Brown, Edgar H. jun.; Li, Tian-Jun \(KO\)-obstructions of non-abelian group action on spin manifolds. (English) Zbl 0903.19002 Topology Appl. 80, No. 3, 259-266 (1997). If a compact Lie group \(G\) of positive dimension acts non-trivially on a closed spin manifold \(M\), then by a theorem of M. Atiyah and F. Hirzebruch [Essays Topol. Relat. Top., 18-28 (1970; Zbl 0193.52401)] the \(\widehat A\)-genus of \(M\) vanishes. This can be viewed as a \(K\)-theoretical obstruction for the existence of \(G\)-actions. Now, in the case that \(G\) is non-abelian, the article proves \(KO\)-theoretical obstructions for the existence of effective \(G\)-actions. There are interesting relations to the existence of metrics of positive scalar curvature. Reviewer: Bernd Ammann (Freiburg i.Br.) MSC: 19J35 Obstructions to group actions (\(K\)-theoretic aspects) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) 57S15 Compact Lie groups of differentiable transformations 57N80 Stratifications in topological manifolds Keywords:spin manifolds; real Dirac operator; non-abelian group action; \(KO\)-theory; positive scalar curvature Citations:Zbl 0193.52401 PDFBibTeX XMLCite \textit{E. H. Brown jun.} and \textit{T.-J. Li}, Topology Appl. 80, No. 3, 259--266 (1997; Zbl 0903.19002) Full Text: DOI References: [1] Atiyah, M.; Bott, R., Clifford modules, Topology, 3, 3-38 (1964) · Zbl 0146.19001 [2] M. Atiyah and F. Hirzebruch, Spin manifolds and group actions, in: Essays on Topology and Related Topics (Springer, New York) 18-28.; M. Atiyah and F. Hirzebruch, Spin manifolds and group actions, in: Essays on Topology and Related Topics (Springer, New York) 18-28. [3] Browder, W.; Hsiang, W.-G., \(G\)-actions and the fundamental group, Invent. Math., 65, 425-440 (1982) [4] Gromov, N.; Lawson, H. B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. Inst. Hautes Études Sci., 58, 295-408 (1983) [5] Hitchin, N., Harmonic spinors, Adv. in Math., 14, 1-55 (1974) · Zbl 0284.58016 [6] Lawson, B.; Michelson, M.-J., Spin Geometry (1989), Princeton Univ. Press: Princeton Univ. Press Princeton [7] Lawson, H. B.; Yau, S.-T., Scalar curvature, nonabelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv., 49, 232-244 (1974) · Zbl 0297.57016 [8] O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13, 459-469 (1966) · Zbl 0145.18602 [9] Schoen, R.; Yau, S.-T., Compact group actions and the topology of manifolds with nonpositive curvature, Topology, 18, 361-380 (1979) · Zbl 0424.58012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.