If $\pi \colon X \longrightarrow Y$ is a cyclic cover of complex projective manifolds of dimension $n$ branched along an ample divisor $B \subset Y$, then by a variation of Lefschetz Hyperplane Theorem one can show that the pullback map $$\pi^* \colon H^i(Y, \, \mathbb{Z}) \longrightarrow H^i(X, \, \mathbb{Z})$$ is an isomorphism for $i \leq n-1$ and is injective for $i=n$, see Proposition 1.1 in **[1]**.

The corresponding dual result for the push-forward map in integral homology is also true, see Proposition 1.11 in **[2]**.

Using Poincaré duality, from this it follows that all Betti numbers of $X$ and $Y$ are equal, except possibly the middle Betti number, for which we have
$b_n(X) \geq b_n(Y)$.

On the other hand, in your case we can easily find the topological Euler number $\chi_{\mathrm{top}}(X)$ by additivity. In fact the ramification locus $R \subset X$ is isomorphic to the (smooth) branch locus $S \subset \mathbb{P}^n$ and the restriction $\pi \colon X - R \longrightarrow \mathbb{P}^n - S$ is an unramified double cover, so we obtain
$$\chi_{\mathrm{top}}(X) = 2(\chi_{\mathrm{top}}(\mathbb{P}^n) - \chi_{\mathrm{top}}(S))+ \chi_{\mathrm{top}}(R) = 2\chi_{\mathrm{top}}(\mathbb{P}^n) - \chi_{\mathrm{top}}(S)$$
and this allows us to recover $b_n(X)$ as well.

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**References**

**[1]** J. A. Wisniewski: **On topological properties of some coverings. An addendum to a paper of Lanteri and Struppa**, *Can. J. Math.* **44**, No.1, 206-214 (1992). ZBL0766.14012.

**[2]** A. Lanteri, D. C. Struppa: **Topological properties of cyclic coverings branched along an ample divisor**, *Can. J. Math.* **41**, No.3, 462-479 (1989). ZBL0699.14019.