I want to show that $[\Delta^n_1-\Delta^n_2]$ is a generator of $H_n(\mathbb{S}^n)$ for $n \geq 1$, where $\Delta^n_i$ are identification of a small neighborhood of the upper hemisphere and the lower hemisphere with the standard simplex $\Delta^n$.
My attempt is to consider the Mayer-Vietoris sequence, which gives rise to the isomorphism $$\partial:H_n(\mathbb{S}^n) \to H_{n-1}(\mathbb{S}^{n-1})$$ where $\partial$ is the connecting homomorphism in the Mayer-Vietoris sequence.
Then I apply induction. I have proved the base case $n=1$. Now, consider $n=k$. My thought of using the inductive hypothesis is that since $\partial$ is an isomorphism, to show that $[\Delta^k_1-\Delta^k_2]$ is a generator of $H_k(\mathbb{S}^k)$, I need to show that $\partial[\Delta^k_1-\Delta^k_2]=[\Delta^{k-1}_1-\Delta^{k-1}_2]$. But I stuggle in doing so.
My question is: Is $\partial[\Delta^k_1-\Delta^k_2]=[\Delta^{k-1}_1-\Delta^{k-1}_2]$ true, and if yes, how can I show it? If not, is it possible that I can do it using the Mayer-Vietoris sequence argument above?
Thank you very much!