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2
votes
1answer
359 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 ...
2
votes
1answer
273 views

First cohomology of the space of long knots in R^4

Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...
2
votes
1answer
893 views

Generalized Beilinson spectral sequences

Assume we are workling on $\mathbb{P}^n$ for some $n\geq 1$ and we have a coherent sheaf $F$ on it. Then there are two (well known?) spectral sequences $E_r^{p,q}$ with $E_1$-term: ...
0
votes
0answers
299 views

spectral sequence of cohomology with compact support

Let k be a field, and A, B is subscheme of C over k, F is a sheaf over C. Question 1. what is the relation between $H_c(A\cap B,k)$, $H_c(B,F)$, and $H_c(A\cap B,F)$ ? Question 2. Let A be a strata ...
5
votes
2answers
889 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 ...
14
votes
1answer
651 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} ...
6
votes
0answers
553 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 ...
2
votes
0answers
760 views

Grothendieck spectral sequence [duplicate]

Possible Duplicate: Composing left and right derived functors Hi, probably this question is obvious. I apologize for this. Given functors $F$ and $G$ left exact, with as good properties as ...
6
votes
2answers
778 views

How does one view the De Rham spectral sequence as a Grothendieck spectral sequence?

I was rereading basic results on de Rham cohomology, and this led me inevitably to the fact that $H^q(X,\Omega^p)$ converges to $H^*(X)$ for any smooth proper variety (over any field). How does one ...
8
votes
1answer
561 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 ...
6
votes
0answers
354 views

The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra

Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak ...
3
votes
1answer
644 views

Special case of Leray spectral sequence

I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the ...
2
votes
2answers
470 views

Vanishing cohomology of line bundles on the Springer resolution

My question is regarding Broer's paper "Line bundles on the cotangent bundle of the flag variety" (see http://www.springerlink.com/content/t41418q436524515/). Given the Springer resolution, and its ...
6
votes
1answer
320 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 $H_3(BG)$ ...
11
votes
1answer
752 views

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 ...
10
votes
2answers
2k views

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 ...
5
votes
2answers
680 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 ...
8
votes
1answer
487 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 ...
83
votes
11answers
6k views

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. ...
0
votes
0answers
505 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 ...
3
votes
2answers
279 views

Is the first filtration Hausdorff?

Maybe this is too technical and elementary, but I cannot make up my mind, nor find a reference. The situation is the following: let $X$ be a double cochain (right half-plane) complex of abelian ...
2
votes
1answer
137 views

Colimit of intersections

Let $B_i^p$ be a family of sets, where $p\in \mathbb{N}$ and $i \in I$, $I$ being a directed set, and such that, for every $i$, we have a descending chain of inclusions $$ \dots \supset B_i^{p-1} ...
15
votes
2answers
947 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 ...
1
vote
0answers
464 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 ...
3
votes
1answer
445 views

Convergence of right half-plane spectral sequence bounded on the right

This is a sequel to my previous question colimits of spectral sequences . I think I've found the answer in S.A. Mitchell's paper "Hypercohomology spectra and Thomason's descent theorem". There the ...
11
votes
2answers
777 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} ...
4
votes
1answer
498 views

colimits of spectral sequences

I'm looking for some references about colimits of spectral sequences. More precisely: let $X : I \longrightarrow \cal{C}$ be a functor from a filtered category $I$ to the category of double cochain ...
5
votes
1answer
841 views

Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...
3
votes
2answers
780 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, $E, F$ ...
11
votes
3answers
885 views

Multiplicative structure on spectral sequence

Let $E$ be a spectral sequence and assume that there is a product $E^{r}_{p_1,q_1} \times E^r_{p_2,q_2} \to E^r_{p_1+p_2,q_1+q_2}$ which satisfies the Leibniz rule (for all $p_i,q_i$, but $r$ ...
35
votes
5answers
5k views

Simple examples for the use of spectral sequences

I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex. All I know are certain "extreme cases", where the ...
28
votes
16answers
8k views

introductory book on spectral sequences

I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand. Are there books or web resources that serve as ...
3
votes
1answer
1k views

Question about hypercohomology / spectral sequence of a complex of “almost-acyclic” sheaves

I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
7
votes
0answers
1k views

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 ...
7
votes
1answer
853 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. ...
5
votes
1answer
695 views

Serre spectral sequence with spectra

A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
12
votes
2answers
459 views

How do you compute the space of lifts of an E-infinity map?

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps ...
5
votes
1answer
703 views

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, ...
42
votes
5answers
5k views

Why are spectral sequences so ubiquitous?

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
18
votes
5answers
2k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...