nLab MU

Idea

$MU$ 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 $\mathbb{Z}_2$-equivariant cohomology theory version of $MU$ complex cobordism cohomology theory.

The $M U$ spectrum

The spectrum denoted $M U$ is, as a sequential spectrum, in degree $2 n$ 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 $U(n)$ on $\mathbb{C}^n$ to the $U(n)$-universal principal bundle:

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

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

$$M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U$$

The cohomology ring $M P({*})$ 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 $PMU$.

Properties

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.

Theorem

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

$$\pi_\ast MU \simeq MU_\ast(pt) \simeq \mathbb{Z}[x_1, x_2, \cdots] \,,$$

where the generator $x_i$ is in degree $2 i$.

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 $MU$ as with any complex oriented cohomology theory is classified by a ring homomorphism $L \longrightarrow \pi_\bullet(MU)$ out of the Lazard ring.

Theorem

This canonical homomorphism is an isomorphism

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

between the Lazard ring and the $MU$-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 $M U$

There is a canonical complex orientation on $MU$ obtained from the map

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

For $E$ a homotopy-commutative ring spectrum there is a bijection between complex orientation of $E$ and ring spectrum maps of the form

$$MU \longrightarrow E \,.$$

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

See also at complex orientation and MU.

$MU$-homology of a manifold: bordisms in $X$

For $X$ a manifold or a topological space, the $MU$-homology group $MU_\ast(X)$ of its underlying homotopy type is the group of equivalence classes of maps $\Sigma \to X$ from manifolds $\Sigma$ 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.

$MU$-cohomology of a manifold: cobordisms in $X$

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

For more information, see the article cobordism cohomology theory.

$MU$-homology of $MU$: Hopf algebroid structure on dual Steenrod algebra

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

Nilpotence 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 $\beta$:

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

$p$-Localization and Brown-Peterson spectrum

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

Related concepts

• differential cobordism cohomology

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

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

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

General

• Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)

• Pierre Conner, Edwin Floyd, Section 12 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Robert Stong, Chapter VII of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)

• Pierre Conner, Larry Smith, On the complex bordism of finite complexes, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (numdam:PMIHES_1969__37__117_0)

• Larry Smith, On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres, American Journal of Mathematics Vol. 92, No. 4 (Oct., 1970), pp. 793-856 (doi:10.2307/2373397)

• Peter Landweber, On the complex bordism and cobordism of infinite complexes, Bull. Amer. Math. Soc. Volume 76, Number 3 (1970) (Euclid)

• Daniel Quillen, Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations, Advances in Mathematics 7 (1971) 29–56 [doi:10.1016/0001-8708(71)90041-7]

• John Frank Adams, Stable homotopy and generalized homology, Chicago Lectures in Mathematics, The University of Chicago Press (1974) [ucp:bo21302708, pdf]

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres (1986)

• Stanley Kochman, Section 4.4 of: Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs, American Mathematical Society, 1996 (cds:2264210)

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, section VIII of Rings, modules and algebras in stable homotopy theory 1997 (pdf)

• Mike Hopkins, section 4 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)

• Dai Tamaki, Akira Kono, Section 3.7 and Chapter 6 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes), Lecture 5 Complex bordism

  (pdf)

• Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes) Lecture 6 MU and complex orientations (pdf)

• Jacob Lurie, Chromatic Homotopy Theory Lecture series (lecture notes), Lecture 7 The homology of MU (pdf)

• Manifold Atlas, Complex bordism

• Neil Strickland, Products on $MU$-modules (pdf)

• Jesse McKeown, Complex Cobordism vs. Representing Formal Group Laws (arXiv:1605.09252)

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

On the Chern-Dold character on complex cobordism:

• Victor Buchstaber, The Chern–Dold character in cobordisms. I,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (mathnet:3530)

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 (doi:10.1070/SM1970v012n04ABEH000939)

• Victor Buchstaber, A. P. Veselov, Chern-Dold character in complex cobordisms and abelian varieties (arXiv:2007.05782)

Differential and Hodge-filtered cobordism cohomology

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

• Ulrich Bunke, Thomas Schick, Ingo Schroeder, Moritz Wiethaup, Landweber exact formal group laws and smooth cohomology theories, Algebr. Geom. Topol. 9 (2009) 1751-1790 [arXiv:0711.1134, doi:10.2140/agt.2009.9.1751]

The notion of Hodge filtered differential complex cobordism theory:

• Michael J. Hopkins, Gereon Quick, §5 in: Hodge filtered complex bordism, Journal of Topology 8 1 (2014) 147-183 [arXiv:1212.2173, doi:10.1112/jtopol/jtu021]

Introduction and survey:

• Gereon Quick, Geometric Hodge filtered complex cobordism, talk at CQTS (March 2023) [video:YT]

A geometric cocycle model:

• Knut B. Haus, Gereon Quick, Geometric Hodge filtered complex cobordism [arXiv:2210.13259]

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

• Gereon Quick, An Abel-Jacobi invariant for cobordant cycles, Documenta Mathematica 21 (2016) 1645–1668 [arXiv:1503.08449]

On Umkehr maps in this context:

• Knut Bjarte Haus, Gereon Quick, Geometric pushforward in Hodge filtered complex cobordism and secondary invariants [arXiv:2303.15899]

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:

• Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467-493 (doi:10.1090/S0894-0347-97-00232-4)

• MO discussion