Gromov-Lawson theorem on spin structure

Statement

A closed simply connected manifold of dimension at least 5 satisfies at least one of these two conditions:

• It has a Riemannian metric admitting a spin structure
• It has a Riemannian metric with positive scalar curvature

This is a corollary of the Gromov-Lawson theorem on surgery.

References

• The clasification of simply connected manifolds of positive scalar curvature by Mikhail Gromov and H. Blaine Lawson, Jr., The Annals of Mathematics, 2nd Ser., Vol. 111, No. 3 (May, 1980), pp. 423-434
• Positive scalar curvature with symmetry by Bernhard Hanke
• Simply connected manifolds of positive scalar curvature by Stephan Stolz, The Annals of Mathematics, 2nd Ser., Vol. 136, No. 3 (Nov., 1992), pp. 511-540