group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a manifold, the group $MU_\ast(X)$ 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)
(…)
Cobordism cohomology theory, denoted $M O$ for oriented cobordism cohomology , $M U$ for complex cobordism cohomology etc, is the generalized (Eilenberg-Steenrod) cohomology theory represented by the universal Thom spectrum.
This spectrum, also denoted $M U$ is the spectrum is in degree $2 n$ given by the Thom space of the vector bundle that is associated by the defining representation of the unitary group $U(n)$ on $\mathbb{C}^n$ to the universtal $U(n)$-principal bundle:
The periodic complex cobordism theory is given by adding up all the even degree powers of this theory:
There is a canonical orientation? on this obtained from the map
(???)
this is the universal even periodic cohomology theory with orientation
The cohomology ring $M P({*})$ is the Lazard ring which is the universal coefficient ring for formal group laws.
The periodic version is sometimes written $PMU$.
The refinement of cobordism cohomology theory to differential cohomology is differential cobordism cohomology.
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 1 with the complex cobordism ring.
This is Quillen's theorem on MU. (e.g Lurie 10, lect. 7, theorem 1)
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.
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)
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 $MU$ decomoses into the Brown-Peterson spectra.
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
Original articles include
Textbooks accounts include
with an emphasis on the use of $MU$ in the Adams-Novikov spectral sequence, and
Lecture notes include
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)
For further context see also the discussion at
See also
For general discussion of equivariant complex oriented cohomology see at equivariant cohomology – References – Complex oriented cohomology