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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. [X,Y] is the abelian group of morphisms from the spectrum X to the spectrum Y, given (roughly) as homotopy classes of maps. [X,Y]n = [SnX,Y] [X,Y]* is the graded abelian group given as the sum of the groups [X,Y]n. πn(X) = [Sn, X] = [S, X]n is the nth stable homotopy group of X. π*(X) is the sum of the groups πn(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. Xn(Y) = [S, X∧Y]n = [Sn, X∧Y] is the generalized homology of Y, Xn(Y) = [Y, X]−n = [S−nY, X] is the generalized cohomology of Y |
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