nLab
Landweber-Novikov theorem

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

The Landweber-Novikov theorem describes the structure of Hopf algebroid over a commutative base on the dual generalized Steenrod algebra of the complex cobordism cohomology theory spectrum MU.

By Quillen's theorem on MU we have π (MU)L\pi_\bullet(MU) \simeq L, the Lazard ring. Moreover the dual Steenrod algebra is

MU (MU)LBL[b 1,b 2,] MU_{\bullet}(MU) \simeq L B \coloneqq L \otimes \mathbb{Z}[b_1, b_2, \cdots]

where on the right we have the polynomial ring on generators b ib_i in degree 2i2i.

Here (L,LB)(L, L B) carries a natural structure of a Hopf algebroid over a commutative base (and apparently this example is what made Haynes Miller introduce Hopf algebroids in the first place). The Landweber-Novikov theorem asserts that this is the Hopf algebroid structure on the dual MU-Steenrod algebra

(π (MU),MU (MU))(L,LB). (\pi_\bullet(MU), MU_\bullet(MU)) \simeq (L, L B) \,.

The analog for BP is the Adams-Quillen theorem.

References

The original articles are

The Hopf algebroid structure on (L,LB)(L, L B) is reviewed in

The Landweber-Novikov theorem is reviewed in

Last revised on February 5, 2016 at 05:36:02. See the history of this page for a list of all contributions to it.