# 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 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 ...
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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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### 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 ...
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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 ...
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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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### 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 ...
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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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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 ...
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 ...