# Tagged Questions

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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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89 views

### 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 ...

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235 views

### 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 ...

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**1**answer

167 views

### spectral sequence with non-trivial action on coefficients

Set-up:
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}_2\xrightarrow{p}\mathbb{Z}_2\rightarrow 1$$
Define ...

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109 views

### spectral sequence differential for cobordism

From page 6 of these solutions:
the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the ...

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103 views

### Hochschild-Serre spectral sequence and non-trivial action on coefficients

Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action ...

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263 views

### Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...

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271 views

### Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?

Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:
We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by ...

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218 views

### The spectral sequence of a tower of principal fibrations

Assume we have a tower of fibrations (of simplicial sets, let's say):
$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$
Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...

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127 views

### “Cut-off” of the Adams exact couple

(This question has been asked on Math.StackExchange where it attracted a few upvotes, but - unfortunately - no answer.)
I have been reading Chapter 2. of A. Hatcher's draft of "Spectral Sequences in ...

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530 views

### 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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**1**answer

246 views

### 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 ...

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762 views

### Serre Spectral Sequence of Representations

Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...

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441 views

### 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 ...

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487 views

### 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.
...

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243 views

### spectral sequence for cobordism without leaving smooth category

In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as ...

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**1**answer

774 views

### K(r)-localization and monochromatic layers in the chromatic spectral sequence

While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...

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233 views

### 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 ...

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**1**answer

283 views

### Image of J in the classical Adams Spectral Sequence

Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...

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**1**answer

302 views

### 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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227 views

### Cohomology groups of quotient by finite group

I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question.
Let $G$ be a finite group acting on a manifold $M$ without fixed ...

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426 views

### 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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**1**answer

277 views

### How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$

Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's Spectral Sequences book, and trying to understand ...

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**1**answer

354 views

### 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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350 views

### 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 ...

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370 views

### What is the first interesting matric Toda bracket in the stable homotopy of the sphere?

Feel free to gloss â€˜interestingâ€™ as you see fit. One way:
1. What is the lowest degree matric Toda bracket in $\pi_\ast(S)$ that doesn't contain zero?
By â€˜degreeâ€™ I mean total homotopical ...

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365 views

### Is the 4-line of the E_2 term of the classical Adams spectral sequence known?

In other words:
What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
If the 4-line is not known, how much is known about it?
Here, $\mathcal{A}$ is the 2-primary Steenrod ...

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642 views

### 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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**1**answer

283 views

### 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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347 views

### Profiniteness Condition for Hochschild-Serre Spectral Sequence?

This question may seem elementary to experts but I am quite confused about it:
According to the entry of Lyndonâ€“Hochschildâ€“Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to ...

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683 views

### 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 ...

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940 views

### 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 ...

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391 views

### Computing the homology groups of spaces in a fibration

Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...

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343 views

### Spectral sequence for H-space bundles

Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle.
One ...

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740 views

### 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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528 views

### Multiplicativity in the descent spectral sequence

For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} ...

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466 views

### On the multiplicative structure in spectral sequences.

Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...

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477 views

### Where does the primary obstruction of a fibration show up in its spectral sequence?

Let $f\colon\thinspace E\to B$ be a Serre fibration whose fibre $F$ is $k-1$-connected, $k\geq 1$. Assume $B$ is a connected CW complex. Then the primary obstruction to the existence of a cross ...

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305 views

### 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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### Slick Proof of Kudo Transgression Theorem

The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal ...

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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 ...

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622 views

### Proof of the ''trangression theorem''

Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...

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462 views

### Technology for various models of spectra

There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of ...

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### Spectral sequences: opening the black box slowly with an example

My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...

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428 views

### Leray-Hirsch for HOMOLOGY?

Let $E\to B$ be a fibre bundle. The Leray-Hirsch theorem states under suitable assumptions, the cohomology of $E$ is an $H^*(B)$-module generated by suitable cohomology classes in $E$.
Is there any ...

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772 views

### 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 ...

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396 views

### Spectral sequence for reduced homology

In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is:
If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if
$\tilde E^2_{pq}=\tilde ...

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666 views

### 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} ...

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697 views

### 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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**1**answer

660 views

### 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. ...