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.)

2022-06-07 14:39:05
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Answers: 2

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.

2022-06-07 15:05:38

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.

2022-06-07 15:05:02