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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A sufficient condition for nonabelianness of fundamental groups of differentiable manifolds


Author: Kuo-tsai Chen
Journal: Proc. Amer. Math. Soc. 26 (1970), 196-198
MSC: Primary 57.31
MathSciNet review: 0279822
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Abstract: In this paper we prove that, if $ {H^\gamma }(X)$ denotes the $ r$th deRham cohomology group of a connected manifold $ X$ and if the cup product $ {H^1}(X){ \wedge _R}{H^1}(X) \to {H^2}(X)$ is not injective, then $ {\pi _1}(X)$ is not abelian. As a corollary, if $ {b_r}$ is the $ r$th Betti number, then $ \frac{1}{2}{b_1}({b_1} - 1) > {b_2}$ implies $ {\pi _1}(X)$ being nonabelian.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0279822-7
PII: S 0002-9939(1970)0279822-7
Keywords: Fundamental groups, de Rham cohomology groups, iterated path integrals
Article copyright: © Copyright 1970 American Mathematical Society