4
votes
1answer
128 views

AdicCompletion$\dashv$Torsion adjunction on spectra?

It seems to me that in slight paraphrase the central result of the article Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386) (theorems 6.11 and ...
1
vote
2answers
198 views

If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if ...
1
vote
0answers
145 views

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 ...
1
vote
0answers
76 views

generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...
0
votes
0answers
84 views

Dimension of category of sheaves [duplicate]

Let $k$ be a field. Consider the category $Shv(\mathbb{R}^n)$ of sheaves of $k$-vector spaces on $\mathbb{R}^n$. What is the cohomological dimension $d$ of $Shv(\mathbb{R}^n)$? I know that $d \in ...
2
votes
1answer
155 views

Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
1
vote
0answers
109 views

Reference needed: Homology of the blow-up

Given an algebraic variety $X$ over the complex numbers. Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$ be the blow-up of $X$ at $V$. It is posible in general to compute the ...
1
vote
0answers
236 views

Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written: Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...
5
votes
1answer
282 views

Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep. I'm ...
1
vote
0answers
99 views

(Co)homology of classifying space of spin group $BSpin(n)$

In the answer for question: Homology of classifying space of spin group BSpin(n), it was shown that $H_i(BSpin(\infty),Z)$ is $0,0,0,Z$, for $i=1,2,3,4$. What is $H_i(BSpin(\infty),Z)$ or ...
1
vote
0answers
81 views

Homology of the fixed points of the singular complex of a G-space

I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me. Suppose $X$ is a topological space and $G$ a ...
9
votes
1answer
387 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
3
votes
1answer
95 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
2
votes
1answer
444 views

A question about the universal coefficients theorems

This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...
11
votes
0answers
361 views

Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
7
votes
2answers
280 views

Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence $$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...
21
votes
4answers
1k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
1
vote
0answers
220 views

Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...
3
votes
1answer
267 views

What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem: Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
7
votes
1answer
240 views

Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes. In the example I have in mind all chain complexes are concentrated in some fixed degree n. There is a canonical map lim D → holim D ...
11
votes
0answers
293 views

Bar construction vs. twisted tensor product

One may study the cohomology of a space $E$ expressed as a homotopy pullback of $X$ and $Y$ over $Z$ using either the Eilenberg-Moore spectral sequence or the Serre spectral sequence for the fibration ...
3
votes
0answers
153 views

Hochschild homology of a tensor algebra modulo a two-sided ideal

Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
4
votes
0answers
160 views

What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact $ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
4
votes
0answers
158 views

Homotopy unites of a differential graded algebra

I apologize in advance if the question is too basic. Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An ...
0
votes
1answer
327 views

A generalization of cochain complex: quasi-cochain complex

It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology. Definition: A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...
3
votes
0answers
178 views

The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules

Short Version Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...
3
votes
0answers
101 views

minimal model of $A_\infty$ structure

Hi all, I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman. 1) The construction of these ...
0
votes
0answers
192 views

Homology of the dg-nerve vs Hochschild homology of the dg-category

Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the ...
6
votes
0answers
296 views

Lyndon-Hochschild-Serre spectral sequence and cup products

First here is my setup: Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particular we get on action ...
3
votes
1answer
221 views

Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...
1
vote
1answer
243 views

Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...
4
votes
4answers
196 views

Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
8
votes
2answers
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 ...
4
votes
1answer
320 views

Mayer-Vietoris sequence in homology with local coefficients

Background. I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients. Question 1. What does the Mayer-Vietoris sequence look like when using ...
7
votes
1answer
242 views

Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...
8
votes
3answers
759 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 ...
5
votes
2answers
998 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
6
votes
3answers
418 views

A lost lemma about periodicity in a grid of long exact sequences?

This is a question about finding references and hopefully a larger context for a lemma in homological algebra I proved recently. The motivation is to understand properties of characteristic classes of ...
4
votes
1answer
340 views

Homology of classifying space of spin group BSpin(n)

While dealing with $BO(n)$, $BSO(n)$ and $BSpin(n)$ with the universal coefficient theorem and Künneth formula, I came to have the following question: The universal coefficient says $H^n(X;M)\cong ...
6
votes
1answer
323 views

Exceptional collections of objects in topological triangulated categories?

People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in ...
3
votes
1answer
222 views

If a t-truncation of the unit object in a stable homotopy category is a ring object up to homotopy, can it be lifted to a ring spectrum? What about the Postnikov t-truncations of the sphere spectrum?

Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to ...
3
votes
1answer
126 views

Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq 0$?

It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion. So I wonder ...
8
votes
4answers
752 views

References for Eilenberg-Zilber shuffle product

Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I ...
6
votes
0answers
258 views

homotopy pullbacks of tensor product chain complexes (towards Kunneth formula in diff cohomology)

I have editted this question from the previous version which did not obtain much attention. Suppose I have two diagrams of chain complexes: $A^* \rightarrow C^* \leftarrow B^*$ $\tilde{A}^* ...
5
votes
2answers
783 views

On the difference between a projective chain complex and a level-wise projective chain complex

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...
10
votes
1answer
326 views

finite complex with non-finitely generated homology with local coefficients

I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a ...
11
votes
1answer
480 views

Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives)

I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the author uses cosheaf ...
0
votes
1answer
215 views

Existence of a chain map lifting the identity; Alexander-Whitney/Eilenberg-Zilber maps

Some preliminary definitions: Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ ...
1
vote
0answers
163 views

Two-point desuspension for augmented chain complexes?

Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = ...
0
votes
1answer
362 views

Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question

Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...