On Two Classical Theorems of Algebraic Topology

Abstract

For theorems which have stimulated so much further research, beginning with the work of H. Hopf and continuing to the present, and whose content is so clear and easy to state, they are surprisingly difficult to prove, even in the simplest cases-the unit disk and the ordinary 2-sphere. Proofs are customarily given in standard courses in algebraic topology, but only after a fairly extensive theory is developed. In a brief, but very readable and elegant book [3 ], J. Milnor gave relatively simple proofs, based in part on a very original approach due to M. Hirsch [4]. Since this book goes into many generalizations of these theorems, it introduces and uses some techniques of differential topology which we wish to avoid, for example Sard's theorem. In this paper it is our purpose to make use of the fact that several advanced calculus texts discuss both manifolds and exterior differential forms in the presentation of multiple integrals and Stokes' theorem, and thus to rewrite the standard classical proofs using these techniques to avoid overt use of algebraic topology. In fact the present treatment goes very little beyond material to be found in the texts of either Fleming [6] or Devinatz [7] or in the more specialized book of Flanders [5], and the proofs of these theorems become, then, a somewhat delicate exercise in advanced calculus.