Questions tagged [spectral-sequences]
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388 questions
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Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the ...
- 1,229
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143
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Maps between Leray spectral sequences
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
- 1,227
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Computation of stable homotopy groups of $RP^2$
I would like to compute the first few stable homotopy groups of $RP^2$.
I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is what I computed for ...
- 1,162
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112
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Reference request for Leibniz rule and spectral sequences
Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...
- 41
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Explicit 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$
I would like to know the explicit expression of 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ with $U(1)\equiv \mathbb{R}/\mathbb{Z}$ coefficient, or namely $H^2[...
- 10.5k
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cohomology version of Cartan-Leray spectral sequence that deduces cup product
On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group $\...
- 1,990
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$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?
I was trying to understand this interesting question by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
- 755
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0
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122
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maps between two Leray spectral sequences based on maps on cochain complexes
Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
- 1,227
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
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Deformations of nilpotent parts of superalgebras
I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...
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Some calculations with the Adams spectral sequence and the cobar complex
I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask ...
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Relating bordism invairants in $d$ and $d+2$ dimensions
Are there some relationship between mapping the bordism invairants of eq.1 and eq.2?
$$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$
and
$$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...
- 2,147
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Torsion in K-theory versus torsion in cohomology
Inspired by this question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus_{k=0}^\infty H^{2k} ...
- 8,803
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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 sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 1,219
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358
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Differentials in the Lyndon-Hoschild-Serre Sequence for p=0
I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact ...
- 631
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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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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
- 1,716
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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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Do people still use Massey Products for computations in the Adams Spectral Sequence
Hey everyone,
It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations.
...
- 1,147
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523
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A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
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Are the homology and cohomology Serre spectral sequences dual to each other?
If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
- 8,167
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91
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Continuous maps vs filtrations construction of the Leray spectral sequence
The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...
- 1,227
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234
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How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m}:...
- 1,716
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1
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Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$
Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
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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 ...
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Condition for a map to carry over to Leray spectral sequences
I am trying to understand the conditions for two Leray spectral sequences to be related by a map.
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...
- 1,227
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Does the degeneracy of the Frölicher spectral sequence vary in families?
I would like to know if there are any known examples of families of complex manifolds for which the Frölicher spectral sequence of one fibre degenerates on the $E_m$ page and the spectral sequence of ...
- 1,211
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cohomology module of unit tangent vector bundles over spheres
Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\...
- 2,223
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Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
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Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755
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Adams spectral sequence for loop spaces
Let $X = \Omega_0^3S^3$ a connected component of $\Omega^3S^3$. I am interested in explicit construction of spectral sequence converging to odd prime torsion in homotopy groups of $X$.
There is a ...
- 1,129
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Fibrations with isomorphic Leray-Serre spectral sequences and non-isomorphic cohomology ?
Are there fibrations $F_i \to X_i \to B_i$ $(i=1,2)$ with path-connected bases $B_i$ and connected fibres $F_i$ such that their corresponding Leray-Serre spectral sequences (integral coefficients) are ...
- 16.2k
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What kind of spectral sequences come from double complexes?
Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.
My ...
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Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...
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Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
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Seifert Fibrations and their associated Spectral Sequence
In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into ...
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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, $E, F$ ...
- 12.5k
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How to Compute Transgressions in a Serre Spectral Sequence?
For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...
- 1,108
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A spectral sequence problem in Alejandro Adem's Paper
I am reading Adem's paper Periodic Complexes and Group actions. But I can't give an argument about a statement on spectral sequences.
Suppose you have an orientable fibration of CW-complexes like ...
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Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
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cohomology algebra of braid spaces, configuration spaces
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$...
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Spectral sequence with a column isomorphic to its homology
I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^...
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Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
- 10.5k
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How to compute the cohomology of the general linear group with integral entries
Q: So how does one compute the cohomology groups $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$?
First note that $H^*(GL_n(\mathbf{Z}),\mathbf{Z})$ is isomorphic to $H_B^*(Y/GL_n(\mathbf{Z}),\mathbf{Z})$ (Betti ...
- 7,609
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A version of Leray Hirsch better for local coefficients
Let $F \hookrightarrow E \to B$ be a fibration, and let everything be over a field $k$.
The leray hirsch theorem in its usual form says that the (homology) cohomology spectral sequence degenerates at (...
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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 @...
- 2,995
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Khovanov-Rozansky homology and spectral sequences
In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...
- 1,474
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Construction of Serre Spectral Sequence
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
- 305
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1
answer
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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 $M$ ...
- 235
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0
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Cohomology of fiber bundles with non constant coefficients
Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the ...
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