Celebratio Mathematica

M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 7–38 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. Republished in Algebraic topology: A student’s guide (1972). MR 139181 Zbl 0108.17705 incollection

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M. F. Atiyah and F. Hirzebruch: “Cohomologie-Operationen und charakteristische Klassen” [Cohomology operations and characteristic classes], Math. Z. 77 : 1 (1961), pp. 149–187. Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday. MR 156361 Zbl 0109.16002 article

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M. F. Atiyah and F. Hirzebruch: “Analytic cycles on complex manifolds,” Topology 1 : 1 (January–March 1962), pp. 25–45. MR 145560 Zbl 0108.36401 article

Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \)-dimensional complex analytic subspace of \( X \). Then \( Y \) defines or “carries” a \( 2k \)-dimensional integral homology class \( y \) of \( X \), although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \). These conditions are that certain cohomology operations should vanish on \( u \), for example \( \mathrm{Sq}^3u = 0 \): they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

M. Atiyah and F. Hirzebruch: “Spin-manifolds and group actions,” pp. 18–28 in Essays on topology and related topics (Mémoires dédiés à Georges de Rham) [Essays on topology and related topics (Memoirs dedicated to Georges de Rham)] (Geneva, 26–28 March 1969). Edited by R. Narasimhan and A. Haefliger. Springer (New York), 1970. MR 278334 Zbl 0193.52401 incollection

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Friedrich E. P. Hirzebruch

Friedrich Hirzebruch (1927–2012)

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