List of Cohomology Theories - Notation

Notation

• S = π = S0 is the sphere spectrum.
• Sn is the spectrum of the n-dimensional sphere
• SnY = Sn∧Y is the nth suspension of a spectrum Y.

• is the abelian group of morphisms from the spectrum X to the spectrum Y, given (roughly) as homotopy classes of maps.

• _n =
• _* is the graded abelian group given as the sum of the groups _n.
• π_n(X) = = _n is the nth stable homotopy group of X.
• π_*(X) is the sum of the groups π_n(X), and is called the coefficient ring of X when X is a ring spectrum.
• X∧Y is the smash product of two spectra.

If X is a spectrum, then it defines generalized homology and cohomology theories on the category of spectra as follows.

• X_n(Y) = _n = is the generalized homology of Y,
• Xn(Y) = _−n = is the generalized cohomology of Y