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5
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1answer
211 views
+50

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
1
vote
0answers
54 views

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 $...
3
votes
0answers
86 views

Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum

This is a following up question of Sphere spectrum, Character dual and Anderson dual. What are the differences and the significances of the following: (1). Homotopy classes of maps from a Thom ...
6
votes
2answers
390 views

Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. However, could you help me to appreciate the mathematical meanings of the following: What is the significance of ...
2
votes
0answers
92 views

Edge map in derived categories

Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
3
votes
0answers
74 views

Twisted spin cobordism v.s. KO theory in low dimensions

Based on the background info and this this webpage, here is a more advanced problem: Question: If we consider a different more subtle twisted structure, like $${\Omega_d^{(\mathrm{spin} \times G)/N}},...
5
votes
0answers
114 views

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 $n$) $$\Omega_d(BG)_p.$$ ...
3
votes
0answers
121 views

Reference for specific detail on Serre spectral sequence

In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...
8
votes
0answers
88 views

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 ...
6
votes
0answers
93 views

Bordism groups and a short exact sequence

Let us consider a short exact sequence: $$ 1\to N\to G\to Q \to 1, $$ where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups). Suppose I have the data and the computations ...
5
votes
0answers
99 views

The $E_2$-page of the May spectral sequence

I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS. At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
5
votes
0answers
113 views

A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?

This question is a follow-up to my previous question: "Rotated" version of the Atiyah-Hirzebruch spectral sequence In that question, I discussed two different spectral sequences for ...
3
votes
0answers
106 views

What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
4
votes
1answer
129 views

Transgression image and Serre spectral sequence for tori

Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...
1
vote
0answers
46 views

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 ...
4
votes
1answer
158 views

How are p-primary parts determined for odd p?

When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away. How are odd primary part calculations done in relation ...
2
votes
0answers
47 views

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 ...
5
votes
0answers
84 views

Does Serre spectral sequence degenerate for fiber bundles that are trivial restricted to the boundary?

Let $W$ be an $n$-dimensional manifold with boundary. Assume $W$ admits an exhausting Morse function with indexes smaller or equal to $k$. Let $\pi: E\to W$ be fiber bundle with compact fiber $F$ and $...
2
votes
1answer
103 views

Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...
2
votes
1answer
141 views

Why only consider decreasing filtrations on cochain complexes?

When reading various literature on spectral sequences one always comes across two setups: A chain complex with an increasing filtration A cochain complex with a decreasing filtration My question is ...
2
votes
0answers
189 views

Degeneration of relative Hodge-de Rham spectral sequence

$$\require{AMScd}$$ $$\newcommand{\CC}{\mathbb{C}} \newcommand{\RR}{\mathbb{R}} \newcommand{\Hdr}{H_{\mathrm{dRh}}} \newcommand{\tensor}{\otimes} \newcommand{\Ohol}{\mathcal{O}}$$ Please excuse that ...
3
votes
1answer
277 views

Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
3
votes
0answers
95 views

Gel'fand's and Fuks' calculation of cohomology of formal vector fields - isomorphic spectral sequences yield isomorphic cohomology?

In this book (and, in what seems to be an equivalent fashion, in this article), Gel'fand and Fuks calculate the Lie algebra cohomology of the formal vector fields $W_n$ on $\mathbb{R}^n$ with trivial ...
8
votes
1answer
127 views

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 ...
6
votes
2answers
261 views

Homology spectral sequence for function space

The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...
3
votes
1answer
355 views

What bigrading is used in this spectral sequence?

I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...
6
votes
1answer
209 views

to compare cohomologies of fibers 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 ...
2
votes
2answers
209 views

homology of a base space of a a fiber sequence

Suppose we have a fiber sequence of connected spaces $A\rightarrow B\rightarrow C$ and suppose we know the homology of A and B, is there a homological spectral sequence converging to the homology of $...
1
vote
0answers
52 views

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 ...
5
votes
0answers
90 views

Spectral Sequence for Twisted K-theory

Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...
1
vote
0answers
92 views

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 ...
4
votes
1answer
144 views

The converse of Vietoris-Begle theorem

It is well known the following result: Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;% %...
8
votes
2answers
334 views

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 ...
15
votes
1answer
420 views

“Rotated” version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
2
votes
1answer
107 views

Leray spectral sequence for continuous functions on pairs of topological spaces

Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$. The Leray spectral sequence (with complex ...
5
votes
1answer
214 views

On the Leray spectral sequence and sheaf cohomology

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
2
votes
0answers
87 views

Local coefficients system

Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow ...
5
votes
1answer
128 views

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_{...
10
votes
0answers
139 views

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 ...
7
votes
1answer
145 views

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{...
11
votes
1answer
304 views

Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence

Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence $$E_{pq}^2 = \tilde{H}_p( X, \pi_q(E)) \Rightarrow \pi_{p+...
2
votes
1answer
184 views

Étale cohomology of tensor product

Let $X$ be a smooth projective variety over a field $k$. Suppose we have étale abelian sheaves $A, B$ on $X_{\rm ét}$ such that $$H^j(X_{\rm ét}, A),\ H^j(X_{\rm ét}, B)$$ are finitely generated ...
8
votes
1answer
293 views

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 ...
5
votes
0answers
151 views

spectral sequence for a complex with two filtrations

Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
2
votes
1answer
163 views

Hochschild-Serre filtration and etale cohomology

I encountered the Hochschild-Serre spectral sequence in étale cohomology $$H^i(\text{Gal}(\overline{k}/k), H^j_{et}(X_{\overline{k}}, F))\Rightarrow H^{i+j}_{et}(X_{{k}}, F)$$ How is the filtration ...
6
votes
1answer
212 views

Calculating topological index

Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological ...
6
votes
1answer
458 views

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}(...
7
votes
1answer
183 views

Does a filtered A_N algebra give rise to a multiplicative spectral sequence?

The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728....
3
votes
1answer
160 views

Strong convergence of whole-plane spectral sequences

I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman: https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-...
4
votes
0answers
169 views

Cohomology of double complex with exact rows

Let $(C^{p,q},d_h,d_v)$ be an unbounded double complex of modules of some algebra $A$ in an abelien category. Let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that (1) ...