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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
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Abstract

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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

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