Singular homology on an untriangulated manifold
Marston Morse, Stewart Scott Cairns · Journal of Differential Geometry · 1972
ObjectivesThis paper is concerned with singular homology over Z on a compact, connected difϊerentiable manifold M n of class C°°.We suppose that there is given on M n a polar nondegenerate 1 function / of class C°% and that n > 1.This paper continues a program with two objectives.The first objective is to relate the existence and characteristics of critical points of / to invariants (the Betti numbers and torsion coefficients of the different dimensions) of the singular homology groups of M n sufficient to determine these homology group up to an isomorphism.The second objective is to accomplish this without any global triangulation of M n .This is a prelude to a similar study of topological manifolds which admit no triangulatoin.The cogency of the second objective became evident in Morse's study of global variational analysis.The function spaces thereby arising are in general not even locally compact.To make the global theory depend on triangulations imposes difficulties which obscure the relations between the critical elements and the topology.This first historical reason was reinforced by the conviction that topological manifolds which admit topologically ND functions (see [3]) are more general than those which admit triangulations (see [1]).This last conviction is being further substantiated by current research of R. C. Kirby and L. C. Siebenmann.See [2].The present paper continues the development in [5] of singular homology over Z on M n .In [5] the following condition was imposed on /.Condition C o on /.Under condition C o , f has different values a at different critical points.The following theorem was proved in [5].Its terms are there defined.Theorem 0.1 of [5].Under Condition C o on f there exists an inductive group-theoretic mechanism by virtue of which relative numerical invariants, associated with the critical points of f on each subset