The -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

DOI:
https://doi.org/10.1090/S0002-9947-99-02314-4

Published electronically:
July 26, 1999

MathSciNet review:
1608269

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Abstract | References | Similar Articles | Additional Information

Abstract: Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an -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 -theory of all toric manifolds and certain singular toric varieties.

**1.**D. W. Anderson,*Universal Coefficient Theorems for -theory*, Berkeley notes, (unpublished), (1968).**2.**D. W. Anderson, E. H. Brown and F. P. Peterson,*The Structure of the Spin Cobordism Ring*, Annals of Math.**86**, (1967), 271-298.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 -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

Email:
bahri@rider.edu

**Martin Bendersky**

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

Email:
mbenders@shiva.hunter.cuny.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02314-4

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