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Questions tagged [cohomology]

A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...

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Cohomology class of fiber bundle

Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$. I ...
Alexander Woo's user avatar
7 votes
0 answers
199 views

Relation beween Chern-Simons and WZW levels, and transgression

3d Chern-Simons gauge theories based on a Lie group $G$ are classified by an element $k_{CS}\in H^4(BG,\mathbb{Z})$, its level. Via the CS/WZW correspondence the theory is related with a 2d non-linear ...
Andrea Antinucci's user avatar
3 votes
0 answers
97 views

English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?

Is there an English translation of this text, or at least some English language paper that proves the same results? I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
user2945539's user avatar
1 vote
0 answers
113 views

Vanishing result on cohomology with support

Let $X$ be a locally contractible topological space of dimension at least $2$ and $x \in X$ a point. Is it true that $H^i_x(X)=0$ for all $i>1$? If necessary, assume $X$ is a projective variety and ...
user45397's user avatar
  • 2,313
4 votes
2 answers
265 views

Loop-space functor on cohomology

For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$. More concretely, $\omega$ is given by the Puppe sequence $$\...
Leo's user avatar
  • 643
2 votes
0 answers
158 views

Renormalization from cohomology point of view

In order to construct a Euclidean quantum filed theory one usually needs to take care of the renormalization problem. Let us consider a simple model like $\phi^4$ in dimension two. In this case just ...
Azam's user avatar
  • 101
1 vote
0 answers
129 views

Hodge numbers of a complement

Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that $$ H^i(X) \to H^i(...
cll's user avatar
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3 votes
0 answers
84 views

Possible relation between causal-net condensation and algebraic K theory

Causal-net condensation is a natural construction which takes a symmetric monoidal category or permutative category $\mathcal{S}$ as input date and produces a functor $\mathcal{L}_\mathcal{S}: \mathbf{...
xuexing lu's user avatar
6 votes
1 answer
352 views

Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$

$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
wonderich's user avatar
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1 vote
1 answer
136 views

Confusion regarding the vanishing of certain relative cohomology groups

Let $X$ be a projective variety of dimension $n$ and $D \subset X$ is a proper subvariety. Embed $X$ into a projective space $\mathbb{P}^{3n}$. The following argument implies that $H^i(X,X\backslash D)...
user45397's user avatar
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4 votes
0 answers
181 views

Cohomology of a differentiable stack: evaluation at a point

I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page. Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
Kandinskij's user avatar
2 votes
0 answers
140 views

Variant of Leray-Hirsch for complex-oriented cohomology theory

I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a complex-oriented cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to ...
onefishtwofish's user avatar
2 votes
0 answers
120 views

The tautness property and the continuity property of cohomology theory

Let $H^{\ast }$ denote the Čech cohomology or Alexander-Spanier cohomology. Definition: (Tautness property of cohomology) Let $X$ be a paracompact Hausdorff space and $A$ be a closed subspace of $X$. ...
Mehmet Onat's user avatar
  • 1,301
4 votes
0 answers
110 views

Minimal Model for $\mathbb{CP}^2 \# \mathbb{CP}^2 \# \mathbb{CP}^2$

I'm a grad student studying non-negative curvature on simply-connected manifolds and the conjectured relationship with rational homotopy theory. The rational homotopy groups (and so the number of ...
Russ Phelan's user avatar
1 vote
0 answers
58 views

How can we construct a non-trivial central extension of a Lie group

Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
Mahtab's user avatar
  • 277
2 votes
0 answers
91 views

Koszul cohomology associated with a regular sequence

Let $A$ be a local Noetherian ring and $M$ be an $A$-module. Let $\mathfrak{a}$ be an ideal of $A$ generated by a regular $M$-sequence $s_1,\cdots,s_r$. Let $K_\bullet(s_1,\cdots,s_r;M)$ be the Koszul ...
Li Li's user avatar
  • 419
6 votes
1 answer
355 views

Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?

I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24: Now let us recall that the ...
John Baez's user avatar
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5 votes
1 answer
417 views

Does coproduct preserve cohomology in differential graded algebra category

Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
wer's user avatar
  • 159
2 votes
0 answers
196 views

Are there any relations between perverse t-structure (cohomologies) and standard t-structure (cohomologies)?

I'm reading the Corollary 3.2.3. in Exponential motives by J. Fresan and P. Jossen. The authors use the following statement in the proof of Corollary 3.2.3: let $C$ be any object in the derived ...
Mathstudent's user avatar
2 votes
0 answers
92 views

Cohomological dimension of functors from fields to vector spaces

Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are pairs $F,j$ consisting of a field $F$ and the embedding $...
Galois group's user avatar
1 vote
1 answer
128 views

About Čech cohomology in transformation groups

I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
Ludwik's user avatar
  • 237
9 votes
0 answers
514 views

In Mann's six-functor formalism, do diagrams with the forget-supports map commute?

One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
Gabriel's user avatar
  • 1,139
5 votes
1 answer
119 views

The Salvetti complex of a non-realizable oriented matroid

Given a real hyperplane arrangement, the Salvetti complex of the associated oriented matroid is homotopy equivalent to the complement of the complexification of the arrangement. In particular, its ...
Nicholas Proudfoot's user avatar
5 votes
0 answers
335 views

Reference request: cohomology of BTOP with mod $2^m$ coefficients

I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where $${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
Baylee Schutte's user avatar
2 votes
0 answers
176 views

Splitting of de Rham cohomology for singular spaces

I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
Thomas Kurbach's user avatar
3 votes
1 answer
301 views

Exact functor in syntomic cohomology

By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site. Is it also true for a finite flat ...
prochet's user avatar
  • 3,442
5 votes
0 answers
145 views

Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
Zhenhua Liu's user avatar
2 votes
0 answers
235 views

Finite generation of stack cohomology

Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra? For instance, $\text{H}^*(\text{B}\mathbf{G}...
Pulcinella's user avatar
  • 5,575
13 votes
1 answer
488 views

Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
Zhenhua Liu's user avatar
4 votes
1 answer
361 views

Reference for isomorphism between group cohomology and singular cohomology

Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that $$ H^i(G, ...
Aidan's user avatar
  • 498
3 votes
1 answer
198 views

Čech cohomology refinement mapping

Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...
Alexander Mrinski's user avatar
1 vote
0 answers
91 views

On the equivalence of two definitions of cohomological dimension for locally compact topological spaces

$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual ...
Rabi Kumar Chakraborty's user avatar
1 vote
2 answers
241 views

Generalized cohomology on the one point space

I am reading Hatcher's algebraic topology for an assignment on generalized cohomology theories, and in section 4.E p. 447 he says the following The wedge axiom implies that $h(\textit{point})$ is ...
Dani Jaen's user avatar
2 votes
0 answers
73 views

Tangent $(\infty,1)$ topos

I am trying to understand the tangent $(\infty,1)$ category. It is the fiberwise stabilization of the codomain fibration (which is a functor from the arrow category to the category). But, intuitively ...
Pinak Banerjee's user avatar
2 votes
2 answers
292 views

When is $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$ for a pair $(X,A)$?

I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\...
Harsh Patil's user avatar
0 votes
0 answers
125 views

Cohomology ring of $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O})$

Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Let $\mathcal{O}$ be the trivial rank one bundle. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\...
asv's user avatar
  • 21.3k
3 votes
0 answers
81 views

When does homology preserve inverse limits of Eilenberg-MacLane spaces?

Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
willie's user avatar
  • 499
3 votes
0 answers
103 views

A question on the averages of Kloosterman sums

Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is, For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound $$\sum_{...
hofnumber's user avatar
  • 553
3 votes
0 answers
174 views

Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following: "In some sense, the operator $\psi$ applied to a power series gives it "better growth ...
Vik78's user avatar
  • 528
2 votes
0 answers
161 views

Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory

$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2. ...
JeCl's user avatar
  • 1,001
3 votes
0 answers
81 views

Explicit examples of 4-cocycles over finite 2-groups

By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...
Andi Bauer's user avatar
  • 2,911
2 votes
0 answers
73 views

G-modules vs. $\Delta(NG)$-modules

Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...
Antoine's user avatar
  • 143
2 votes
1 answer
205 views

When is a (co)edge trivial in graph cohomology?

Let $G$ be a connected graph and let $e$ be an edge in this graph. I would like to know if there are necessary and sufficient questions so that $e^{\vee}=0$ in $H^1(G)$? The question must be easy to ...
divergent's user avatar
1 vote
0 answers
116 views

A question about cohomology with local coefficient

Let's consider the next theorem. Theorem [The cohomology Leray-Serre Spectral sequence] Let $R$ be a commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{% \rightarrow }B$, ...
Mehmet Onat's user avatar
  • 1,301
2 votes
0 answers
68 views

Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
wonderich's user avatar
  • 10.4k
1 vote
1 answer
368 views

Why should we study the total complex?

Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
mrtaurho's user avatar
  • 165
2 votes
1 answer
261 views

Exact sequence for relative cohomology + normal crossing divisors

Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc. Is it true that there is an exact sequence $$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
Galois group's user avatar
11 votes
1 answer
1k views

Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions

Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions: Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
KStarGamer's user avatar
8 votes
1 answer
294 views

Fiber product of spaces and cohomology

Let $Y \to X \leftarrow Z$ be a cospan of topological spaces and let $W = Y \times_X^h Z$ be their homotopy fiber product. I am interested in sufficient conditions on the map $Y \to X$ that ensure ...
Matthias Ludewig's user avatar
0 votes
0 answers
95 views

Is there a "cohomology theory" for involutive algebras?

I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
wlad's user avatar
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