nLab MP

Idea

The 2-periodic version of complex cobordism cohomology theory MU is usually denoted $MP$ (e.g. Lurie lecture 18, example 5).

Properties

As a Thom spectrum

$MP$ may be realized as the Thom spectrum of the universal virtual complex bundle of arbitrary rank over the space $B U \times \mathbb{Z}$ (Lurie lecture 18, Example 5).

Landweber exact functor therem

There is an analogue of the Landweber exact functor theorem for even 2-periodic cohomology theories, with MU replaced by MP (Lurie lecture 18, prop. 11).

Related concepts

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

• Jacob Lurie, in Chromatic Homotopy Theory, Lecture series 2010 , Lecture 18 Even periodic cohomology theories (pdf)