nLab MU



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 a homotopy-commutative ring spectrum there is a bijection between complex orientation of EE and ring spectrum 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: bordisms in XX

For XX a manifold or a topological space, 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.

See Ravenel chapter 1, section 2.

For more information, see the article bordism homology theory, which treats the oriented case; the case of (stable almost) complex structure is similar.

MUMU-cohomology of a manifold: cobordisms in XX

MUMU-cohomology groups of a manifold MM can be expressed in terms of bordisms given by proper complex-oriented maps into MM.

For more information, see the article cobordism cohomology theory.

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.

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory\;M(B,f) (B-bordism):

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:




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

On the Chern-Dold character on complex cobordism:

Differential and Hodge-filtered cobordism cohomology

On differential cobordism cohomology (enhancement of cobordism cohomology to differential cohomology):

The notion of Hodge filtered differential complex cobordism theory:

Introduction and survey:

A geometric cocycle model:

Refinement of the Abel-Jacobi map to Hodge filtered differential MU-cobordism cohomology theory:

On Umkehr maps in this context:

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)

Relation to divisors

Relation of complex cobordism cohomology with divisors, algebraic cycles and Chow groups:

Last revised on June 9, 2023 at 17:45:22. See the history of this page for a list of all contributions to it.