Related papers: Multiplicative structure on Real Johnson-Wilson th…

Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson-Wilson spectrum E(2) to deduce certain non-immmersion results for real projective spaces. ER(n) is a $2^{n+2}(2^n-1)$-periodic…

We consider real spectra, collections of Z/(2)-spaces indexed over Z oplus Z alpha with compatibility conditions. We produce fibrations connecting the homotopy fixed points and the spaces in these spectra. We also evaluate the map which is…

We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to ER(n)^*(pt) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E(n), the cohomology ring ER(n)^*(Z)…

We completely compute the Real Johnson-Wilson cohomology of $\mathbb{C}P^\infty$. Applying techniques from equivariant stable homotopy theory to the Bockstein spectral sequence, we produce permanent cycles and solve extension problems to…

We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory…

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)^*(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2^{n+1}] is ER(n)-orientable (but not…

The completion tower of a nonunital commutative ring is a classical construction in commutative algebra. In the setting of structured ring spectra as modeled by algebras over a spectral operad, the analogous construction is the homotopy…

Let $E\mathbb{R}$ be an even-periodic Real Landweber exact $C_2$-spectrum, and $ER$ its spectrum of fixed points. We compute the $ER$-cohomology of the infinite stunted projective spectra $P_j$. These cohomology groups combine to form the…

This is a survey paper on cohomology theories for $A_\infty$ and $E_\infty$ ring spectra. Different constructions and main properties of topological Andr\'e-Quillen cohomology and of topological derivations are described. We give sample…

Let $k$ be a perfect field of characteristic $p$. Associated to any (1-dimensional, commutative) formal group law of finite height $n$ over $k$ there is a complex oriented cohomology theory represented by a spectrum denoted $E(n)$ and…

We show that the homotopy groups of a connective $E_k$-ring spectrum with an $E_k$-cell attached along a class $\alpha$ in degree $n$ are isomorphic to the homotopy groups of the cofiber of the self-map associated to $\alpha$ through degree…

A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact…

It is demonstrated that when the bundle of 2-forms on a four-dimensional manifold M admits an almost-complex structure any choice of "real + imaginary" subspace decomposition of the bundle defines a conjugation map, as well as a Hermitian…

For strongly even $\mathbb{E}_{\infty}^{C_2}$-rings $E$ we show that any homotopy ring map $\mathrm{MU} \to E^e$ lifts to an $\mathbb{E}_{\rho}$-map $\mathrm{MU}_{\mathbb{R}} \to E$. This refines the Hahn-Shi Real orientations of Lubin-Tate…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…

We study the $K$-theory and Swan theory of the group ring $R[G]$, when $G$ is a finite group and $R$ is any ring or ring spectrum. In this setting, the well-known assembly map for $K(R[G])$ has a companion called the coassembly map. We…

Exceptional points (EPs) are non-Hermitian spectral degeneracies marking a simultaneous coalescence of eigenvalues and eigenvectors. Despite the fact that multiband $n$-fold EPs (EP$n$s) generically emerge as special points on manifolds of…

Homotopy type theory is a version of Martin-L\"of type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this…

We prove that E_n-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by…

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in…