Resumo

The main goal of this work is to generalize a fundamental result due to Smale, which states that if \(M\) is a closed smooth n-manifold equipped with a Morse-Smale flow \(\varphi\), then
\[
\chi(M)=\sum_{i=0}^n (-1)^ic_i,
\]
where \(c_i\) denotes the number of singularities of \(\varphi\) with Morse index \(i\). We extend this result to \(n\)-pseudomanifolds with isolated singularities whose links are homeomorphic to finite disjoint unions of \((n-1)\)-spheres, which we define as pseudomanifolds with spherical-cone singularities. More precisely, we show that if \(X\) is a closed \(n\)-pseudomanifold with spherical-cone singularities, equipped with a singular Morse-Smale flow \(\varphi\), then
\[
\chi(X) = \sum_{i=0}^n (-1)^i c_i + \sum_{k \geq 2 } (1-k) \# {M}(\mathcal{C}_k),
\]
where \(\#{M}(\mathcal{C}k)\) is the number of isolated singularities of \(X\) whose link consists of \(k\) \((n-1)\)-spheres, and \(c_i = \sum_{p \in {Sing}(\varphi)} \eta_{i}(p)\) for all \(i=0, 1, \dots, n\). Here, the natural number \(\eta_i(p)\) counts the number of sheets of \(p\) on which the restriction of the flow presents a hyperbolic singularity of Morse index \(i\).