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\(KO\)-obstructions of non-abelian group action on spin manifolds. (English) Zbl 0903.19002

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.

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

Citations:

Zbl 0193.52401
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References:

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