Manifolds with a Boundary and Their Trans- Formations*

This is the continuation of the paper which appeared in the January, 1926, number of these Transactions.! Its chief object is to extend to an Mn with a boundary the results already obtained for transformations of manifolds without boundary. To these already treated in full we devote a few pages chiefly to elucidate and simplify certain points of importance for the extension. We have succeeded in deriving coincidence and fixed points formulas for the two types of transformations that are alone amenable to anything like a general treatment and extended the formulas of this and the preceding paper to transformations between two different manifolds with or without a boundary. As an incidental acquisition there should be pointed out some highly interesting topological propositions obtained in Parts II, III. Of importance also is the fact that by means of ample use of matrices we have been able to put all coincidence formulas of this and the previous paper in very simple and manageable form.