Computation of $\tilde{H}_k(\bigvee_{j = 1}^N S^n)$.

Question

In my algebraic topology course, we state the following proposition $$ \tilde{H}_k(\bigvee_{j = 1}^N S^n) \cong \begin{cases} \mathbb Z^N & \text{if } k = n,\\ 0 & \text{else}, \end{cases} $$ which should be a direct corollary of the following isomorphism $$\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y).$$ However, by this isomorphism, shouldn't we have $$\tilde{H}_0(\bigvee_{j = 1}^N S^n) \cong \mathbb Z^N$$ as well ? I think my teacher made a mistake here, I just want to be sure I am not missing something.

Answer 1

Note that $\tilde{H}_i$ denotes reduced homology, so $\tilde{H}_0(S^n) = 0$ and hence $$\tilde{H}_0\left(\bigvee_{j=1}^NS^n\right) = \bigoplus_{j=1}^N\tilde{H}_0(S^n) = \bigoplus_{j=1}^N0 = 0.$$