Higher chromatic analogues of twisted \(K\)-theory (Q2299563)

The paper is devoted to constructing so called higher chromatic analogues of twisted \(K\)-theory and to studying some properties of the theory. Let \(X\) be a topological space and \(H\in H^{n+2}(X,\mathbb Z_p)\) be a cohomology class, \(P \to X\) a \(K(\mathbb Z_p,n+1)\)-bundle, \(Q_n = \Sigma^\infty K(\mathbb Z_p,n+1)\) and \(T_n\) the localized spectrum \(L_{K(n)}K(\mathbb Z_p,n+1)_+\) splitting in two parts \(Z \vee Z^\perp\) with a natural map \(i: Z \to \Sigma^\infty K(\mathbb Z_p,n+1)_+\) and the Bockstein homomorphism \(\beta : L_{K(n)}\Sigma^\infty K(\mathbb Z_p,n)_+ \to L_{K(n)}\Sigma^\infty K(\mathbb Z_p,n+1)_+ = T_n\), \(\alpha = \beta\circ i\), \(T_n[\alpha^{-1}]\) the homotopy colimit of the system \(T_n \to L_{K(n)}(A \wedge Z) \to L_{K(n)}(A \wedge A \wedge Z) \to \cdots\), \(\gamma_n: T_n \to T_n[\alpha^{-1}] \to L_{K(n)}T_n[\alpha^{-1}] \simeq R_n\). \(X^H = \Sigma^\infty P_+\wedge_{Q_n} R_n\) the generalized Thom spectra, then the \(H\)-twisted \(R_n\)-(co)homology of \(X\) is the \textit{higher chromatic twisted theories} \(R_{n*}(X,H) = \pi_*(X^H)\) and \(R_n^*(X,H) = \pi_{-*}F_{R_n}(X^H,R_n)\) (Definition 5.2). These theories have many good properties, namely: \(R_{n*}(X,H=0) = R_{n*}(X)\), the induced homomorphism \(R_{n*}(X, f^*H) \to R_{n*}(Y,H)\) for any continuous map \(f: X \to Y\), the cup product homomorphism \(R^*_n(X,H_1) \otimes R^*_n(X,H_2) \to R^*_n(X,H_1+H_2)\), the \(R_n^*(X)\)-module structure on \(R^*_n(X,H)\) (Theorem 6.1), the isomorphism from \(R_1(X,H)\) to the twisted \(p\)-adic \(K\)-theory \(\hat{K}_*(X,H)\) (Theorem 6.2) and the existence of the bar spectral sequence \(\mathrm{Tor}^{s,t}_{R_{n*(Q_n)}}(R_{n*}(P),R_{n*}) \Longrightarrow R_{n*}(X,H)\) (Theorem 6.3). The paper is well exposed.