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Elliptic genera, modular forms over \(KO_ *\), and the Brown-Kervaire invariant. (English) Zbl 0714.57013
In this paper, a refined elliptic genus \(\beta_ q; \Omega_*^{spin}\to KO_*[[q]]\) is introduced. For any spin manifold \(M^ n\), \(\beta_ q[M^ n]\) is shown to be a modular form over \(KO_*\) for \(\Gamma_ 0(2)\). In particular, if \(n=8m+2\), one has \(\beta_ q[M^ n]=a_ 0+a_ 1{\bar \epsilon}+...+a_ m{\bar \epsilon}^ m\) \((a_ i\in {\mathbb{F}}_ 2)\), where \({\bar \epsilon}=\sum q^{(2n-1)^ 2}\). It turns out that an \(a_ 0\) is the Atiyah invariant of \(M^ n\) while \(a_ m=k(M)\) is the KO-part of the Brown-Kervaire invariant. This gives a new expression for k(M) as a KO-characteristic number.
Reviewer: S.Ochanine

MSC:
57R20 Characteristic classes and numbers in differential topology
55N15 Topological \(K\)-theory
57R90 Other types of cobordism
58J26 Elliptic genera
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