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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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 ...
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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 $...
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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-...
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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 ...
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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 ...
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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^...
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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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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^...
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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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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 ...
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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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Mayer-Vietoris Sequence with Pairs Applied to (Oriented) Connected Sum of Manifolds
I was trying to figure out why the (oriented) connect sum of two oriented manifolds should be oriented. Here the oriented connect sum $M_1\# M_2$ of two $n$-manifolds $M_1$ and $M_2$ is taken to be ...
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natural exact triple sequence, Mayer-Vietoris sequence
I have some questions on the proof of the following lemma, which is used to deduce the exactness of triple sequences, of the rows in the Mayer-Vietoris sequence.
Let $(\mathcal{H}_\ast, \partial_\ast)$...
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