Homology operations in the topological cyclic homology of a point
Topological Cyclic homology of a point, \(TC(*;p)\) has a commutative \(\mathbb{S}\)-algebra structure. Let \(C_{p^n}\) be the cyclic subgroup of order \(p^n\) of the circle group \(\mathbb{T}\). Then \(TC(*;p)\) is the homotopy inverse limit of a diagram involving the Frobenius and restriction maps \( R,F: \mathbb{S}^{C_{p^n}}\to\mathbb{S}^{C_{p^{n-1}}}\), where \( \mathbb{S}^{C_{p^n}}\) is the \(C_{p^n}\) fixed points of the \(\mathbb{T}\) equivariant sphere spectrum. These spectra are all \(E_{\infty}\) ring spectra. With \(TC^{(1)}(*,p)\) the homotopy inverse limit of \(R,F: \mathbb{S}^{C_{p}}\to \mathbb{S}\), we get canonical maps of \(E_{\infty}\) ring spectra \(TC(*,p)\to TC^{(1)}(*,p)\to \mathbb{S}^{C_{p}}\). The main result of the present paper is a calculation of the Dyer-Lashof operations \(Q^i\) in \(H_*(TC^{(1)}(*,2))\) and \(H_*(\mathbb{S}^{C_2})\), where \(H_*(TC^{(1)}(*,2))\) is mod 2 spectrum homology. As a corollary, the Dyer-Lashof operations \(Q^{2i}\) for \(TC(*,2)\) follow and in particular, they are non-trivial. \(Q^{2i+1}\) are all zero. Moreover, the \(G\)-fixed point of the sphere spectrum \(\mathbb{S}^G\) and the algebraic \(K\)-theory spectrum of the category of finite \(G\)-sets are compared as structured spectra
scientific article; zbMATH DE number 5673942
scientific article