 21
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Brick, S.G., ‘Normal-convexity and equations over groups’, Invent. math. 94 (1988), 81–104.CrossRefGoogle Scholar
[2]Burns, R.G. and Hale, V.W.D., ‘A note on group rings of certain torsion-free groups’, Canad. Math. Bull. 15 (1972), 441–45.CrossRefGoogle Scholar
[3]Gersten, S.M., ‘Reducible diagrams and equations over groups’, in Essays in group theory (MSRI Publ., Vol. 8), pp. 15–74 (Springer-Verlag, Berlin, Heidelberg, New York, 1987).CrossRefGoogle Scholar
[4]Gersten, S.M., ‘Branched coverings of 2-complexes and diagrammatic asphericity’, Trans. Amer. Math. Soc. 303 (1987), 689–706.CrossRefGoogle Scholar
[5]Howie, J., ‘On pairs of 2-complexes and systems of equations over groups’, J. Reine Angew. Math. 324 (1981), 165–174.Google Scholar
[6]Howie, J., ‘On locally indicable groups’, Math. Z. 180 (1982), 445–461.CrossRefGoogle Scholar
[7]Howie, J., ‘Spherical diagrams and equations over groups’, Proc. Camb. Philos. Soc. 96 (1984), 255–268.CrossRefGoogle Scholar