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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0279822-7
MathSciNet review:
0279822
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Kuo-tsai Chen, An algebraic dualization of fundamental groups, Bull. Amer. Math. Soc. 75 (1969), 1020–1024. MR 260834, DOI https://doi.org/10.1090/S0002-9904-1969-12345-1 ---, Algebras of iterated path integrals, (to appear).
- P. A. Smith, Manifolds with abelian fundamental groups, Ann. of Math. (2) 37 (1936), no. 3, 526–533. MR 1503296, DOI https://doi.org/10.2307/1968475
- Kurt Reidemeister, Kommutative Fundamentalgruppen, Monatsh. Math. Phys. 43 (1936), no. 1, 20–28 (German). MR 1550506, DOI https://doi.org/10.1007/BF01707583
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57.31
Retrieve articles in all journals with MSC: 57.31
Additional Information
Keywords:
Fundamental groups,
de Rham cohomology groups,
iterated path integrals
Article copyright:
© Copyright 1970
American Mathematical Society