group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
$MU$ 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 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:
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:
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 $PMU$.
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 $MU$ on the point yields the complex cobordism ring, whose underlying group is
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.
This canonical homomorphism is an isomorphism
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)
There is a canonical complex orientation on $MU$ obtained from the map
For $E$ an E-infinity ring there is a bijection between complex orientation of $E$ and E-infinity ring maps of the form
(e.g Lurie 10, lect. 6, theorem 8, Ravenel, chapter 4, lemma 4.1.13)
See also at complex orientation and MU.
For $X$ a manifold, 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.
e.g (Ravenel chapter 1, section 2)
(…)
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.
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 $\beta$:
The p-localization of $MU$ decomposes into the Brown-Peterson spectra.
Frank Adams, Stable homotopy theory and generalized homology, Chicago lectures in mathematics, 1974
Peter Landweber, On the complex bordism and cobordism of infinite complexes, Bull. Amer. Math. Soc. Volume 76, Number 3 (1970) (Euclid)
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986
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)
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)
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
A relation to 2d CFT over Spec(Z) was suggested in
Last revised on December 22, 2016 at 05:31:27. See the history of this page for a list of all contributions to it.