"In layman's terms, it [the genus] is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A doughnut, or torus, has 1 such hole. A sphere has 0."
So maybe hollowness isn't really a hole but a nested space with the same topological properties as the nesting object. For example a hollow ball is two balls, one inside the other, except the inner one is made of space and is therefore hollow all the way through. That inner one still has a surface with just one closed curve, just as the inner spatial torus inside a hollow torus has three closed curves on its surface.