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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End cohomology and space of ends
I have started learning about end cohomology, and as far as I understand, the zeroth end cohomology $H_e^0(M; \mathbb{Z})$ is isomorphic to the zeroth Čech cohomology $\check{H}^0(e(M); \mathbb{Z})$, ...
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Nilpotency of generalized cohomology
$\newcommand\pt{\mathrm{pt}}$Let $(X,\pt)$ be a connected, pointed, finite CW complex and let $h$ be a generalized cohomology theory. Let $\smash{\tilde{h}}^*(X)$ denote the kernel of restriction $h^*(...
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Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
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Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$
...
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Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?
Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
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2
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Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
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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 ...
- 2,908
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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 ...
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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 ...
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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 ...
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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
$$\...
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Renormalization from cohomology point of view
In order to construct a Euclidean quantum filed theory one usually needs to take care of the renormalization problem. Let us consider a simple model like $\phi^4$ in dimension two. In this case just ...
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Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
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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{...
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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 ...
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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)...
- 2,313
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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 ...
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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 ...
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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 $X$. ...
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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 ...
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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-...
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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 ...
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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 ...
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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 "...
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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 ...
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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 $...
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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'...
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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 ...
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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 ...
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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
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
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Splitting of de Rham cohomology for singular spaces
I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
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Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,452
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Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
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Finite generation of stack cohomology
Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...
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Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 467
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Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
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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 ...
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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 ...
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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 ...
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74
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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 ...
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2
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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)\...
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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)\...
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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 &...
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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_{...
- 553
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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 ...
- 538
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0
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162
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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.
...
- 1,001
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84
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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 ...
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74
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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 $...
- 215
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1
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205
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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 ...
- 63
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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 $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
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