On the Minimal Ordered Morse Functions on Compact Simply-Connected Manifolds

Abstract

Throughout this paper, (W ; V, V) denotes a compact smooth manifold triad of dimension n in the sense that dW=V(jV and VftV' = , and & the space of C°°-functions of (W; V, V) into (I; 0, 1) without critical points on the boundary. The codimension 0 stratum J^° of the natural stratification of J^ is the space of excellent functions ( = Morse functions with distinct critical values). By definition, jfej^ is called to be ordered and minimal, if the critical values are ordered in the order of their indices and the number of critical points is minimal with respect to the homology group structure of H#(W, V). Let / be an excellent function with m critical values v19. .., vm and put u0=v1/2, um — and u{ = (vt+vi+1)/2 for l^i^m — l. Then, (W, V) has a filtration Wf=(W»...9 WJ defined by W^f-^Q, w,] so that Wm = Wand Wf has a presentation of the handle-attaching form (Wf-!, f~(ui_-^ ; W such that (goh) | W,: Wt->Wf and (hog) IW,: W^W; are homotopy equivalences for every /.