Morse function and attaching map (Q1896021)

Let \(M\) be a closed manifold and \(f\) be a Morse function, a differentiable function on \(M\) with isolated, non-degenerate, critical points. Associated to a Morse function there is a cell complex \(L\) which is homotopy equivalent to \(M\) [\textit{J. Milnor}, Morse theory, Princeton University Press (1963; Zbl 0108.104)]. It is assumed that the Morse function \(f\) on the manifold \(M\) has no critical points of indices \(j+1, \dots, j+\ell-1\), where \(\ell\) is a pointwise integer. The purpose of the reviewing article is to prove the following theorem and to give its application to the projective spaces. Theorem. The attaching map of the \(j+\ell\)-cell \(\alpha\) to the \(J\)-cell \(\beta\) is described as the (transversal) intersection manifold of the unstable manifold of \(- \text{grad } f\) at \(\alpha\) with the stable manifold \(-\text{grad } f\) at \(\beta\) considered as a framed manifold embedded in the unstable manifold.