Cobordism theory



Special and general types

Special notions


Extra structure





MUMU is the universal Thom spectrum for complex vector bundles. It is the spectrum representing complex cobordism cohomology theory. It is the complex analog of MO.

MR cohomology theory, or real cobordism, (Landweber 68, Landweber 69) is the 2\mathbb{Z}_2-equivariant cohomology theory version of MUMU complex cobordism cohomology theory.

The MUM U spectrum

The spectrum denoted MUM U is, as a sequential spectrum, in degree 2n2 n given by the Thom space of the underlying real vector bundle of the complex universal vector bundle: the vector bundle that is associated by the defining representation of the unitary group U(n)U(n) on n\mathbb{C}^n to the U(n)U(n)-universal principal bundle:

MU(2n)=Thom(standardassociatedbundletouniversalbundleEU(n) BU(n)) M U(2n) = Thom \left( standard\;associated\;bundle\;to\;universal\;bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right)

A priori this yields a sequential S2-spectrum, which is then turned into a sequential S 1S^1-spectrum by taking the component spaces in odd degree to be the smash product of the circle S 1S^1 with those in even degree.

This represents a complex oriented cohomology theory and indeed the universal one among these, see at universal complex orientation on MU.

The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:

MP= nΣ 2nMU M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U

The cohomology ring MP(*)M P({*}) is the Lazard ring which is the universal coefficient ring for formal group laws, see at Milnor-Quillen theorem on MU .

The periodic version is sometimes written PMUPMU.


Homotopy groups: Cobordism and Lazard ring

The graded ring given by evaluating complex cobordism theory on the point is both the complex cobordism ring as well as the Lazard ring classifying formal group laws.


Evaluation of MUMU on the point yields the complex cobordism ring, whose underlying group is

π *MUMU *(pt)[x 1,x 2,], \pi_\ast MU \simeq MU_\ast(pt) \simeq \mathbb{Z}[x_1, x_2, \cdots] \,,

where the generator x ix_i is in degree 2i2 i.

This is due to (Milnor 60, Novikov 60, Novikov 62). A review is in (Ravenel theorem 1.2.18, Ravenel, ch. 3, theorem 3.1.5).

The formal group law associated with MUMU as with any complex oriented cohomology theory is classified by a ring homomorphism Lπ (MU)L \longrightarrow \pi_\bullet(MU) out of the Lazard ring.


This canonical homomorphism is an isomorphism

Lπ (MU) L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU)

between the Lazard ring and the MUMU-cohomology ring, hence by theorem with the complex cobordism ring.

This is Quillen's theorem on MU. (e.g Lurie 10, lect. 7, theorem 1)

Universal complex orientation on MUM U

There is a canonical complex orientation on MUMU obtained from the map

ω:P MU(1)MU(P ) \omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty)

For EE an E-infinity ring there is a bijection between complex orientation of EE and E-infinity ring maps of the form

MUE. MU \longrightarrow E \,.

(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)

See also at complex orientation and MU.

MUMU-homology of a manifold: Cobordisms in XX

For XX a manifold, the MUMU-homology group MU *(X)MU_\ast(X) of its underlying homotopy type is the group of equivalence classes of maps ΣX\Sigma \to X from manifolds Σ\Sigma with complex structure on the stable normal bundle, modulo suitable complex cobordisms.

e.g (Ravenel chapter 1, section 2)


MUMU-homology of MUMU: Hopf algebroid structure on dual Steenrod algebra

Moreover, the dual MUMU-Steenrod algebra MU (MU)MU_\bullet(MU) forms a commutative Hopf algebroid over the Lazard ring. This is the content of the Landweber-Novikov theorem.

Nilpotence theorem

Snaith’s theorem

Snaith's theorem asserts that the periodic complex cobordism spectrum is the ∞-group ∞-ring of the classifying space for stable complex vector bundles (the classifying space for topological K-theory) localized at the Bott element β\beta:

PMU𝕊[BU][β 1]. PMU \simeq \mathbb{S}[B U][\beta^{-1}] \,.

pp-Localization and Brown-Peterson spectrum

The p-localization of MUMU decomposes into the Brown-Peterson spectra.



For general discussion of equivariant complex oriented cohomology see at equivariant cohomology – References – Complex oriented cohomology

Relation to CFT

A relation to 2d CFT over Spec(Z) was suggested in

  • Toshiyuki Katsura, Yuji Shimizu, Kenji Ueno, Complex cobordism ring and conformal field theory over \mathbb{Z}, Mathematische Annalen March 1991, Volume 291, Issue 1, pp 551-571 (journal)

Last revised on December 22, 2016 at 05:31:27. See the history of this page for a list of all contributions to it.