Mappings of Manifolds and the Notion of Degree

This paper has two main purposes. In the first place, we shall generalize Brouwer's original definition of degree 2 for a mapping f: Mn Qn of one nmanifold into another to what we shall call the twisted degree. Geometrically the new definition is as elementary and intuitive as the old (see (3.8) below); it agrees with it for mappings of orientable manifolds and is more natural and gives a great deal more information in the non-orientable case. We shall denote the twisted degree of f simply by deg f. Part A is devoted to this definition. Secondly, we shall continue the program inaugurated by Brouwer 3 at the International Congress of Mathematicians in 1912 when he expressed his belief that in many cases the degree of a mapping would characterize its homotopy class, i.e., that two mappings of one orientable n-manifold M' into another Qf would be homotopic if and only if they had the same degree. He proved this for the mappings of an n-sphere S' into itself and subsequently 4 for the mappings of a 2-manifold into the 2-sphere. Hopf 11 then proved Brouwer's assertion for the case where M' is arbitrary and Qf = S' is the n-sphere; if M' is nonorientable the mod 2 degree is used here instead of the degree. Later the work of Hurewicz 13c showed that the theorem remains true if S' is replaced by any n-manifold Qf for which the homotopy groups irr(Q') = 0 for r < n; (of course, S8 may be the only such n-manifold). We shall extend these results to the case where M' is arbitrary and ir,(Q') = 0 for 1 < r < n. It is immediately apparent that the degree alone is not enough here; something is required which will reflect algebraically the 1-dimensional portion of the homotopy. For this purpose, however, another simple invariant is at hand, namely the homomorphism (or homomorphism-class, see (3.2)) 0 of the fundamental group of Mn into that of Qf induced by f. Our principal theorem, Theorem II (see also Theorem I), asserts that for orientable M and Qf these two invariants, the Brouwer degree and the induced 0, do characterize the homotopy class of f completely. Indeed, replacing the Brouwer degree by our twisted degree, the result holds also for mappings of non-orientable manifolds M' and Qf provided that the homomorphism 0 is (see definition (3.1)). We show furthermore that if 0 is not orientation-true then (Theorem III) there are in any case not more than two homotopy classes for each 0; these are in some cases distinguished by the twisted degree, in some cases not, The detailed statements of all of these results, for both homotopy relative to a fixed point and free-homotopy, are given in Part B of the paper. Finally, we give a general theorem (Theorem IV) on the homotopy-types of n-manifolds whose homotopy groups between 1 and n vanish, All proofs are deferred to Part D.