Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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Hatcher $3.1.12$ Show that $H^k(X,X^n;G)=0$ for $k \leq n$

In Hatcher it is written in a theorem that by Universal Coefficient Theorem we get $H^k(X,X^n;G)=0$ where $X,X^n$ are CW complexes. But to use UCT, we have to show $H_k(X,X^n;G)=0$ $k \leq n$. ...
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2answers
75 views

Example of $H^n(X,R)$ not equal to $Hom(H_n(X,R),R)$

The universal coefficient theorem shows that under suitable assumptions, the cohomology groups with coefficients in $R$ are simply the morphisms between the homology groups and $R$. In general, ...
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1answer
60 views

dual basis of cohomology algebra

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients. Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$. Let the dual basis be ...
2
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1answer
49 views

The higher cohomologies of a quasi-coherent sheaf on the intersection of two affine open subsets.

It is well-known in algebraic geometry that if $X$ is affine and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $$ H^i(X,\mathcal{F})=0,~ \forall ~i\geq 1. $$ Now let $X$ be an arbitrary ...
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63 views

(Co)homology theory and electrical circuit

I have read that one of the origins of the theory of (co)homology is the study of electrical circuits by Poincare. I'd like to know more about that. Could someone sugest any reference on this subject? ...
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1answer
21 views

Definition of exterior derivative from a connection

I fail to see what is the meaning of the symbol $d_{\nabla}$ in (1.2) of http://arxiv.org/pdf/hep-th/9712042v2.pdf I know the meaning of that symbol in the context of forms taking values on some ...
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0answers
35 views

Cohomology ring of classifying space

I am looking for $H^*(BZ/2p , Z/2p)$ where $p$ is odd prime.We can calculate cohomology groups by using gysin exact sequence and universal coefficient theorem.But I am unable to calculate the ring ...
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20 views

rational cohomology of finite dimensional grassmannian

Let $G$ denote the grassmannian. It is known that the cohomology ring $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Q})=\mathbb{Q}[p_1,p_2,\cdots,p_{[n/2]}]. $$ What is $$ ...
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21 views

Deformations of associative algebras and Hochschild cohomology.

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A,\mu)$ be a commutative associative algebra ...
3
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1answer
55 views

constant stalks but not constant sheaf?

In the coherent world we have the following: if X is reduced and F is a coherent sheaf on it, then if the rank of all fibres in constant then F is locally free. I thought something similar held in ...
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58 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
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34 views

Shapiro's Lemma-Finding the inverse of an isomorphism.

Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is $\phi(f+B^n(G,Hom_{ZH}(ZG, A) ...
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1answer
32 views

requirement of openess of the subset of a manifold for mayer-vietoris theorem

So for a given manifold $M$ and two of $U,V$ open sets that can be used to cover $M$, one can use to mayer-vietoris theorem to relate the decomposed de rham cohomology of $M$ with that of $U$ and $V$. ...
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0answers
44 views

$\mathrm{Ext}^i(-,A/\mathfrak{m})$ in $(A,\mathfrak{m})$ noetherian regular local ring

Dealing with $\mathrm{Ext}^i(\mathcal{F},k(x))$ on a smooth variety over a field $k$, with $\mathcal{F}$ coherent and $k(x)$ skyscraper sheaf of a closed point I foundin a proof that for $i=2,3$ (and ...
2
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1answer
39 views

What exactly is meant by “an integer basis of the $\mathbb{Z}$-module $H^*(M)$”?

I thought I understood the concept of a cohomology ring, but am confused by the following statement found in a textbook. Context: M is a symplectic manifold of dimension $2n$. "Let us choose an ...
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0answers
21 views

What tools in algebraic topology help me capture the connectivity structure of a weighted graph as a REAL number?

All - This is a follow-up to a previous question about cohomology. I am researching a problem and, as with so much problem-solving, this has led me into parts of math well beyond where I went in ...
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1answer
83 views

How can I understand cohomology theories in the context of basic homology theory?

Please pardon the ignorance in advance -- I'm doing research, trying to solve a specific problem, so naturally I'm led down paths in mathematics I never had the opportunity to study in depth. I ...
2
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1answer
59 views

ordinary cohomology from equvariant cohomology

Is it possible that the ordinary cohomology of a space can be obtained from its equivariant cohomology? action is algebraic torus action and space is nonsingular complete complex algebraic variety ...
0
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1answer
25 views

Jordan regular Representative of $H_1(\Omega)$ with coefficients $\mathbb Z/ 2 \mathbb Z$

Consider the first homology group $H_1(\Omega)$ with coefficients in $\mathbb Z/2\mathbb Z$ for a bounded, open subset $\Omega\subset \mathbb C$. Then I should be able to find a representative path, ...
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0answers
12 views

Rationally graded singular cohomology

In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by $H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$ ...
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1answer
29 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
3
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1answer
55 views

Cohomology of Segre varieties

Let $\Sigma_{n,m}$ be a Segre variety, i.e. the image of the Segre map $\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^{(n+1)(m+1)-1}$. Then how can I calculate the first cohomology group of its ...
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56 views

Compute homology groups of space $\Bbb RP^2$ attached with Mobius band using Mayer Vietories

(This is exercise 2.2.28 from Hatcher) Consider the space $X$ (say) obtained from a $\Bbb RP^2$, by attaching a Mobius band $M$ via a homeomorphism from the boundary circle of the Mobius band to the ...
2
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1answer
27 views

Singular homology of discrete space

Let $X$ be a space with discrete topology. How to calculate singular homology of $X$, if: a) $|X|$ is finite b) $|X|$ is countable c) $|X|$ is uncountable. For $|X|=1$ it is obvious, but I have ...
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1answer
37 views

(Co)homology question

I'm learning homology and cohomology by myself, and I've stumbled upon a nice introductory paper here (http://www3.nd.edu/~mbehren1/18.904/Heffern_project.pdf). On page 1, under Motivation, second ...
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50 views

Is There a de Rham Homology

For differential manifold category, we can introduce the differential form to make up a cochain, and then get the de Rham cohomology group. My question is that if we use $\text{Hom}$ functor to get ...
2
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1answer
53 views

What kind of cohomology is meant?

What kind of cohomology is meant in Deligne's work about mixed hodge structure on cohomology groups of an complex algebraic variety? I think it refers to the singular cohomology with coefficients in ...
0
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1answer
59 views

Topological modules and relative homological algebra.

This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ...
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0answers
24 views

Bicomplexes-reference request

I'm not an expert in homological algebra, I would say that I have gathered only preliminary knowledge. I would like to learn more in particular about bicomplexes and homology and cohomology of such ...
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2answers
93 views

can we derive integral cohomology from rational cohomology and mod p cohomology?

Let $X$ be a topological space. If we know that for $\mathbb{F}=\mathbb{Q}$ and $\mathbb{Z}/p$, for any prime $p$, $$ H^*(X;\mathbb{F})=0$$ for any $*\geq n+1$, can we conclude that $$ ...
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1answer
56 views

A simpler definition of the snake map?

I would like to ask whether the following definition of the connecting morphism in the long exact sequence in homology of a pair $(X,A)$ is correct. First, define relative cycles and boundaries via ...
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1answer
58 views

Hatcher, Thm 2.13

Theorem 2.13. If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $$\cdots \longrightarrow \tilde H_n ...
3
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1answer
34 views

Higher homology groups relative a lower dimensional subspace

One often works with reduced homology, which (in the case of say, smplicial homology) is defined as the homology relative a point. Now at every grade except zero, the reduced homology objects are ...
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0answers
26 views

Reduced homology isomorphic to homology relative to a point

I'm going over reduced homology following tom Dieck's Algebraic Topology. I don't understand where the short exact sequence in the excerpt below comes from. $j:(X,\emptyset)\hookrightarrow (X,P)$ is ...
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0answers
90 views

Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand ...
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0answers
15 views

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$?

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$? $H^{1}(K)$ is the first de Rham cohomology group of $K$.
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2answers
196 views

Intuition of non-free homology groups?

According to Soft Question - Intuition of the meaning of homology groups and Wikipedia, homology groups can be thought of as counting the number of holes in a given dimension. For instance, a wedge of ...
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0answers
33 views

Morphism induced in cohomology of a covering space

It is a basic question but I'm stuck. If $p:M\rightarrow N$ is a $m$-fold unramified covering between surfaces, why the morphism induced by $p$ in cohomology at level 2 with coefficients in ...
3
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1answer
48 views

Isomorphism on top homology for closed, orientable manifolds

If $M$ and $N$ are orientable, closed $n$-manifolds, then $H_n(M)$ and $H_n(N)$ are generated by the fundamental classes $[M]$ and $[N]$. We can think of them as the sum of the faces of triangulations ...
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1answer
34 views

Discussion of local homology groups on Hatcher's Algebraic Topology

In p. 126 of Hatcher's Algebraic Topology, there is a discussion after theorem 2.26 about local homology groups. In particular, he says that the local homology groups of $X$ at a point $x \in X$ are ...
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1answer
66 views

Hochschild (co)homology and derived functors

Suppose that $A$ is (complex) unital algebra. We will consider $A-A$ bimodules $M$: such a bimodule is the same as (say) left $A \otimes A^{op}$ module. Let us define $C_n(A,M)$ as $M \otimes ...
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0answers
76 views

Relative (co)homology of closed submanifolds

Let $M$ be a closed orientable manifold of dimension $n$ and $A$ a closed submanifold. I obtained these equivalences on relative homology and cohomology groups (with coefficients in arbitrary ring): ...
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Vector space identity from Chow's “You Could Have Invented Spectral Sequences”

In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces: $$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus ...
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33 views

What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that ...
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0answers
59 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
2
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2answers
79 views

First Cohomology Group

Is it true that the first cohomology group of a differentiable manifold with finite fundamental group is trivial? If so, could you explain why? Thanks very much
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1answer
28 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
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59 views

Homology of $\mathbb{R}\setminus A_+$. [duplicate]

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that ...
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30 views

Relative singular homology $H(M,\partial M)$ for a manifold $M$?

Let $M$ be an orientable manifold. What can be said about the relative homology $H(M,\partial M)$? Perhaps one can calculate the homology using excision?
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22 views

Suspension of a cup product

Suppose we have a multiplicative cohomology theory $E$. Hence we have suspension isomorphisms $E^n(X, o) \to E^{n+1}(\Sigma X, o)$. Take two elements $x, y \in E^*(X, o)$ of degree's $i,j$ with ...