Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a finite-sheeted cover of \( Y \). Prove or disprove that if \( Y \) has a nontrivial first Betti number, then \( X \) also has a nontrivial first Betti number.
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The finite sheetedness condition can be dropped as it is implied from being a covering between closed manifolds