SPHERES AS FROBENIUS OBJECTS

Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension d = 1, all the spheres are commutative Frobenius objects in categories whose arrows are (d + 1)-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres-they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer.