Flag manifolds and the Landweber–Novikov algebra

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.

Keywords

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

Mathematical Subject Classification

Primary: 57R77

Secondary: 14M15, 14M25, 55S25

References

Forward citations

Publication

Received: 23 October 1997

Revised: 6 January 1998

Accepted: 1 June 1998

Published: 3 June 1998

Proposed: Haynes Miller

Seconded: Gunnar Carlsson, Ralph Cohen

Authors

Victor M Buchstaber
Department of Mathematics and Mechanics Moscow State University 119899 Moscow Russia

Nigel Ray
Department of Mathematics University of Manchester Manchester M13 9PL United Kingdom

Volume 2, issue 1 (1998)

Victor M Buchstaber and Nigel Ray