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cobordism cohomology theory

Context

Manifolds and cobordisms

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Homology groups

For XX a manifold, the group MU *(X)MU_\ast(X) 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)

(…)

As represented by a spectrum

Cobordism cohomology theory, denoted MOM O for oriented cobordism cohomology , MUM U for complex cobordism cohomology etc, is the generalized (Eilenberg-Steenrod) cohomology theory represented by the universal Thom spectrum.

This spectrum, also denoted MUM U is the spectrum is in degree 2n2 n given by the Thom space of 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 universtal U(n)U(n)-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)

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

There is a canonical orientation? on this 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)

(???)

this is the universal even periodic cohomology theory with orientation

The cohomology ring MP(*)M P({*}) is the Lazard ring which is the universal coefficient ring for formal group laws.

The periodic version is sometimes written PMUPMU.

Differential cohomology refinement

The refinement of cobordism cohomology theory to differential cohomology is differential cobordism cohomology.

Properties

On the point: 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 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.

Theorem

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 1 with the complex cobordism ring.

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

On itself: 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.

Universal complex orientation

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)

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 decomoses into the Brown-Peterson spectra.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyEll EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References

General

Original articles include

  • John Milnor, On the cobordism ring ­Ω \Omega^\bullet and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
  • Sergei Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).
  • Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442.

Textbooks accounts include

with an emphasis on the use of MUMU in the Adams-Novikov spectral sequence, and

  • Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)

Lecture notes include

For further context see also the discussion at

See also

Higher algebra over MUMU

Equivariant theory

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

Revised on April 4, 2014 11:03:31 by Urs Schreiber (89.204.153.232)