Related papers: Symmetric Powers and Eilenberg--Maclane Spectra

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of…

Let $\mathcal{H}$ be a separable Hilbert space and $P$ be an idempotent on $\mathcal{H}.$ We denote by $$\Gamma_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P\}$$ and $$\Delta_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and…

For a completely regular space $X$ and a non-vanishing self-adjoint closed subalgebra $H$ of $C_B(X)$ which separates points from closed sets in $X$ we construct the Gelfand spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the…

We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based…

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^{\alpha} f(n) : = \sum_{m \neq 0} \frac{e^{2 \pi i\alpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $\alpha \in…

The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On…

Inspired by the work of Hahn-Raksit-Wilson, we introduce a variant of the even filtration which is naturally defined on $\mathbf{E}_{1}$-rings and their modules. We show that our variant satisfies flat descent and so agrees with the…

In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds $O(V; W)$ and $U(V; W)$ of orthogonal and unitary, respectively, maps $V \to V \oplus W$ stably split as a wedge sum of Thom spaces defined…

We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential…

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed…

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or…

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the…

We compute the $C_p$-equivariant dual Steenrod algebras associated to the constant Mackey functors $\underline{\mathbb{F}}_p$ and $\underline{\mathbb{Z}}_{(p)}$, as $\underline{\mathbb{Z}}_{(p)}$-modules. The $C_p$-spectrum…

It is well known that the moduli space of flat connections on a trivial principal bundle MxG, where G is a connected Lie group, is isomorphic to the representation variety Hom(\pi_1(M), G)/G. For a tiling T, viewed as a marked copy of R^d,…

We display a symmetric monoidal equivalence between the stable $\infty$-category of filtered spectra, and quasi-coherent sheaves on $\mathbb{A}^1 / \mathbb{G}_m$, the quotient in the setting of spectral algebraic geometry, of the flat…

We first show the existence of a weight filtration on the equivariant cohomology of real algebraic varieties equipped with the action of a finite group, by applying group cohomology to the dual geometric filtration. We then prove the…

We propose a new method to compute the $C_{2^n}$-equivariant homotopy groups of the Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ as a $RO(C_{2^n})$-graded Green functor using the generalized Tate squares. As an example, we…

For a finite group G and a finite G-CW-complex X, we construct groups H_\bullet(G,X) as the homology groups of the G-invariants of the cellular chain complex C_\bullet(X). These groups are related to the homology of the quotient space X/G…

Given any finite quiver, we consider a complete flag of vector spaces over each vertex. Consider the unipotent invariant subalgebra of the coordinate ring of the filtered quiver representation subspace. We prove that the dimension of the…

We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the…