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Topological Methods in Algebraic Geometry


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Introduction Chapter 1: Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter 2: The cobordism ring 5. Pontrjagin numbers 6. The ring /ss(/Omega) /oplus //Varrho 7. The cobordism ring /omega 8. The index of a 4k-dimensional manifold 9. The virtual index Chapter 3: The Todd genus 10. Definiton of the Todd genus 11. The virutal generalised Todd genus 12. The t-characteristic of a GL(q, C)-bundle 13. Split manifolds and splitting methods 14. Multiplicative properties of the Todd genus Chapter 4: The Riemann-Roch theorem for algebraic manifolds 15. Cohomology of Compact complex manifolds 16. Further properties of the (/chi)x characteristics 17. The virtual (/chi)x characteristics 18. Some fundamental theorems of Kodaira 19. The virtual (/chi)x characteristics for algebraic manifolds 20. The Riemann-Roch theorem for algebraic manifolds and complex analytic line bundles 21. The Riemann-Roch theorem for algebraic manifolds and complex analytic vector bundles Appendix 1 by R.L.E. Schwarzenberger 22. Applications of the Riemann-Roch theorem 23. The Riemann-Roch theorem of Grothendieck 24. The Grothendieck ring of continuous vector bundles 25. The Atijah-Singer index theorem 26. Integrality theorems for differentiable manifolds Appendix 2 by A. Borel A spectral sequence for complex analytic bundles Bibliography Index

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About the Author

Biography of Friedrich Hirzebruch Friedrich Hirzebruch was born on October 17, 1927 in Hamm, Germany. He studied mathematics at the University of Munster and the ETH Zurich, under Heinrich Behnke and Heinz Hopf. Shortly after the award of his doctoral degree in 1950, he obtained an assistantship in Erlangen and then a membership at the Institute for Advanced Study, Princeton, followed by an assistant professorship at Princeton University. In 1956 he returned to Germany to a chair at the University of Bonn, which he held until his retirement in 1993. Since 1980 he has been the Director of the Max Planck Institute for Mathematics in Bonn. Hirzebruch's work has been fundamental in combining topology, algebraic and differential geometry and number theory. It has had a deep and far-reaching influence on the work of many others, who have expanded and generalized his ideas. His most famous result is the theorem of Riemann-Roch-Hirzebruch.

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