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cobordism cohomology theory

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Manifolds and cobordisms

manifolds and cobordisms

Definitions

Genera and invariants

Theorem

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

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Contents

Definition

Cobordism cohomology theory, denoted MO for oriented cobordism cohomology , MU for complex cobordism cohomology etc, is the generalized (Eilenberg-Steenrod) cohomology theory represented by the Thom spectrum.

This spectrum, also denoted MU is the spectrum is in degree 2n given by the Thom space of the vector bundle that is associated by the defining representation of the unitary group U(n) on ℂ n to the universtal 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=∨ n∈ℤΣ 2nMUM P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U

There is a canonical orientation? on this obtained from the map

ω:ℂP ∞→≃MU(1)MU(ℂP ∞)\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(*) is the Lazard ring which is the universal coefficient ring for formal group laws.

Differential cohomology refinement

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

Properties

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:

MU≃𝕊[BU][β −1].MU \simeq \mathbb{S}[B U][\beta^{-1}] \,.

References

for further context see the discussion at

Lecture 5 in

Revised on June 17, 2013 11:47:52 by Urs Schreiber (89.204.155.252)
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