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Elliptic genera, involutions, and homogeneous spin manifolds. (English) Zbl 0712.57010
The authors study the normalized elliptic genus \(\Phi (X)=\phi (X)/\epsilon^{k/2}\) for 4k-dimensional homogeneous spin manifolds X, and show that they are constant (zero for odd k) modular functions. When \(\Phi\) (X) is a constant it equals the signature of X. For example, on quaternionic projective spaces \(P_ k({\mathbb{H}})\) one finds by this method that \(\Phi (P_ k({\mathbb{H}}))=1\) for k even and vanishes for k odd. The authors derive a general formula for sign(G/H), \(G\supset H\) compact Lie groups, and determine its value in some cases by making use of the theory of involutions on compact Lie groups. The basic tool is a reduction formula relating \(\Phi\) (X) to the value of \(\Phi\) on the self- intersection of the fixed point set of an involution g on X. Under appropriate hypotheses the formula takes the simple form \(\Phi (X)=\Phi (X^ g\circ X^ g)\).
Reviewer: P.Landweber

MSC:
57R20 Characteristic classes and numbers in differential topology
57S25 Groups acting on specific manifolds
57R19 Algebraic topology on manifolds and differential topology
57S17 Finite transformation groups
58J26 Elliptic genera
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