Vol. 107, No. 2, 1983

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There are no phantom cohomology operations in K-theory

Donald Werner Anderson

Vol. 107 (1983), No. 2, 279–306

Let h1, h2 be cohomology theories defined on the category of finite CW-complexes, and suppose that h1(point), h2(point) are both countable. Then by Brown’s [5] Representability Theorem, there are Ω-spectra Z1, Z2 such that h1(X) = [X,Zi], the graded group of homotopy classes of maps of X into the terms of the spectrum. If we exercise some care in the choice of Z1, we shall see that every stable cohomology operation φ : h1(X) h2(X) defined for X finite extends to a map φ : Z1 Z2 of spectra. We shall examine the question: How many choices, up to homotopy, are there for φ, given φ? As an intermediate question, we shall also investigate: How many extensions are there to infinite CW-complexes are there of φ?

Mathematical Subject Classification 2000
Primary: 55S25
Secondary: 55P42, 55T25
Received: 16 May 1977
Revised: 20 September 1981
Published: 1 August 1983
Donald Werner Anderson