The Kervaire conjecture and the minimal complexity of surfaces

Lvzhou Chen, Purdue University

Abstract: We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain injectivity of the map G->(G*Z)/<> when w is a proper power for arbitrary G. Both results generalize to certain HNN extensions.

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