Let $A = C \times T^{n - 1}$ and $B = D \times T^{n - 1}$, where $C$ and $D$ are two overlapping arcs of one of the $S^1$'s satisfying that $S^1 = C \cup D$ and $T^n = S^1 \times \cdots \times S^1$. Now consider the Mayer-Vietoris sequence
$$ \cdots \to H_n(A) \oplus H_n(B)\to H_n(T^n) \to H_{n - 1}(A \cap B) \to H_{n - 1}(A) \oplus H_{n - 1}(B) \to \cdots$$
Assume that $H_{n - 1}(A) = H_{n - 1}(B) \cong \mathbb{Z}$. It is not hard to see that $A \cap B \simeq T^{n - 1} \sqcup T^{n - 1}$, which implies that $\varphi_* \colon H_{n - 1}(A \cap B) \to H_{n - 1}(A) \oplus H_{n - 1}(B)$ is a homomorphism between two groups isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$.
My question
Is $\ker \varphi_{*} \cong H_{n - 1}(A)$?
I believe the answer is yes because, as far as I understand, $\varphi\colon C_{n - 1}(A \cap B) \to C_{n - 1}(A) \oplus C_{n - 1}(B)$ sends $x$ to $(x, -x)$, which means that $\ker \varphi_*$ are... the cycles in $C_{n - 1}(A)$?