Celebratio Mathematica — Atiyah

Works authored by Atiyah

Works connected to Kunihiko Kodaira

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M. F. Atiyah: “The signature of fibre-bundles,” pp. 73–84 in Global analysis: Papers in honor of K. Kodaira. Edited by S. Iyanaga and D. C. Spencer. Princeton Mathematical Series 29. University of Tokyo Press, 1969. MR 0254864 Zbl 0193.52302 incollection

For a compact oriented differentiable manifold \( X \) of dimension \( 4k \) the signature (or index) of \( X \) is defined as the signature of the quadratic form in \( H^{2k}(X;\mathbb{R}) \) given by the cup product. Thus \[\operatorname{Sign}(X) = p - q \] where \( p \) is the number of \( + \) signs in a diagonalization of the given quadratic form and \( q \) is the number of \( - \) signs. If \( \operatorname{dim} X \) is not by divisible by 4 one defines \( \operatorname{Sign}(X) \) to be zero. Then one has the the multiplicative formula \[\operatorname{Sign}(X \times Y) = \operatorname{Sign}(X)\cdot\operatorname{Sign}(Y)\] In [Chern, Hirzebruch and Serre 1957] it was proved that this multiplicative formula continues to hold when \( X \times Y \) is replaced by a fibre bundle with base \( X \) and fibre \( Y \) provided that the fundamental group of \( X \) acts trivially us on the cohomology of \( Y \).

In this paper we exhibit examples which show that this restriction on the action of \( \pi_1(X) \) is necessary, and that the signature is not multiplicative in general fibre-bundles.

Shōkichi Iyanaga	Related
Donald Clayton Spencer	Related
Kunihiko Kodaira	Related

M. F. Atiyah: “Obituary: Kunihiko Kodaira,” Bull. London Math. Soc. 31 : 4 (1999), pp. 489–493. MR 1687532 Zbl 0928.01020 article