Brown–Peterson cohomology explained

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by, depending on a choice of prime p. It is described in detail by .Its representing spectrum is denoted by BP.

Complex cobordism and Quillen's idempotent

Brown–Peterson cohomology BP is a summand of MU_(p), which is complex cobordism MU localized at a prime p. In fact MU_(p) is a wedge product of suspensions of BP.

For each prime p, Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ_(p) to itself, with the property that ε([CP^''n'']) is [CP^''n''] if n+1 is a power of p, and 0 otherwise. The spectrum BP is the image of this idempotent ε.

Structure of BP

The coefficient ring

\pi _{*(BP)} is a polynomial algebra over \Z _{(p)} on generators v _{n} in degrees 2 (p ^{n-1)} for n\ge 1 .BP _{*(BP)} is isomorphic to the polynomial ring \pi _{*(BP)[t} _{1,} t _{2,} \ldots] over \pi _{*(BP)} with generators t _{i} in BP _{2 (p ^{i-1)}} ( BP ) of degrees 2 (p ^{i-1)} .(\pi _{*(BP),} BP _{*(BP))} is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres.

BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.

See also

• List of cohomology theories#Brown–Peterson cohomology

References

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