Morse theory without critical points (Q1812739)

Let \(X\) be an \(n\)-dimensional differentiable manifold and \(f: X\to\mathbb{R}\) a real-valued \(C^ \infty\) function without critical points. The topological pair \((X,A)\) is said to be regular if \(H_ .(X,A)=0\). The fiber \(f^{-1}(c)\) is said to be critical if for every positive \(\varepsilon\) there is some \(\delta\), \(0<\delta < \varepsilon\), such that \((f^{-1}[c,c+\delta],f^{-1}(c))\) is not regular. In this case, for small \(\delta\) the morphisms \(i_ p: H_ p(f^{-1}(c))\to H_ p(f^{-1}[c,c+\delta])\) induced by inclusion do not depend upon \(\delta\), hence the following type numbers are well defined: \(m_ p(c)=\text{rank coker }i_ p+\text{rank ker }i_ p\). Given that \(f\) is bounded and has only a finite number of critical fibers one can set \(M_ p=\sum_ cm_ p(c)\). The main result of this paper is that these coefficients satisfy the following Morse inequalities: \(M_ 0\geq R_ 0\), \(M_ 1-M_ 0\geq R_ 1-R_ 0,\dots,M_{n-1}-M_{n- 2}+\dots\pm M_ 0=R_{n-1}-R_{n-2}+\dots\pm R_ 0\), where \(R_ p\) denotes the \(p\)-dimensional Betti number of \(X\). In particular, if \(M\) is compact, \(h: M\to\mathbb{R}\), is a Morse function, \(X=M-\{\text{critical points of }h\}\), \(f=h| X\), then these inequalities reduce to the usual Morse inequalities.