# 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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### 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 ...
1 vote
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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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