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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145 views

$0$-Dimensional $k$-varieties in the Grothendieck ring $K_0(V_k)$

Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. Let $K_0(V_k^0)$ be the Grothendieck ring of $0$-dimensional $k$-varieties. I also assume that $k$ is perfect. (1) I ...
1 vote
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175 views

Cohomology ring of special linear group over finite fields

I am trying to find about the cohomology ring $H^*(SL_n(\mathbb{F}_q),\mathbb{Z}/2\mathbb{Z})$ where $q$ is odd. For $n=2$, an explicit description is given. But for $n>2$, I didn't come across a ...
2 votes
1 answer
344 views

Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
8 votes
2 answers
1k views

Ideals generated by regular sequences

In Vasconcelos' paper (Ideals generated by R-sequences), he proved If $R$ is a local ring, $I$ an ideal of finite projective dimension, and $I/I^2$ is a free $R/I$ module, then $I$ can be ...
1 vote
1 answer
196 views

Steenrod operations from the delooping viewpoint

Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation $$ H^*(-;F) \to H^{*+i}(-;F).$$ By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the ...
4 votes
0 answers
186 views

Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...
3 votes
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279 views

Why the scissor relations in Grothendieck rings?

Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. One type of relation which defines $K_0(V_k)$ is the following: if $A$ is a $k$-variety and $C$ a closed subset of $A$,...
8 votes
1 answer
362 views

On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513 Woodward proposed a classification of $\mathrm{PU}...
9 votes
1 answer
525 views

About the cohomology of $BG^\delta$. Making a Lie group discrete

Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\...
3 votes
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333 views

mod $p$ homology of Thom spectra MSU

Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big(...
2 votes
1 answer
107 views

Effective semi-group of a singular abelian surface

Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
6 votes
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208 views

Generators for unitary bordism ring $\pi_*(MU)$

I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”. He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...
4 votes
0 answers
159 views

Reference for equivariant derived Künneth formula

I'm looking for a reference for the following statement in as much generality as possible, assuming it is correct. Let's $X$ and $Y$ be "spaces" with a $G$-action. We can take the $G$-product defined ...
19 votes
2 answers
3k views

Why do gerbes live in H^2?

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Čech cohmology w.r.t....
14 votes
2 answers
5k views

Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution, $\require{AMScd}$ \begin{CD} \cdots @>N>> \mathbb ...
11 votes
3 answers
3k views

Betti number and harmonic forms

On a compact, boundaryless, Riemannian manifold, the dimension of the space of harmonic k-forms is equal to the k-th Betti number. Is this correct (by Hodge theory)? For example, on the surface of a ...
14 votes
1 answer
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Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$?

Hopefully this question is of an appropriate level for this site: I'm reading some notes by Claire Voisin titled Géométrie Algébrique et Géométrie Complexe. Let $X$ be a smooth $k-$scheme. In these ...
6 votes
0 answers
239 views

Signature of a non-compact manifold

Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...
4 votes
0 answers
304 views

Beilinson regulator: a road map

I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...
12 votes
1 answer
874 views

De Rham and Koszul complexes

Consider the algebraic de Rham complex of the $n$-dimensional plane: this is merely $$\ldots\rightarrow Sym(V^*)\otimes\bigwedge^{k}V^*\rightarrow Sym(V^*)\otimes\bigwedge^{k+1}V^*\rightarrow\ldots $$...
16 votes
4 answers
3k views

The homology of the universal covering space, why so difficult to compute

Let suppose that we are given a connected CW-complex $X$, such that we know All its homology groups. All its homotopy groups, in particular we know $\pi_{1}(X)$. As far as I know there is no ...
5 votes
0 answers
166 views

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,\...
5 votes
0 answers
250 views

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 $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
7 votes
0 answers
235 views

$X$ with $H^*(X)=$affine Verma module?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $...
8 votes
0 answers
497 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
7 votes
1 answer
1k views

How to identify cup product with intersection

What's the standard generalization and reference for the following statement: If two oriented submanifolds $L$, $L'$ of an oriented compact manifold $M$ intersect transversally, then the Poincare ...
7 votes
0 answers
219 views

Homological and homotopical equivalence of complex analytic varieties

Consider a map between two complex analytic varieties of finite type $f:X\to Y$. Suppose that $f$ induces isomorphisms on cohomology with (constant) integral coefficients. Under what reasonable ...
2 votes
1 answer
263 views

Middle cohomology of very general hyperplane sections

Let $X$ be a smooth, projective variety over $\mathbb{C}$ of dimension $n$ satisfying the property that for every $i \ge 0$, $H^{i,i}(X,\mathbb{C}) \cap H^{2i}(X,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}...
2 votes
1 answer
365 views

Derived category of singular varieties

Let $X$ be a projective variety with only normal crossing singularity. Is there a description of the derived category or the category of perfect complexes? What about the existence of semiorthogonal ...
15 votes
4 answers
4k views

Program for computing group cohomology

Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space? I mainly care about infinite groups.
2 votes
0 answers
93 views

Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support

Let $(Z,A)$ a compact ENR pair, then $$\chi(Z)=\chi_c(Z-A)+\chi(A)$$ where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
3 votes
1 answer
332 views

Cohomology with compact support

Let $X$ be a smooth, projective variety, $V_1, V_2$ smooth, closed subvarieties of the same dimension and $E$ a locally free sheaf on $X$. There exist natural morphisms $$r_1: H^i_{V_1}(E) \to H^i(E) ...
8 votes
1 answer
1k views

When homology isomorphism implies homotopy isomorphism

Let's suppose that $f:X\rightarrow X$ is a continuous map such that $H_{\ast}(f): H_{\ast}(X)\rightarrow H_{\ast}(X)$ is a homology isomorphism (with integral coefficients) $X$ is a finite ...
3 votes
0 answers
377 views

Stalks of perverse cohomology sheaves?

For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...
9 votes
1 answer
313 views

Weyl map for $SU(n)$

Let $G= SU(n)$ and let $\mathbb{T}$ be the maximal torus in $G$ given by diagonal matrices. We have $$ H^*(G,\mathbb{Q}) \cong \Lambda_{\mathbb{Q}}[x_3, x_5, \dots, x_{2n-1}] \ . $$ Now consider the ...
1 vote
0 answers
195 views

Find torsion classes using flat bundles

My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
3 votes
3 answers
310 views

Homologically trivial fibre

Let us consider a homotopy fibre sequence of connected spaces $A\rightarrow B\rightarrow C$ and let $K$ be a fixed field. Assume that the homology $H_{\ast}(A, K)$ is trivial and that $C$ is a ...
7 votes
1 answer
494 views

Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
3 votes
1 answer
804 views

Chern classes of complex vector bundle

I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows: $E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
3 votes
2 answers
375 views

Alexander duality and homology equivalence

While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation. In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...
5 votes
1 answer
337 views

When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?

Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
2 votes
1 answer
381 views

An action of the symmetric group $S_n$ on group cohomology $H^n(G, A)$ of abelian groups

Let $H$, $A$ be discrete abelian groups, and for simplicity suppose $A$ is given the trivial $H$-action. When considering the second cohomology group $H^2(H,A)$, it is natural to talk about the ...
7 votes
0 answers
254 views

Integral cohomology of compact Lie groups and their classifying spaces

Let $G$ be a compact Lie group and let $BG$ be its classifying space. Let $\gamma\colon \Sigma G \to BG$ be the adjoint map for a natural weak equivalence $G \xrightarrow{\sim} \Omega BG$. Taking ...
11 votes
0 answers
489 views

Chromatic Homotopy Theory and Physics

Chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating ...
6 votes
1 answer
463 views

Category of spaces/sheaves

Consider the following category $\mathcal C$: An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$. A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
7 votes
1 answer
433 views

Imperfect Tate (cup product) pairing in Galois cohomology?

Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$ and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$. Let $M$ be a finite $\Gamma$-module, that is, a finite abelian ...
5 votes
0 answers
182 views

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 ...
9 votes
1 answer
446 views

Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product

Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$. ...
2 votes
1 answer
532 views

Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
1 vote
0 answers
115 views

3rd Cohomology of a fibration with Flag varieties as fibers

Let $X$ be a smooth projective rational variety over $\mathbb{C}$, let $Y$ be another smooth projective variety, both of dimension bigger than 2, and let $\pi : Y \rightarrow X$ be a locally trivial ...

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