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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Example request: seriously deficient homogeneous spaces
In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
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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 ...
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how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
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Hodge decomposition for Bott-Chern cohomology
$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
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Cocycles for right- and left- regular representations on $\ell_2(G)$
Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on $...
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local-global principles in group cohomology
Let $G$ be a (profinite) group. It is known that if $H^n(G_p,A) = 0$ for all $p$, $S_p$ the Sylow subgroups of $G$, then $H^n(G,A) = 0$.
Are there other local-global principles for different sets of ...
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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
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
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Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
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Appropriate notion of derived category over condensed set
If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\...
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Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$
Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
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Is there a "Kunneth isomorphism" for internal hom of chain complexes?
If $X^\bullet$ and $Y^\bullet$ are chain complexes over a field, we know from Kunneth theorem that
$$H^*((X\otimes Y)^\bullet)\cong H^*(X^\bullet)\otimes H^*(Y^\bullet) $$
I want to know if there is a ...
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Extension groups in quotient categories
Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
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Stacky points detect nilpotent cohomology
Take a finite type scheme $X/\mathbf{C}$ acted on by a reductive group $G$. Then $X/G$ is an Artin stack. Let $\alpha\in H^*(X/G)$ be a rational cohomology class on it.
Question: Is it true that $\...
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Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
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Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?
Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
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Integration on an non-orientable manifold [closed]
Suppose $M_n$ is a $n$ dimensional non-orientable manifold.
I am interesting in knowing whether the following statements are true:
A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
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Compactly supported cohomology of a topological DM stack
Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
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Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in the $n$-line of the $E_2$ of the ASS of the sphere?
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element $...
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Legendrian surgery and invertible elements in zeroth degree symplectic cohomology
Is there anything known about the relation between Legendrian handle attachment and invertible elements in $\mathit{SH}^0(M)$?
As the simplest interesting case, take $M_0$ to be the cotangent bundle $...
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Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
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Applications of one of Serre's Theorems
This theorem is due to Serre:
Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no
element of order $p$ and let $H \leq G$ be an open subgroup. Then
$cd_p(G) = cd_p(H)$.
Where $cd_p(...
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Atiyah class and coboundary map
Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...
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Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
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finiteness of syntomic cohomology of proper varieties over algebraically closed and finite fields
Let $k$ be an algebraically closed field or a finite field and $X/k$ be a proper variety. Let $G/X$ be a finite flat group scheme. Is $\mathrm{H}^i_\mathrm{syn}(X,G)$ finite for all $i$? (So: are ...
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Pull back group cohomology onto handle decomposition
A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients:
An oriented, (assumed here to be smooth) manifold $M^n$
A finite group $G$ (and a field, chosen to be $\...
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Golod-Shafarevich groups and L_2- Betti numbers
Is it something known about $L^2$-Betti numbers for Golod-Shafarevich groups?
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Torsors for discrete groups in the etale topology
Let $S$ be a smooth variety over $\mathbb C$ or a smooth quasi-projective integral scheme over Spec $\mathbb{Z}$.
Let $G$ be an (abstract) discrete group. For instance, $G =\mathbb{Z}^n$ or $G$ a ...
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rational cohomology of classifying spaces of complex reductive Lie groups
I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
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Equivariant cohomology of grassmannian
The Grassmannian $X=Gr(2,\mathbb C^4)$ has an involution $\sigma:V \mapsto V^{\perp}$ with respect to the symplectic form $antidiag(1,1,-1,-1)$. The torus equivariant cohomology of the grassmannian ...
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Torsion-free sheaf cohomology over discrete valuation rings
Let $R$ be a Henselian discrete valuation rings with algebraically closed residue field and $X$ be a regular, flat, proper $R$-scheme. Assume that the generic fiber to the natural morphism from $X$ to ...
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A course on modern algebraic geometry from "The Stacks Project"
I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't.
For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
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Analytic refinement of generalized cohomology theories
Recently, U.Bunke and others developed in a number of papers (such as this, this or this) an approach to smooth extensions of cohomology theories based on stable homotopy theory. In this approach a ...
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Generalization of de Rham cohomology, or cohomology for non-smooth case
Let $\Omega\subseteq \mathbb{R}^{3}$ be a contractible region and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$ ...
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Relation of BRST model of equivariant cohomology and BRST cohomology?
I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes \...
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Two natural maps asssociated with the nerve of a cover
Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
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Cocycle condition for 2-groups
I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
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cohomology ring of configuration spaces on $S^2$ and the projective plane
For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely,
$$
B(M,n)=\{(...
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Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953
There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here).
Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related by$...
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Comparison of sheaves of modular forms
Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and $\...
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Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack
Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...
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configuration space of Riemannian manifolds with a parameter on the distance of distinct points
Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...
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cohomology ring of stable configuration spaces
Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...
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Possible Betti numbers of smooth complex varieties
Given a smooth projective complex variety $X$ of dimension $n$, there are various restrictions on its sequence of Betti numbers $b_0, b_1, ..., b_{2n}$. Of course, $b_0=b_{2n}=1$ and $b_i=b_{2n-i}$ by ...
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Hochschild Cohomology of the Quantum Torus
I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
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Identifying a cohomology class arising from a Postnikov decomposition of BU(2)
For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface $...
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The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities
The Bullet-Macdonald identity (c.f. On the Adem relations)is the following:
$$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$
where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the
Adem ...
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Which ring spectra have some kind of exponential map turning addition into multiplication?
This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time.
What will follow is sort of vernacular but whether it can be ...
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extension of cohomology theories
In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be ...
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cohomology of a blowup: reference needed
Does anybody know a reference in which the computation of the cohomology of a blow-up is made in detail?
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Defining the cup product in Ext using a Kunneth formula
I want to make a Kunneth product of sorts on Ext. In particular, letting $C_*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in $Hom_R(...