On the global gravitational instanton and soliton that are homotopy spheres

Abstract

A generalization of the Dirac string trick for higher-dimensional manifolds is presented. As a rather unexpected result, the procedure is shown to yield homotopy spheres. Because of their relevant interest to physical theories, particular emphasis is given to even-dimensional homotopy spheres. For instance, it is demonstrated that a nontrivial representation of 6–10-dimensional exotic spheres in dimensions 4k+2 can be performed by using the Pontrjagin–Thom construction. The physical content of the results is consistent with Witten’s hypothesis [Commun. Math. Phys., 100, 197–229 (1985)] that n–10-dimensional exotic spheres can be interpreted as gravitational instanton and/or soliton. It is also shown that three mutually diffeomorphic elements can be isolated from the 6–10-dimensional exotic spheres and their spectrum is provided.