Changes of cohomology after destroying a cycle

Question

Suppose $M$ is a manifold and $i: S^3 \hookrightarrow M$ is an embedding whose image induces a non zero homology class of $H_3(M,\mathbb{Q})$. If we now destroy this cycle: take the quotient $M/i(S^3)$ where the 3 sphere $i(S^3)$ shrinks to a point. How does the homology and cohomology change in this quotient process? Do we have $\text{dim}\,H^3(M/i(S^3),\mathbb{Q})=\text{dim}\,H^3,\mathbb{Q})-1$ in general?