Some Elementary Applications of Topological Degree

In this paper we consider the degree theory for a mapping f from an oriented compact connected manifold X to a oriented, compact, connected manifold Y of the same dimension and some elementary problems of fixed points. 1. Topological degree on manifolds. There are many approaches to the introduction of the notion of degree and the background involved in each of them may differ considerably.We briefly introduce the notion of topological degree on manifolds and for more details we refer the reader to Nirenberg,L., [N] and Berger,M. & Gostiaux,B., [BG]. Let X be an n−dimensional C ′ manifolds and q, r integers such that 0 ≤ q ≤ q ′ − 1 and 0 ≤ r ≤ n. Let ω be a C differential form of degree r, or r − form, on X. The space of r − forms of class C on X will be denoted by Ωq(X). In this section everything is of class C , and Ω = Ω∞(X). Definition. Let X and Y be two oriented compact connected manifolds of same dimension d, and f ∈ C(X;Y ) a map. There exists an integer, called the degree of f and denoted by deg(f), such that: (i) if ω ∈ Ω(Y ) we have Received: June, 2001