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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Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
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1answer
37 views

Computing the homology groups of a quotient space of the sphere

I want to solve following question: Let $A$ denote the union of equatorial circle and the north pole on $S^2$. Let $X=S^2 / A$. Compute the homology groups of X. I calculated that $H_2(X) = \Bbb ...
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54 views

Reduced homology groups of a space which is the union of finitely many open subsets

This is exercise 33 (p.158) from section 2.2 in Hatcher's Algebraic Topology: Suppose the space $X$ is the union of open sets $A_1, \ldots, A_n$ such that each intersection $A_{i_1} \cap \cdots ...
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1answer
45 views

Poincaré lemma on a space with trivial homology group

Today I read about Poincaré's lemma from do Carmo's book Differential Forms and Applications. It says that A closed differential $k$-form on a contractible space is exact. I wonder if the ...
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40 views

Basic Question on Mayer-Vietoris Sequence

On Pg 449 of Lee's Introduction to SMooth Manifolds (2nd Edition), the Mayer-Vietoris Theorem is given: Let $M$ be a smooth manifold. Let $U$ and $V$ be open in $M$ such that $U\cup V=M$. Then ...
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1answer
19 views

Contractible chain complex

Suppose that we have a chain complex $... \to P_n \to ... \to P_1 \to P_0 \to 0$ (denote by $d_k:P_K \to P_{k-1}$ the differential of this complex) and suppose that we have $D_k:P_k \to P_{k+1}$ such ...
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1answer
62 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
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2answers
78 views

Finding the de Rham cohomology of an open subset of $ \Bbb{R}^{n} $ minus a point.

Here’s my question: Let $ n \in \mathbb{N}_{\geq 2} $. Suppose that $ U \subseteq \Bbb{R}^{n} $ is an open set and that $ x \in U $. Then show that $$ {H_{\text{dR}}^{n - 1}}(U \setminus \{ x ...
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1answer
49 views

computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...
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1answer
25 views

first Chern class of E is first Chern class of det E

Let $\pi:E\to M$ be a vector bundle, and $\nabla$ a connection. My definition of the first Chern class is $$c_1(E)=\left[tr\left(\frac{i}{2\pi}F^\nabla\right)\right],$$ where $F^\nabla$ is the ...
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1answer
62 views

Understanding Hatcher's proof for $\chi(M)=0$ for non-orientable manifolds $M$ of odd dimension

In the Corollary 3.37 Hatcher proves that for a closed odd-dimensional manifold $M$, its Euler characteristic is zero. The first part of the proof deals with orientable manifolds, and uses Poincare ...
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1answer
83 views

Cohomology Group of $CP^2 \wedge CP^2$

Calculate the cohomology group of $CP^2 \wedge CP^2$ To do this, at first I am trying to calculate the homology group and then use Universal Coefficient Theorem. To do this, at first I have ...
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1answer
46 views

Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
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3answers
41 views

Relative homology $H_n(S^2,S^0)$, or other examples

I've been reading Hatcher and think I understand the idea of relative homology, but he only provides two (fairly trivial) examples, homology relative to a point computing $H(S^n)$ using $D^n$s. My ...
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0answers
29 views

Cohomology of two pieces of torus

In an exercise from an old exam, I found myself confronted with $M=\{(\sqrt{x^2+y^2}-2)^2+z^2=1\}$, $U=M\cap\{x\neq0\vee y>0\}$, $V=M\cap\{x\neq0\vee y<0\}$ and $U\cap V$, all subsets of ...
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1answer
48 views

Compute Euler characteristic of a compact manifold

We have the manifold embedded in $\mathbb{R}^4$ given by $$M=\{(x,y,z,w)|2x^2+2=2z^2+w^2,3x^2+y^2=z^2+w^2\}$$ How could I compute the Euler characteristic? I've no idea computing the homology group of ...
2
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1answer
51 views

Calculating $H_1(\mathbb{R})$

Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't ...
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24 views

The Homomorphism Induced by a Continuous Map

'' §3. The Homomorphism Induced by a Continuous Map Homology theory associates with every topological space X the sequence of groups Hn(X), n = 0, 1,2, .... Equally important, it associates with ...
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1answer
50 views

Proving that $H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$

In the article about the Reduced Homology it's stated that $$H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$$ but I don't know how to prove that. I know $$H_0(X)=\bigoplus_{\alpha\in ...
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0answers
34 views

How to compute perverse sheaves?

In the video, from 49:00 to the end of the video, there is an example of computing $IC(S, L)$ and equivariant local systems. I don't understand some parts of the computations. Let $X$ be the variety ...
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0answers
43 views

Find cohomology ring of $H^*(S^n \times X)$ [closed]

I know that $H^* (S^1 \times X) \equiv H^*(S^1) \otimes H^*(X)$ . Now how to generalize it for $S^n$. Please state in brief and without using category and Kanneth formula.
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How to show that the boundary of an antipodally symmetric 1-chain contains an even number of antipodal pairs?

This is an exercise in Jiri Matousek's book 'Using the Borsuk-Ulam Theorem' which I'm going through. A 1-chain is of course a collection of 1 dimensional simplices (edges). A chain is antipodally ...
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1answer
32 views

Does an Homotopy between local homeos preserve orientation behaviour?

Suppose we have two compact, orientable $n$ manifolds $M,N$, and two homotopic local homeos $f,g \colon M \to N$. Suppose moreover that $f$ is orientation preserving. Is it true that $g$ is ...
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1answer
46 views

Expressing generalized cohomology by ordinary cohomology

I'd like to ask for either pointing an error or confirming correctness of the following reasoning. Theorem: let $h^* \colon CW \to Ab$ be a cohomology theory, then there exist abelian groups $ ...
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54 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
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0answers
45 views

Commutative diagram of cohomology (to show Albanese variety is a torus)

Suppose $X$ is a compact Kahler manifold of complex dimension $n$, define $H_1(X,\mathbb{Z})\to H^0(X,\Omega_X^1)^*$ by $[\alpha]\to \int_\alpha\cdot-$. We want to show the image of ...
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1answer
27 views

Does acyclic resolution induces morphism of cohomologies?

Consider sheaves $F,G$ of abelian groups on a topological space $X$. Fix some $f^0\colon F\to G$. Given chain map between injective resolutions $0\to F\to I^*$, $0\to G\to J^*$, denoted by $f^0\colon ...
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1answer
25 views

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
94 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
77 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 ...
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1answer
50 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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1answer
82 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
27 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
38 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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31 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 ...
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1answer
58 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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68 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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35 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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47 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 ...
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1answer
43 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
22 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
94 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 ...
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1answer
65 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 ...
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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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13 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
32 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 ...
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1answer
58 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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101 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 ...
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1answer
30 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 ...