## Duke Mathematical Journal

### The fundamental group of manifolds of positive isotropic curvature and surface groups

#### Abstract

In this article, we study the topology of compact manifolds with positive isotropic curvature (PIC). There are many examples of nonsimply connected compact manifolds with PIC. We prove that the fundamental group of a compact Riemannian manifold of dimension at least $5$ with PIC does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface. The proof uses stable minimal surface theory

#### Article information

Source
Duke Math. J. Volume 133, Number 2 (2006), 325-334.

Dates
First available: 21 May 2006

http://projecteuclid.org/euclid.dmj/1148224042

Digital Object Identifier
doi:10.1215/S0012-7094-06-13325-2

Mathematical Reviews number (MathSciNet)
MR2225695

Zentralblatt MATH identifier
1110.53027

#### Citation

Fraser, Ailana; Wolfson, Jon. The fundamental group of manifolds of positive isotropic curvature and surface groups. Duke Mathematical Journal 133 (2006), no. 2, 325--334. doi:10.1215/S0012-7094-06-13325-2. http://projecteuclid.org/euclid.dmj/1148224042.

#### References

• D. B. A. Epstein, Ends'' in Topology of 3-manifolds and Related Topics (Athens, Ga., 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, 110--117.
• A. M. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2) 158 (2003), 345--354.
• M. Gromov and H. B. Lawson Jr., Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math (2) 111 (1980), 209--230.
• W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, 2nd ed., Dover, New York, 1976.
• M. J. Micallef and J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), 199--227.
• M. J. Micallef and M. Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1992), 649--672.
• R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127--142.
• Y. T. Siu and S. T. Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), 189--204.
• J. R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312--334.