Solution
= Solution
\b[No.] Collapse the one-skeleton of the torus summand to obtain a map $X\to S^2$ inducing degree one on $H_2$. A null-homotopic map induces zero on reduced homology.
= Solution
\b[No.] Collapse the one-skeleton of the torus summand to obtain a map $X\to S^2$ inducing degree one on $H_2$. A null-homotopic map induces zero on reduced homology.