Geometry & Topology

Flag manifolds and the Landweber–Novikov algebra

Victor M Buchstaber and Nigel Ray

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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 Poincaré duality with respect to double cobordism theory; these lead directly to our main results for the Landweber–Novikov algebra.

Article information

Source
Geom. Topol., Volume 2, Number 1 (1998), 79-101.

Dates
Received: 23 October 1997
Revised: 6 January 1998
Accepted: 1 June 1998
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882903

Digital Object Identifier
doi:10.2140/gt.1998.2.79

Mathematical Reviews number (MathSciNet)
MR1623426

Zentralblatt MATH identifier
0907.57025

Subjects
Primary: 57R77: Complex cobordism (U- and SU-cobordism) [See also 55N22]
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14M25: Toric varieties, Newton polyhedra [See also 52B20] 55S25: $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx]

Keywords
complex cobordism double cobordism flag manifold Schubert calculus toric variety Landweber–Novikov algebra

Citation

Buchstaber, Victor M; Ray, Nigel. Flag manifolds and the Landweber–Novikov algebra. Geom. Topol. 2 (1998), no. 1, 79--101. doi:10.2140/gt.1998.2.79. https://projecteuclid.org/euclid.gt/1513882903


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References

  • J Frank Adams, Stable Homotopy and Generalized Homology, Chicago Lectures in Mathematics, University of Chicago Press (1974)
  • Martin Aigner, Combinatorial Theory, Springer–Verlag (1979)
  • Anthony Bahri, Martin Bendersky, The KO–theory of toric manifolds, preprint, Rider University (1997)
  • Raoul Bott, Hans Samelson, Application of the theory of Morse to symmetric spaces, American J. Math. 80 (1958) 964–1029
  • Paul Bressler, Sam Evens, Schubert calculus in complex cobordism, Transactions of the AMS 331 (1992) 799–813
  • V M Buchstaber, A B Shokurov, The Landweber–Novikov algebra and formal vector fields on the line, Funktsional Analiz i Prilozhen 12 (1978) 1–11
  • Victor M Buchstaber, Semigroups of maps into groups, operator doubles, and complex cobordisms, from: “Topics in Topology and Mathematical Physics”, S P Novikov, editor, AMS Translations (2) 170 (1995) 9–31
  • Victor M Buchstaber, Nigel Ray, Double cobordism and quantum doubles, preprint, University of Manchester (1997)
  • Michael W Davis, Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417–451
  • A D Elmendorf, The Grassmannian geometry of spectra, J. of Pure and Applied Algebra 54 (1988) 37–94
  • William Fulton, Introduction to Toric Varieties, Annals of Math. Studies, No. 131, Princeton UP (1993)
  • Phillip Griffiths, Joseph Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley and Sons (1994)
  • Howard H Hiller, Geometry of Coxeter Groups, Research Notes in Mathematics, No. 54, Pitman (1982)
  • Christian Kassel, Quantum Groups, Graduate Texts in Mathematics, volume 155, Springer–Verlag (1995)
  • Toshiyuki Katsura, Yuji Shimizu, Kenji Ueno, Complex cobordism ring and conformal field theory over ${\bf Z}$, Mathematische Annalen 291 (1991) 551–571
  • Ian G Macdonald, Notes on Schubert Polynomials, Publications du Laboratoire de Combinatoire et d'Informatique Mathématique, volume 6, Université du Québec a Montréal (1991)
  • Peter Magyar, Bott–Samelson varieties and configuration spaces, preprint, Northeastern University (1996)
  • Sergei P Novikov, Various doubles of Hopf algebras: Operator algebras on quantum groups, and complex cobordism, Uspekhi Akademii Nauk SSSR 47 (1992) 189–190
  • Daniel G Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971) 29–56
  • Nigel Ray, SU and Sp Bordism. PhD thesis, Manchester University (1969)
  • Nigel Ray, On a construction in bordism theory, Proceedings of the Edinburgh Math. Soc. 29 (1986) 413–422
  • Nigel Ray, William Schmitt, Combinatorial models for coalgebraic structures, preprint, University of Memphis (1997)
  • Nigel Ray, Robert Switzer, On $SU\!\times\! SU$ bordism, Oxford Quarterly J. Math. 21 (1970) 137–150
  • Robert E Stong, Notes on Cobordism Theory, Princeton UP (1968)