Intersections and transformations of complexes and manifolds

In writing this paper my first objective has been to prove certain formulas on fixed points and coincidences of continuous transformations of manifolds. To this proof for orientable manifolds without boundary is devoted most of the second part, the remainder of which is taken up by a study of product complexes in the sense of E. Steinitz, as they are the foundation on which the proof rests. With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion. It may be stated that our formulas include and completely generalize the early results due to Brouwer and whatever has been obtained since along the same line.t No such generality would have been possible without that powerful instrument, the product complex. The principle of the method is best explained by means of a very simple example. Letf(x) and so(x) be continuous and uni-valued functions over the interval 0, 1, and let their values on the interval also lie between 0 and 1. It is required to find the number of solutions of f(x) = (x), 0 < x < 1. Graphically the problem is solved by plotting the curvilinear arcs y =(x), y = (x), 0 < x<1 and taking their intersections. A slight modification of the functions may change tlle number of solutions, even make them become infinite in number. However, the difference between the numbers of positive and negative crossings of sufficiently close polygonal approximations to the arcs is a fixed number, their Kronecker index. Its determination is then a partial answer to the question, and indeed seemingly the only possible general answer.