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Abstract In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm by permutation summands. These quotients are of interest because of their close relationship with higher real ‐theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories . These spectra serve as natural equivariant generalizations of connective integral Morava ‐theories. We provide a complete computation of the ‐localized slice spectral sequence of , where is the real sign representation of . To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the ‐based Adams spectral sequence in the category of ‐modules. Furthermore, we provide a full computation of the ‐localized slice spectral sequence of the height‐4 theory . The ‐slice spectral sequence can be entirely recovered from this computation. 

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MUR. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the π_{**}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_{\star}MUR-module structure on the Real Morava K-theories. 

We study modules over the commutative ring spectrum 𝐻𝔽₂∧𝐻𝔽₂, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ_{k} in the category of associative algebras freely kills the higher generators ξ_{k+n}. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative 𝐻𝔽₂∧𝐻𝔽₂-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum. 

We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.