Questions tagged [spectral-sequences]
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388 questions
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Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?
Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...
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Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
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$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand
I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
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Adams Spectral sequence for computing some $B$-bordism groups
As the title suggests, I'm trying to apply the Adams Spectral sequence to get some insights of the bordism group
$$ \Omega_4(\xi)= \pi_4(M\xi)$$
where $\xi \colon BSpin \times K(D_{2n},1) \to BSO$ is ...
- 2,018
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Leray-Hirsch theorem for Dolbeault cohomology
In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this:
Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
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Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
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Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
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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,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
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Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
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How to Compute Transgressions in a Serre Spectral Sequence?
For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by ...
- 1,108
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Third bordism group of BG, where G is an arbitrary compact Lie group.
Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...
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In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $F$ and $G$ on abelian categories:
\begin{align*}
E_{p,q}^{2}(A)=L_{p}G\circ L_{q}...
- 301
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Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
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Khovanov-Rozansky homology and spectral sequences
In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...
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Is this sequence of Lie algebra cohomology a part of spectral sequence?
There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \...
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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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H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory
Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
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Retrieval of algebra structure from spectral sequence
Suppose we have a spectral sequence of algebras and know that it degenerates at some $E_r$, take for example the cohomology Leray Serre spectral sequence associated to some fibration $F\hookrightarrow ...
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Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?
I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller in the comments, but received no answers. Here I will specialize to an interesting ...
- 1,886
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Mahowald uncertainty outside of homotopy theory
In homotopy theory there is the following informal idea:
The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named ...
user138661
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Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
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Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
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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 ...
- 2,018
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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 sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 1,219
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Can the Bockstein spectral sequence be used to compute cohomology rings ?
If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...
- 2,160
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To compare the total, base and fiber spaces of two fiber bundles
Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...
- 1,367
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Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
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How does one get the short exact sequence in a two-column spectral sequence?
In a two-column double complex, one gets from the associated spectral sequence short exact sequences $0\to E_2^{1,n-1}\to H^n\to E_2^{0,n}\to 0$, where $H^n$ is the cohomology of the total complex, ...
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Conditions under which the preimage of a submanifold in nontrivial in homology
Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...
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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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Convergence of spectral sequences of cohomological type
Following the first chapter of Hatcher's great book "Spectral Sequences in Algebraic Topology", I got into problems with spectral sequences of cohomological type. Fix a ring $R$ once and for all. ...
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Torsion in the integral cohomology of $BPU_{n}$
I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.
We have a fiber sequence of the form $BSU_{...
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Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups
In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:
Let $\mathbb{Z}_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{...
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Third differential in the homology AHSS
I need some guidance in identifying the third differential in the homology AHSS for $\Omega_{\ast}^{\text{Spin}^c}(X)$ in degrees $\leq 4$.
Remember that $\pi_0(M\text{Spin}^c)=\Bbb Z$, $\pi_2(M\...
- 2,018
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Transgressions commute with the Steenrod operations on the base and fiber in a central group extension?
The following sentence is quoted from the paper ON THE COHOMOLOGY OF SPLIT EXTENSIONS by D. J. BENSON AND M. FESHBACH:
In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
...
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Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?)
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
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Elementary computation of direct image sheaves.
I am a physicist and would like to understand the section 1 of
this math paper, which explains how the SYZ conjecture implies topological mirror symmetry. I have some technical problem and would ...
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Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
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Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
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fibrations of classifying spaces - Leray Hirsch Theorem converse
Let $G$ be a topological group and let $H$ be a closed subgroup. Assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces
$$G/H \rightarrow BH \rightarrow ...
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A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence
I've been reading the book 'Cohomology of Number Fields' for years.
But I couldn't check the commutativity of the diagram
on page 126 until now. So I ask for help.
The diagram is induced by taking ...
- 1,013
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Relating bordism generators in d and d+2 dimensions --- an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
- 2,147
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isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?
Let $(C,\partial)$ and $(C',\partial')$ be chain complexes of $R$-modules where $R$ is a (commutative) ring. Let $F$ and $F'$ be finite filtrations of $C$ and $C'$ respectively, i.e., $$\varnothing = ...
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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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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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Hodge Numbers and Leray Spectral Sequence
Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way
$$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 ...
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Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective
I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...
- 1,211
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Basic example of derived descent
I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a ...
- 258
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Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)
We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies 2-...
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