From \(K(n+1)_*(X)\) to \(K(n)_*(X)\) (Q2781319)

This is a contribution to the study of the Morava \(K\)-theories \(K(n)_*\) of \(p\)-local spaces and in particular to vanishing theorems for Morava \(K\)-groups of spaces of finite type. Building on work of \textit{D. C. Ravenel}, \textit{W. S. Wilson} and \textit{N. Yagita} [\(K\)-theory 15, No. 2, 147-199 (1998; Zbl 0912.55002)] and of \textit{A. K. Bousfield} [Contemp. Math. 239, 85-89 (1999; Zbl 0939.55004)] the author obtains the following results:NEWLINENEWLINENEWLINETheorem: Let \(X\) be a space of finite type. Set \(q= 2(p- 1)\) and define the \(\text{mod }q\) support of \(K(n)^*(X)\) as \(S(X, K(n)):= \{m\in\mathbb{Z}/q\mathbb{Z}\mid K(n)^d(X)\neq 0\text{ for some }d\in m+ q\mathbb{Z}\}\). Suppose that \(K(n+ 1)^*(X)\) is sparse, i.e. there is no \(m\in \mathbb{Z}/q\mathbb{Z}\) with \(m,m+1\in S(X, K(n+ 1))\). Then \(S(X, K(n))\subset S(X, K(n+ 1))\).NEWLINENEWLINENEWLINECorollary: If \(X\) is a space of finite type then \(K(n+ 1)^{\text{odd}}(X)= 0\Rightarrow K(n)^{\text{odd}}(X)= 0\).