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I am reading about the Mayer Vietoris sequence

$$\cdots \xrightarrow{\beta}H_c^{q-1}(A\cap B) \xrightarrow{\delta} H^{q}(A\cup B) \xrightarrow{\alpha}H^{q}(A) \oplus H^{q}_c(B) \xrightarrow{\beta}H^{q}_c(A\cap B) \xrightarrow{\delta} \cdots$$

(I attached a screenshot of my reference below). Then I am considering an example where $X = A \cup B$ is a 2-D CW complex, $H^{k}_c(A\cup B)$ is the cohomology of a wedge of spheres, and some of the other groups are known to be zero, which gives:

$$0 \xrightarrow{\alpha}H^{1}_c(A) \oplus 0 \xrightarrow{\beta}H^{1}_c(A\cap B) \xrightarrow{\delta} H^{2}_c(A\cup B) \xrightarrow{\alpha}H_c^{2}(A) \oplus 0 \xrightarrow{\beta} 0$$

Since $A \cap B$ and $A \cup B$ are flipped in this sequence compared to the one in Hatcher (p. 246), I'm assuming the short exact sequence of cochain complexes also change direction.

That is, this must be the cochain map:

$$0 \to C^k(X, (X-K) \cap (X-L)) \to C^k(X, X-K) \oplus C^k(X, X-L) \xrightarrow{\beta'} C^k(X, X- K\cap L) \to 0$$

Then in my situation, the coboundary map $\delta$ arises from this following short exact sequence where $K \subset A$ and $L \subset B$, and $(X, (X-K) \cap (X-L))$ is a wedge of spheres.

Then I assume the map $\beta'$ is defined as surjectively mapping chains in $K_i$ and $L_i$ to the intersection of both of these chains. This is equivalent to all chains in the intersection.

Question

My question is whether someone can point me on a better track as to how the coboundary map $\delta$ defined, particularly in the case where $H^{k}_c(B) = H^1_c(A \cup B) = H^3_c(A \cup B) = 0$. Here is my reasoning so far:

As Hatcher explains (p. 242), the coboundary of a 1-cochain can be nonzero only on 2-cells that have a face on which the 1-cochain is nonzero. That seems to imply that $\delta^{-1}$ of a 2-cocycle are all of the 1-cocycles that are nonzero on at least one 1-cycle within the 2-cell on which the 2-cocycle is nonzero. From this, I believe $\delta$ sends each 1-cocycle in $H^{1}_c(A\cap B)$ (which is nonzero on a 1-cycle $\sigma$ in $K\cap L$) to the 2-cocycle in $H^2_c(A\cup B)$ that is nonzero on the 2-cycle (sphere) in $K \cup L$ on which the 1-cycle $\sigma$ lies.

(A worry is the next page in Hatcher (p. 247) that says $\delta \phi = 0$, and so $\delta \phi_A = \delta \phi_B$. But in my sequence, $\phi_B \in H^{1}_c(B) = 0$, which seems to imply that $\delta$ is the trivial map. But this cannot be true just because $H^{1}_c(B) = 0$, which makes me think the previous paragraph is still correct and $\delta \phi = 0$ refers only to when $\phi \in \text{im} \beta$.)

Update: For the example in which $H^2_c(X)$ is the cohomology of a wedge of spheres and $H_c^1(A \cap B)$ a wedge of circles, the exactness of the sequence offers additional information. I think the coboundary map takes each 1-cocycle in $(X, X- K \cap L)$ to the 2-cocycle in $H^2_c(X)$ that is nonzero on the sphere on which the 1-cocycle is nonzero. (Maybe?) Then the 2-cocycles in $H^2_c(X)$ that are not in the image of the coboundary map will be those that nonzero only on the spheres that do not contain nontrivially evaluated 1-cycles in $(X, X- K \cap L)$. Then those map surjectively to $H_c^2(A)$.

Your help is really appreciated! As are any resources or intuition from one of the teachers or math people here.

enter image description here

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  • $\begingroup$ The page attached is from Cohomology with Supports, E. Spanier, 1986. Although I think I would have the same misunderstanding when looking at a different form of MV sequence in which H*(B)=0. $\endgroup$ Commented Jan 6 at 23:56

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To me it seems that Spanier studies cohomology theories with compact support, rather than "compactly supported cohomology". The two disagree in the following two ways:

  1. Compactly supported cohomology $H_c^*(-)$ is not a cohomology theory. For instance, it does not see contractible spaces as points: $ H^1_c(\mathbb{R}) \cong \mathbb{Z} \neq 0 = H^1_c(*)$.

  2. Assuming that the groups $H^*_c(X)$ may be extended to a cohomology theory with compact support, there will be a Mayer--Vietoris sequence as described. Let us apply this to the cover $(-\infty, 1] \cup [-1,\infty) = \mathbb{R}$. The proposed M.V. sequence contains: $$ H_c^1(\mathbb{R})\to H_c^1((-\infty, 1]) \oplus H_c^1([-1,\infty))\to H_c^1([0,1]) $$ simplifying to $$ \mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z} \to 0.$$ This cannot be exact, providing the unfortunate verdict that no such M.V. sequence can exist.

This means that the "reversed" version of this sequence, which you have already studied, is the only sequence one has!

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