Degree theory and BMO; part I: Compact manifolds without boundaries

In this paper we consider degree theory for mappings u from one compact smooth n-dimensional manifold X to a connected compact smooth manifold Y of the same dimension. These are manifolds without boundary and which are oriented. The classical degree counts the "number of times" Y is covered by u(X), taking into account algebraic multiplicity. For instance, if u E C 1 and y E Y is a regular value of the map u, i.e., u l ( y ) consists of a finite number of points x l , . . . ,xk at each of which the Jacobian of the map, d~, in terms of local coordinates (with the given orientation), is nonsingular, then