# How useless can the Mayer-Vietoris sequence be in general?

In an algebraic topology training course I'm taking we are usually asked to calculate the homology teams of a room $X = A \cup B$ making use of the Mayer - Vietoris series, and also it takes place in all of the instances I've seen until now that it is feasible to do this without recognizing anything concerning the attaching homomorphisms $\partial_{\ast}$ (claim on the degree of chains) ; we just wind up requiring $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ and also perhaps several of the incorporation maps.

My hunch is that this is not a regular scenario ; exists a reasonably straightforward instance of a wonderful room $X$ and also wonderful subspaces $A, B$ such that recognizing $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ is not nearly enough to calculate $H_{\ast}(X)$ without recognizing the details kind of the attaching homomorphisms? (For topmost importance to the training course $X, A, B, A \cap B$ need to be limited simplicial facilities.)

Given a lengthy specific series

$$ \cdots \to C_{i+1} \to A_i \to B_i \to C_i \to A_{i-1} \to \cdots $$

allowed the map $A_i \to B_i$ be represented $f_i$. After that you have that $C_i$ is an expansion

$$ 0 \to coker(f_i) \to C_i \to ker(f_{i-1}) \to 0$$

so approximately that expansion trouble, the maps $f_i$ constantly establish the $C_i$ teams. So if you desire a scenario where the team $C_i$ is unclear, you can have $ker(f_{i-1}) = \mathbb Z_2$ and also $coker(f_i) = \mathbb Z$, in this way $C_i$ can be either $\mathbb Z$ or $\mathbb Z \oplus \mathbb Z_2$.

No matter, the attaching map $\partial_i : C_i \to A_{i-1}$ is established by this expansion trouble, and also it is very easy sufficient to prepare instances either - means.

So I'm a little overwhelmed regarding the nature of your inquiry. I presume what I'm claiming is that you *are * in the regular scenario, and also Grigory is instance is additionally regular because it is the incorporation map that makes the distinctions in between his instances.

Pertaining to just how useful/useless the MVS is for a regular trouble, it actually relies on just how conveniently - expressible your room is as a union of rooms you recognize (and also their junctions). If your room does not fit that account, you've obtained possibly a great deal of job to do. The Serre Spectral Sequence of a Fibration remains in a feeling something of a souped - up Mayer - Vietoris series, and also there are a lot of documents where individuals enjoy simply calculating the $E_3$ - web page, or establishing which web page the SS falls down on, or calculating a differential. These expansion concerns often tend to be really tough and also eat much literary works.

Knowing just $H(A)$, $H(B)$ and also $H(A\cap B)$ is not nearly enough, certainly. As an example, taking $A=B=S^1\times D^2$ and also gluing them by $S^1\times S^1=A\cap B$ one can get either $X_1=S^2\times S^1$ or $X_2=S^3$. This offers 2 Mayer - Vietoris series with the same $H(A)$, $H(B)$ and also $H(A\cap B)$ yet various H (X).

When it comes to the scenario where one additionally recognizes incorporation maps, see Ryan Budney is superb answer.

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