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Cores of spaces, spectra, and \(E_\infty\) ring spectra. (English) Zbl 0987.55009
A construction is given generalizing Priddy’s construction of the Brown-Peterson spectrum from the \(p\)-local sphere spectrum by killing odd dimensional homotopy groups [S. Priddy, Math. Z. 173, 29-34 (1980; Zbl 0417.55009)].
The concept of a core of a \(p\)-local space or spectrum is introduced. For \(Y\) \(p\)-local, \((n_0-1)\)-connected and with \(\pi_{n_0}(Y)\) cyclic, a core is an \((n_0-1)\)-connected ‘nuclear’ CW-complex or CW-spectrum \(X\) and a map \(f:X\to Y\) inducing an isomorphism on \(\pi_{n_0}\) and a monomorphism on all homotopy groups. Existence of cores is proved and an example is given showing that cores need not be unique up to equivalence. Open problems are stated, including the classification of all cores of a given \(Y\) and the identification of the class of spaces or spectra \(Y\) for which the core of \(Y\) is unique. It is shown that \(BP\) is the core of \(MU\).
The concept of core is then translated to the category of commutative \(S\)-algebras [A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47 (1997; Zbl 0894.55001)]. The original motivation was to arrive at a construction of \(BP\) as a commutative \(S\)-algebra. The authors explain why this approach does not work: they prove that \(BP\) is not a commutative \(S\)-algebra core of \(MU\). In fact, for \(p=2\), it is shown that \(M(Sp/U)\) is a commutative \(S\)-algebra core of \(MU\).

55P15 Classification of homotopy type
55P42 Stable homotopy theory, spectra
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55S12 Dyer-Lashof operations
55N22 Bordism and cobordism theories and formal group laws in algebraic topology