Singular Morse-Smale Flows on Pseudomanifolds with Spherical-Cone Singularities: Conley Theory and Intersection Homology
Classical Morse-Conley theory provides powerful tools for relating dynamical and topological invariants of smooth manifolds. In this paper, we extend this perspective to pseudomanifolds with spherical-cone singularities. By introducing and investigating singular Morse-Smale flows on pseudomanifolds with isolated singularities whose links are homeomorphic to finite disjoint unions of spheres. We establish formulas for the Conley indices of spherical-cone singularities in terms of their local dynamics, prove the existence of global Lyapunov functions, and investigate the structure of the associated Lyapunov graphs. These results yield alternative formulas for the Euler-Poincar\'e characteristic expressed in terms of Conley-theoretic invariants. To relate the singular and smooth settings, we introduce a global morsification procedure that associates a smooth manifold $\widetilde{X}$ to a singular pseudomanifold $X$. This construction allows us to compare the topology of $X$ and $\widetilde{X}$ and, in particular, to derive formulas relating their Euler-Poincar\'e characteristics. Finally, we study the intersection homology of pseudomanifolds with spherical-cone singularities. We establish connections between intersection homology, singular homology, and the Morse homology of the morsification, thereby providing a dynamical approach to the computation of intersection homology.