complex bordism

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complex bordism

The complex bordism spectrum \({\operatorname{MU}}\) is a universal cohomolology theories : complex orientations of a spectrum \(E\) are in one-to-one correspondence with multiplicative maps \({\operatorname{MU}}\to E\).

\({\operatorname{MU}}\) is a Thom spectrum

attachments/Pasted%20image%2020220425175711.png

Relation to complex bordism: attachments/Pasted%20image%2020220422153551.png

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Thom spectrum in Projects/2019 AWS/AWS 2019 Background Definitions, -Quick_Notes/2021-05-10, -Unsorted/basic topics in stable homotopy theory, -Unsorted/complex bordism

cohomolology theories in Projects/2019 AWS MOC, -Unsorted/complex bordism
complex bordism

Relation to complex bordism:
chromatic homotopy theory

Relation to complex cobordism, complex oriented cohomology theories, and FGLs via the Landweber exact functor theorem.
Lazard ring

Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, … (where \(c_{i,j}\) has degree \((i + j − 1)\)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard’s universal ring.
Landweber exactness

MU

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