Equivariant Anderson duality and Mackey functor duality (Q2816786)

Complex conjugation on \(BU(n)\) determines a \(\mathbb{Z}/2\)-equivariant spectrum structure on \(MU\). The spectrum thus obtained is called the Real cobordism spectrum and denoted by \(M\mathbb{R}\). It has a coefficient \(r_i\) in degree \(i(1+\alpha)\) for every \(i\geq1\) where \(\alpha\) is the one-dimensional sign representation. Let \(k\mathbb{R}(n)\) denote the \(\mathbb{Z}/2\)-spectrum obtained by killing the ideal generated by all the \(r_i\) except for \(i= 2^n-1\); then we have an \(M\mathbb{R}\)-map \(M\mathbb{R} \to k\mathbb{R}(n)\). When setting \(K\mathbb{R}(n)=k\mathbb{R}(n)[v_n^{-1}]\) where \(v_n=r_{2^n-1}\), it defines a \(\mathbb{Z}/2\)-spectrum, called the \(\mathbb{Z}/2\)-equivariant \(n\)th integral Morava \(K\)-theory with reality, and so we have a natural map \(M\mathbb{R}[v_n^{-1}]\to K\mathbb{R}(n)\). Let \(\nabla\) be the equivariant Anderson duality functor. Then, in this paper the author proves that the map \(M\mathbb{R}[v_n^{-1}] \to \Sigma^{-2+2\alpha}\nabla K\mathbb{R}(n)\) induced by a generator \(1 \in \pi_0(\Sigma^{-2+2\alpha} K\mathbb{R}(n))\) factors through \(K\mathbb{R}(n)\) and yields a weak equivalence NEWLINE\[NEWLINE\phi : K\mathbb{R}(n) \rightarrow{\cong} \Sigma^{-2+2\alpha}\nabla K\mathbb{R}(n).NEWLINE\]NEWLINE Moreover, if we put \(KO(n)=K{\mathbb{R}(n)}^{\mathbb{Z}/2}\), then as a corollary we know that \(KO(n)\) also has self-duality. The proof is done using the slice spectral sequences for \(K\mathbb{R}(n)\) and \(\Sigma^{-2+2\alpha}\nabla K\mathbb{R}(n)\) due to \textit{M. A. Hill, M. J. Hopkins} and \textit{D. C. Ravenel} [The slice spectral sequence for the \(C_4\) analog of real \(K\)-theory, {\url arXiv:1502.07611}]. The author proves that \(\phi\) induces an isomorphism between the \(E_2\)-pages of both these spectral sequences and so between all \(i\)th pages of them for \(i\geq 2\). Thus we know that the above equivalence holds.