On the slice spectral sequence for quotients of norms of Real bordism

In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm $M{U}^{\left(\left(C_{2^n}\right)\right)}$ by permutation summands. These quotients are of interest because of their close relationship with higher real $K$-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories $B{P}^{\left(\left(C_{2^n}\right)\right)}⟨m,m⟩$. These spectra serve as natural equivariant generalizations of connective integral Morava $K$-theories. We provide a complete computation of the $a_{\sigma }$-localized slice spectral sequence of $i_{C_{2^{n-1}}}^{\ast }B{P}^{\left(\left(C_{2^n}\right)\right)}⟨m,m⟩$, where $\sigma$ is the real sign representation of $C_{2^{n-1}}$. To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the $H{F}_2$-based Adams spectral sequence in the category of $H{F}_2\wedge H{F}_2$-modules. Furthermore, we provide a full computation of the $a_{\lambda }$-localized slice spectral sequence of the height-4 theory $B{P}^{\left(\left(C_4\right)\right)}⟨2,2⟩$. The $C_4$-slice spectral sequence can be entirely recovered from this computation.