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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.

References [Enhancements On Off] (What's this?)

  • [1] K. T. Chen, An algebraic dualization of fundamental groups, Bull. Amer. Math. Soc. 75 (1969), 1020-1024. MR 0260834 (41:5455)
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  • [4] K. Reidemeister, Kommutative Fundamentalgruppen, Monatsh. Math. Phys. 43(1936), 20-28. MR 1550506

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Keywords: Fundamental groups, de Rham cohomology groups, iterated path integrals
Article copyright: © Copyright 1970 American Mathematical Society

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