nLab Landweber-Novikov theorem

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 $\pi_\bullet(MU) \simeq L$, the Lazard ring. Moreover the dual Steenrod algebra is

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

where on the right we have the polynomial ring on generators $b_i$ in degree $2i$.

Here $(L, L B)$ 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

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

The analog for BP is the Adams-Quillen theorem.

Related theorems

• Lazard's theorem

• Quillen's theorem on MU

• universal complex orientation on MU

• Landweber exact functor theorem

References

The original articles are

• Sergei Novikov, The metods of algebraic topology from the viewpoint of cobordism theories, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian).

• Peter Landweber, Cobordism operations and Hopf algebras, Trans. Amer. Math. Soc. 129 (1967), 94–110.

• Frank Adams, Part II.11 of Stable homotopy and generalised homology, 1974

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

• Doug Ravenel, appendix 2 of Complex cobordism and stable homotopy groups of spheres

The Landweber-Novikov theorem is reviewed in

• Doug Ravenel, chapter 4, section 1 of Complex cobordism and stable homotopy groups of spheres