Lifting Surgeries to Branched Covering Spaces
Hugh Michael Hilden, José María Montesinos · Transactions of the American Mathematical Society · 1980
It is proved that if ${M^n}$ is a branched covering of a sphere, branched over a manifold, so is ${M^n} \times {S^m}$, but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple covering of ${S^n}$ branched over an orientable manifold. The proof involves the development of a new technique to perform equivariant handle addition. Other consequences of this technique are given.