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Transactions of the American Mathematical Society

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The $KO$-theory of toric manifolds

Authors: Anthony Bahri and Martin Bendersky
Journal: Trans. Amer. Math. Soc. 352 (2000), 1191-1202
MSC (1991): Primary 55N15, 55T15, 14M25, 19L41; Secondary 57N65
Published electronically: July 26, 1999
MathSciNet review: 1608269
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Abstract: Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an $n$-dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the $KO$-theory of all toric manifolds and certain singular toric varieties.

References [Enhancements On Off] (What's this?)

  • 1. D. W. Anderson, Universal Coefficient Theorems for $K$-theory, Berkeley notes, (unpublished), (1968).
  • 2. D. W. Anderson, E. H. Brown and F. P. Peterson, The Structure of the Spin Cobordism Ring, Annals of Math. 86, (1967), 271-298. MR 36:2160
  • 3. Dilip Bayen and Robert R. Bruner, Real Connective $K$-Theory and the Quaternion Group, Transactions of the AMS 348 (1996), 2201-2216. MR 97a:19008
  • 4. V. Buchstaber and N. Ray, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk. 53:2, 135.
  • 5. Michael W. Davis and Tadeusz Januszkiewicz, Convex Polytopes, Coxeter Orbifolds and Torus Actions, Duke Mathematical Journal 62 (1991), 417-451. MR 92i:52012
  • 6. Dale Husemoller, Fibre Bundles, Springer-Verlag, Berlin (1975). MR 51:6805
  • 7. A. Liulevicius, The Cohomology of a Subalgebra of the Steenrod Algebra, Transactions of the AMS 104 (1962), 443-449. MR 26:6964
  • 8. Robert Morelli, The K-Theory of a Toric Variety, Advances in Mathematics 100 (1993), 154-182. MR 94j:14047
  • 9. Robert E. Stong, Notes on Cobordism Theory, Princeton University Press, (1968). MR 40:2108

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Additional Information

Anthony Bahri
Affiliation: Department of Mathematics, Rider University, Lawrenceville, New Jersey 08648

Martin Bendersky
Affiliation: Department of Mathematics, Hunter College, New York, New York 10021

Keywords: Toric manifolds, toric varieties, $KO$-theory, Adams spectral sequence.
Received by editor(s): September 17, 1997
Published electronically: July 26, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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