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Questions tagged [mayer-vietoris-sequence]

The Mayer-Vietoris sequence is a powerful tool to compute the integer coefficients homology of a topological space. It is a long exact sequence relating the homology of a topological space, the homologies of covering subsets and the homologies of their intersections.

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Mayer-Vietoris for Morse homology

Let $M$ be a compact oriented manifold with boundary and $(f,X)$ be a Morse-Smale pair on it. We assume that none of the critical points lie on the boundary, which can then be decomposed as $\partial ...
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Generalizing the relative homology group of the solid torus relative to the hollow torus

In Hatcher's Algebraic Topology textbook, we are given some tools to calculate relative homology groups, the prime example of which, as I have seen, are finding the relative homology group of the ...
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Mayer-Vietoris for Sheaf with support vs Usual Mayer-Vietoris

In Hartshorne Chapter III, Exercise 2.4, (for a topological space $X = Y_1\cup Y_2$) one is asked to prove the exactness of $$ \ldots \to H^i_{Y_1\cap Y_2} (X,\mathcal{F}) \to H^i_{Y_1} (X,\mathcal{F})...
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Homology groups of genus $g$ Riemann surface with $n$ punctures.

I want to know whether one can compute $H_k(\Sigma_{g,n})$ with elementary methods like singular chain complex? By $\Sigma_{g,n}$ I mean a genus $g$ surface with $n$ circular holes. If not, I'm not ...
hossein mohammadi's user avatar
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Mayer Vietoris Sequence of the Circle

Let $\mathcal{H}_*$ be a ordinary homology theory, that is, a homology theory that satisfies the Eilenburg-Steenrod Axioms. I wanted to show that $\mathcal{H}_1(S^1)\cong \mathcal{H}_0(S^1)\cong R$, ...
JenkinsUnderTheSun's user avatar
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Looking for a proof for Mayer-Vietoris sequence for $R$-modules

Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but ...
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Checking reasoning and computations in cohomology

I am trying to compute the cohomology for the complex projective space $P_{\mathbb{C}}^2$ which we know is a manifold of dimension $4$. Recall that $P_{\mathbb{C}}^2 = {\mathbb{C}}^3 - \{0\}/\sim$ ...
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Explanation of these Mayer-Vietoris Maps

I am reading about the Mayer Vietoris sequence $$\cdots \xrightarrow{\beta}H_c^{q-1}(A\cap B) \xrightarrow{\delta} H^{q}(A\cup B) \xrightarrow{\alpha}H^{q}(A) \oplus H^{q}_c(B) \xrightarrow{\beta}H^{q}...
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Examples using this long exact sequence of cohomology [duplicate]

I began reading this paper (also provided below), and I would really appreciate if someone could help me with the premises behind the first sentence. I have not been able to find the given long exact ...
June in Juneau's user avatar
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Help with this short exact sequence of cohomology

My goal is to understand a particular short exact sequence derived from a Mayer-Vietoris Sequence. I will tell you what I have so far, and then I am stuck in defining the last group. We begin with a ...
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Sequence of Inclusions in the Definition of Mayer-Vietoris Sequence is unclear

I'm currently reading Bott and Tu's "Differential Forms in Algebraic Topology". The authors introduce the Mayer-Vietoris Sequence as follows: Suppose $M = U \cup V$, with $U,V$ open. Then ...
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Does Hatcher's barycentric subdivison argument of Prop. 2.21 also hold for smooth singular chains?

Suppose that $X$ is a topological space, covered by open sets $A$ and $B$. When deriving the Mayer-Vietoris sequence for simplicial homology, the standard argument is to argue for the exactness of the ...
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Compute the cohomology of the n dimensional torus with disjoint circles removed

I'm particularly interested in the case of $n=3$ but the most general formulation of the problem I'm trying to tackle is the following: Let $\mathbf{T}^n$ be the n-dimensional torus and $W=L_1\sqcup\...
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Mayer-Vietoris for smooth manifold with boundary

Disclaimer: I am aware that this question is entirely moot, since any smooth manifold with boundary is homotopic to its interior! Nonetheless, I am still interested in learning what goes wrong if we ...
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Homology of exterior of a solid knot in $S^3$

Let $K$ be a knotted solid torus in $S^3$, $T$ be its boundary torus, and $X$ be its exterior, i.e. the closure of $S^3\setminus N$. The goal is to compute $H_n(X)$, the integral homology of $X$. ...
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calculate homology of mapping cone

Here is the problem statement I am concerned of: For a space $A$ let $CA = A\times [0,1]/A\times \{1\}$ be the cone and if $f:X\to Y$ let $C_f = (Y\coprod CX)/\sim$ where we identify $(x,0)\sim f(x)$....
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Contravariant functor $\Omega^*$ from category of smooth manifolds to commutative differential graded algebras

I thought I understood the Mayer-Vietoris Sequence in the context of De Rham cohomology, but I have realized that I have taken something for granted in the set up of the proof, and I can't quite ...
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John Lee's ISM Proposition 18.13 Proof Clarification

In John Lee's Introduction to Smooth Manifolds, Proposition 18.13's proof, the author states "By the characterization of $\partial_*$, we can let $c=\partial f$ where $f,f'$ are smooth $p$-chains ...
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Generalised Mayer-Vietoris long exact sequence

In chapter 8 of Bott/Tu, the authors generalise the standard Mayer-Vietoris sequence to the setting of a countable open cover of $X$. Let's fix a countable cover $\{ U_i\}$ of $X$. According to Prop 8....
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Is the cohomology of a pushout of open sets a pullback of cohomologies?

I recently found an interesting question, and I'd like to ask the co-question. I will differ from what is linked by using de Rham cohomology, since its the easiest way for me to articulate my ...
ArchimedeezNuts's user avatar
2 votes
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Computing the integral cohomology ring of intersection of two spaces

$\newcommand{\C}{\mathbb{C}}\newcommand{\cn}{\colon}\newcommand{\R}{\mathbb{R}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\sm}{\setminus}$I'm trying to compute the integral cohomology ring of $X=\{(x,y)\...
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Injectivity of map into $H_n(S^n \setminus h(X \times [0,\frac{1}{2}]) \oplus H_n(S^n \setminus h(X \times [\frac{1}{2},1]))$

This was an exercise from last week in our topology class. Since there are no solutions I wanted to ask here. Let $h:X \times [0,1] \to S^n$ be an embedding. Where $X$ is connected. Let $\gamma \in ...
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Decomposition of a space as a union of two closed subspaces that dosen't form an Mayer-Vetoris pair.

When applying Mayer-Vietoris sequence in homology theory, one case is where $X=X_1\cup X_2$, $X_1$ and $X_2$ are closed subspaces of $X$ and $X_1\cap X_2$ is a deformation retract of one of its open ...
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Understanding the correlation between the Van Kampen theorem and Mayer Vietoris sequences from the Hatcher Algebraic Topology text

I am trying to understand a sentence from Hatcher's section 2.2 on Mayer Vietoris sequences. It states that: "Mayer-Vietoris sequences can be viewed as analogs of the van Kampen theorem since if $...
Student005's user avatar
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Homology groups of disjoint sets

I have the following question If $ X=A\cup B$ where $A$ and $B$ are open non empty sets and $A\cap B = \emptyset$ then $H_n(X) \cong H_n(A) \oplus H_n(B)$. My attempt is to prove this using Mayer-...
Marwa Zeyad's user avatar
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Homology groups of the quotient space obtained by gluing the boundary of the Möbius band to the circle $x^2 + y^2 =1$ of $S^2$

For homework i have to compute the homology groups of the quotient space obtained by gluing the boundary of the Möbius band to the circle $x^2 + y^2 =1$ of $S^2$. But I find a bit difficult to ...
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Why are (co)homology groups algebraic invariants?

Background & definitions: I am studying algebraic topology, in particular (co)homology groups and Mayer-Vietoris sequences. Important terms: Algebraic invariant = property of a topo. space ...
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Mayer-Vietoris Sequence and Poincare dual

Given a $4$-dimensional simply connected manifold $M$ and open sets $U,V\subseteq M$ such that $U\cup V=M$ we can compute the deRham cohomology in terms of the Mayer-Vietoris sequence: \begin{align*} ...
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Compactly supported cohomology of open disk plus a point

Let $X=\{v\in\Bbb R^2:\|v\|<1\}\cup\{(-1,0)\}$, that is, the open unit disk union a point on its boundary. (Here, $(-1,0)$ refers to a point in the plane, not an interval.) What is $H_c^•(X)$, that ...
Akiva Weinberger's user avatar
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1 answer
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Computing singular homology of a cylinder with a bottom, using Mayer Vietoris Sequence

I am trying to practice using MVS on an easy example, a cylinder with a bottom. Explicitly, something like $S^1 \times [0,1]$ with a copy of $D^2$ glued at one end. Call this object $X$. Then I ...
pyridoxal_trigeminus's user avatar
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1 answer
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Why the torus has open subsets diffeomorphic to a cylinder?

If I take the torus $T$ of revolution in $\mathbb{R}^3$ is very intuitive for me that there is an open subset of $T$ that is diffeomorphic to a cylinder, for example $T$ minus a circle on the $xz$ ...
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Cohomology of a blow up and Mayer-Vietoris

I have been reading the section on the cohomology of a blow-up in Griffths-Harris "Principles of algebraic geometry", and I am a bit confused about a result that their recipe, which uses the ...
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$\mathbb R^n=U \cup V$ where $U, V$ connected open subsets then show that $U\cap V$ connected?

$\mathbb R^n=U \cup V$ where $n\ge 2$ and $U, V$ open and connected is the intersection $U\cap V$ connected? My attempt is: to try Mayer Vietoris sequence, since $\mathbb R^n$ is contractible its ...
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Two fold map while calculating Mayer–Vietoris sequence of $\mathbb RP^2$.

My reference answer is: https://math.stackexchange.com/a/768843/342943 My problem is: While writing MW exact sequence $$\cdots\to H_2(M)\oplus H_2(D^2)\to H_2(\mathbb{R}P^2)\to \underbrace{H_1(S^1)}_{\...
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Proof of $\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$.

In my algebraic topology course, we showed the following statement: Let $X, Y$ two topological spaces, $(x_0, y_0) \in X \times Y$ and $X\lor Y$ the wedge product of $(X, x_0)$ and $(Y, y_0)$. If $x_0$...
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Mayer-Vietoris Exact Sequence for Whitehead groups and projective module groups

I came across the claim that there is an exact sequence $K_1\Lambda\to K_1\Lambda_2\oplus K_1\Lambda_2\to K_1\Lambda'\to K_0\Lambda\to K_0\Lambda_1\oplus K_0\Lambda_2\to K_0\Lambda'$ while reading ...
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A question about Mayer-Vietoris sequence [duplicate]

Let $A$ and $B$ be two open sets in $\mathbb{R}^n$ ($n>1$), with $A \cup B=\mathbb{R}^n$ and $A\cap B\not=\emptyset$. There is a version of the Mayer-Vietoris sequence for reduced homology that ...
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homology group of connected sum of Klein bottle and $RP^2$

How to compute the homology group of connected sum of Klein bottle and $RP^2$? I want to use mayer vietories sequence,I know Klein bottle remove a disk is $S^1 \vee S^1$,and $RP^2$ remove a disk is $S^...
zeng's user avatar
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Prove that $\frac{H_1(\Sigma)}{[\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]} \cong H_1(Y)$

I am looking at Definition 2.11 in this paper: https://arxiv.org/pdf/math/0101206.pdf. In particular, given a Heegaard diagram $(\Sigma, \alpha, \beta)$ for a 3-manifold $Y$, how to prove the ...
CCC's user avatar
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2 answers
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Homology of complement of a $m$-sphere in $\mathbb R^n$, with $m<n$

Trying to solve the problem in the title, I found this post where a particular case is described and the answer gives a generalization that I think could help me. Here is my first doubt: How to see ...
GingFreecss17's user avatar
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1 answer
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Homology of 4-tori $T^4$ minus a point

I have this exercise where I'm asked to compute the homology groups of the 4-tori $T^4$ minus a point $p$, where $T^4\simeq (\mathbb{R}/\mathbb{Z})^4$. I'll denote the space $T^4 \setminus \{p\}$ with ...
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1 answer
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Deriving the long exact sequence of a pair $(X,A)$ from the Mayer-Vietoris sequence

It is an exercise in Hatcher's Algebraic Topology text to derive the long exact sequence of a pair $(X,A)$ from the Mayer-Vietoris sequence applied to $X \cup CA$. We may use the isomorphism $H_n(X,A)...
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The cohomology of the surface bundle over $S^1\vee S^1$

Given a surface bundle $S_g\to E \xrightarrow{f} S^1\vee S^1$, where $S_g$ is a closed surface of genus $g$. I want to calculate $H^3(E)$ by applying Mayer-Vietoris. Name the two circles of $S^1\vee S^...
Danny's user avatar
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How to calculate connecting Homomorphisms in de Mayer-Vietoris long exact sequence

I have a specific problem, but it generalizes to the problem on how to compute connecting homomorphisms of any shape or form. So the problem is the following: Let $S^n$ be the unit sphere, obtined by ...
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Problem in computing $H_*(C(\phi_{n,k})).$

Let $\phi_{n,k}$ denote $\Sigma^{n-1} \phi_k : S^n \longrightarrow S^n.$ Compute $H_*(C(\phi_{n,k})),$ where $C(\phi)$ is the mapping cone of $\phi.$ Here $\Sigma^{n-1}$ denotes the $(n-1)$-fold ...
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Is this proof for Mayer-Vietoris theorem correct or wrong?

Here is a proof for Mayer-Vietoris theorem for cofibre homology theory from TOPOLOGICAL METHODS FOR C*-ALGEBRAS III:AXIOMATIC HOMOLOGY (p426 theorem 4.1) Theorem 4.1. Let $h_*$ be a cofibre homology ...
Yuz's user avatar
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Calculate the homology of the mapping cone of $\Delta.$

Let $\Delta : S^n \longrightarrow S^n \times S^ n$ be the diagonal map given by $\Delta (x) = (x,x),$ $x \in S^n.$ Calculate $H_{*} (C(\Delta)),$ the homology of the mapping cone of $\Delta.$ I have ...
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How to compute the homology of two sphere with two handles

How to compute the homology of two sphere with two handles?? here is the figure two sphere with two handles
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Computing the homology group of the real projective plane using Mayer-Vietoris sequence.

Compute the homology groups $H_* \mathbb RP^2$ using Mayer-Vietoris sequence. I know that $\mathbb R P^2$ is obtained from the Möbius strip by attaching a $2$-cell along the boundary circle of the ...
Fanatics's user avatar
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2 answers
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Reduced homology: Mayer vietoris sequence

I'm reading Hatcher's book "Algebraic topology", the section about Mayer-Vietoris sequences (p150). It is claimed that there is also a version for reduced homology. However, let $X$ be space ...
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