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Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology

The analogue of the Künneth formula for Borel $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback $\require{AMScd}$ \begin{CD} (X \times Y)_G @...
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Tensor product of spectral sequences?

I'm wondering about a cross product for spectral sequences. I've got an idea, and I wonder if it is written up anywhere, or if it even holds water. Let's start with three spectral sequences, $E, F$ ...
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Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
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An example computation of etale cohomology

(edited for clarity) In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
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Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
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Spectral sequence of a bicomplex equipped with a group action

Let $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ be complexes of $\mathbb{C}$-vector spaces. We suppose that $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ are equipped with ...
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completion and convergence of spectral sequence

I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...
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Transgression in terms of k-invariant for chain complexes

I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ ...
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Differentials in the Adams Spectral Sequence for spheres at the prime p=2

How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only ...
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Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set of such a polynomial gives a real curve ...
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generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation

The Atiyah-Hirzebruch spectral sequence \begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*} computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...
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Spectral sequence and HOM functor

I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules ...
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What is the abutment filtration of the second spectral sequence of hypercohomology?

I have been recently learning about spectral sequences, following mainly Illusie's notes and EGA, and I am about to write some expository notes, but there are still some points that I was not able to ...
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On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces, \begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
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stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequenceE^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
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extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequenceE^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
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Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them. It can be shown that $H^d[U(1), Z]$ is $Z$ for ...
Consider an extension$$1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1$$Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action \$a\...