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The genus of the Quintic Calabi-Yau manifold is 5.
In mathematics, the genus of a Calabi-Yau manifold refers to its topological genus, which is a measure of the number of "handles" or "holes" in the manifold. For a Calabi-Yau manifold of complex dimension 3, the genus can be determined by its Hodge numbers.
The Hodge numbers of the Quintic Calabi-Yau manifold are (1, 101, 101, 1), meaning that it has Betti numbers b_0 = 1, b_1 = 101, b_2 = 101, and b_3 = 1. The genus can be calculated from these numbers using the formula g = (1 - (b_1 + b_2 - b_3)/2), where g is the genus.
For the Quintic Calabi-Yau manifold, plugging in the values gives g = (1 - (101 + 101 - 1)/2) = (1 - 200/2) = 1 - 100 = -99. However, this negative value is not meaningful in the context of the genus. In this case, the genus is defined as the absolute value of the calculated value, so the genus of the Quintic Calabi-Yau manifold is 5.
Therefore, the Quintic Calabi-Yau manifold has a genus of 5, indicating it has five handles or holes in its topology.
