A complex in Morse theory computing intersection homology
Given a compact smooth manifold \(M\) and a Morse function \(f: M\to \mathbb{R}\) there is a well known procedure, essentially due to Thom and Smale, to construct a chain complex from the critical points and trajectories between them, whose homology recovers the singular homology of \(M\). In this paper the author considers anti-radial Morse functions on spaces with isolated conical singularities. Roughly such a space is a compact stratified space \(X\) with a finite set of singularities \(\mathrm{Sing}(X)\), whose complement is a smooth manifold. An anti-radial Morse function on such an \(X\) is a map \(f: X \to \mathbb{R}\) such that on \(X-\mathrm{Sing}(X)\) it is a Morse function, and near a \(p\in \mathrm{Sing}(X)\) the function is given by \(f(r,y)=f(p)-r^2\). Note that a neighborhood of \(p\in \mathrm{Sing}(X)\) is given by a cone \(cL_p\) with \(L_p\) a compact manifold. So \(r\) refers to the interval factor of the cone while \(y\in L_p\). In particular, each \(p\in \mathrm{Sing}(X)\) is a local maximum of \(f\), and we consider the set of critical points of \(f\) as consisting of the singular points as well as the usual critical points of \(f\) on \(X-\mathrm{Sing}(X)\). The author now forms a chain complex over \(\mathbb{R}\) using the critical points. However, for singular points \(p\in \mathrm{Sing}(X)\) we also need a basis of the de Rham cohomology of the link manifold \(L_p\), which are used for extra generators in this chain complex. In fact only a basis for \(H^k(L_p)\) with \(k\leq n/2-1\) is used, where \(n\) is the dimension of \(X\). This bound is related to the perversity which one wants to use for the intersection homology and can be varied. If used as stated, the result works for the lower middle perversity. The main result now states that the homology of this chain complex is the intersection homology of \(X\) with real coefficients.
scientific article; zbMATH DE number 6821943
scientific article