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A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwise quarter-pinched sectional curvatures and manifolds with positive curvature operator. By the results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC; namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds with PIC. An important open problem has been to understand the fundamental groups of manifolds with PIC. In this paper we prove a new result in this direction. We show that the fundamental group of a compact manifold $M^n$ with PIC, $n \geq 5$, does not contain a subgroup isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$. The techniques used involve minimal surfaces.
@article{10_4007_annals_2003_158_345,
author = {Ailana M. Fraser},
title = {Fundamental groups of manifolds with positive isotropic curvature},
journal = {Annals of mathematics},
pages = {345--354},
publisher = {mathdoc},
volume = {158},
number = {1},
year = {2003},
doi = {10.4007/annals.2003.158.345},
mrnumber = {1999925},
zbl = {1044.53023},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.345/}
}
TY - JOUR AU - Ailana M. Fraser TI - Fundamental groups of manifolds with positive isotropic curvature JO - Annals of mathematics PY - 2003 SP - 345 EP - 354 VL - 158 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.345/ DO - 10.4007/annals.2003.158.345 LA - en ID - 10_4007_annals_2003_158_345 ER -
%0 Journal Article %A Ailana M. Fraser %T Fundamental groups of manifolds with positive isotropic curvature %J Annals of mathematics %D 2003 %P 345-354 %V 158 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.345/ %R 10.4007/annals.2003.158.345 %G en %F 10_4007_annals_2003_158_345
Ailana M. Fraser. Fundamental groups of manifolds with positive isotropic curvature. Annals of mathematics, Tome 158 (2003) no. 1, pp. 345-354. doi: 10.4007/annals.2003.158.345
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