Geometry & Topology, Vol. 2 (1998) Paper no. 5, pages 79--101.

Flag Manifolds and the Landweber-Novikov Algebra

Victor M Buchstaber, Nigel Ray

Abstract. We investigate geometrical interpretations of various structure maps associated with the Landweber-Novikov algebra S^* and its integral dual S_*. In particular, we study the coproduct and antipode in S_*, together with the left and right actions of S^* on S_* which underly the construction of the quantum (or Drinfeld) double D(S^*). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincare duality with respect to double cobordism theory; these lead directly to our main results for the Landweber--Novikov algebra.

Keywords. Complex cobordism, double cobordism, flag manifold, Schubert calculus, toric variety, Landweber-Novikov algebra.

AMS subject classification. Primary: 57R77. Secondary: 14M15, 14M25, 55S25.

DOI: 10.2140/gt.1998.2.79

E-print: arXiv:math.AT/9806168

Submitted to GT on 23 October 1997. (Revised 6 January 1998.) Paper accepted 1 June 1998. Paper published 3 June 1998.

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Victor M Buchstaber, Nigel Ray
Department of Mathematics and Mechanics, Moscow State University
119899 Moscow, Russia
Department of Mathematics, University of Manchester
Manchester M13 9PL, England
Email: buchstab@mech.math.msu.su, nige@ma.man.ac.uk

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