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, ...

Filter by
Sorted by
Tagged with
2
votes
1answer
114 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 ...
1
vote
1answer
112 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 ...
2
votes
0answers
178 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 $A$,...
4
votes
0answers
156 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 $G$ ...
7
votes
2answers
194 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
votes
0answers
90 views

Commutative square up to sign as indicated, Poincare duality

Let $M$ be smooth oriented manifold with boundary $\partial M $, ${\rm dim}\,M=n$. The two short exact sequences in de Rham cohomology and singular homology $$0\longrightarrow{}\Omega^{*}(M, \...
2
votes
1answer
85 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 ...
5
votes
0answers
168 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 $z_n \in H_n(MO; \mathbb F_2)$...
2
votes
0answers
51 views

Models for computing cohomology of Lie groupoids

Given a Lie groupoid $\mathcal{G}=[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$, let $\mathcal{G}_\bullet$ be the associated simplicial manifold. Let $\Omega^\bullet(\mathcal{G}_\bullet)$ be the ...
4
votes
0answers
118 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 ...
2
votes
0answers
49 views

Understanding top Cech cohomology

For a oriented compact smooth manifold, $X^n$ we can understand the class $1\in \mathbb Z \cong H^n(X; \mathbb Z)$ as a volume form that integrates to $1$ on $X$. Under the isomorphism $H^n(X;\mathbb ...
6
votes
0answers
159 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
0answers
138 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 ...
11
votes
1answer
343 views

De Rham and Koszul complexes

Consider the algebraic de Rham complex of the $n$-dimensional plane: this is merely $$\ldots\rightarrow Sym(V^*)\otimes\bigwedge^{k}V^*\rightarrow Sym(V^*)\otimes\bigwedge^{k+1}V^*\rightarrow\ldots $$...
9
votes
1answer
418 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 $\...
5
votes
0answers
109 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
0answers
109 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 $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
14
votes
4answers
1k 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 ...
7
votes
1answer
246 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 ...
6
votes
0answers
196 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 ...
3
votes
0answers
250 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
1answer
161 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
1answer
228 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 ...
2
votes
0answers
68 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 ...
8
votes
1answer
570 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
0answers
136 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
1answer
192 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
0answers
115 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
1answer
336 views

Motivating the coefficient field of $\ell$-adic cohomology

It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures. It was also ...
3
votes
3answers
229 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 ...
3
votes
1answer
262 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 $E_p$...
2
votes
2answers
318 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
1answer
223 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{...
3
votes
0answers
141 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 ...
6
votes
0answers
189 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 $...
10
votes
0answers
236 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 ...
5
votes
1answer
347 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 $(X,\mathcal F)\to(Y,\mathcal G)$...
4
votes
0answers
121 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 ...
7
votes
1answer
293 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
3answers
483 views

An intuitive explanation for group cohomology via cochains?

I'm fairly new to topology, and so far I've understood cohomology via cochains. First we build an object called a cochain ($C^n$), then define a differential map that takes you from $C^n$ to $C^{n+1}...
2
votes
1answer
164 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
0answers
91 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 ...
0
votes
0answers
95 views

cohomology of curves

Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$ in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle. If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
2
votes
1answer
187 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 ...
2
votes
0answers
82 views

Computation of mod p homology of $MSU$

I am trying to proof Novikov theorem \begin{equation} MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i. \end{equation} This can be proved by using ...
13
votes
3answers
578 views

Unifying “cohomology groups classify extensions” theorems

It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean: 1) Derived functor of hom, $\text{Ext}^1_R(M, N)$. ...
1
vote
1answer
134 views

A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras

Johnson in cohomology of Banach algebra proved the following proposition. I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this ...
11
votes
2answers
317 views

$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$

Cross posted from here after no responses and a bounty being placed on the question. Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-...
0
votes
0answers
129 views

Cohomology of (complex) varieties

I am trying to understand a version of Lotthar Göttsche's computation of the betti numbers of the punctual Hilbert Scheme of a smooth projective surface. However, before I can even begin, I am unsure ...
1
vote
1answer
211 views

Easier ways to compute homology/cohomology by adding extra structure

Suppose $X$ is a topological space and I want to talk about its “homology”. There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute. Suppose ...

1
2 3 4 5
21