A Chain Complex for Computing Intersection Homology of GS Manifolds with n-sheet Cone Singularities

Abstract The aim of this paper is to construct a chain complex for a singular 2-manifold X with n -sheet cone singularities, where $$n \in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> with $$n \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , equipped with a Gutierrez-Sotomayor flow $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> , that computes the intersection homology of X . By combining Morse theory and de Rham cohomology, we introduce a chain complex $$(C_{*}(X,\varphi ; \mathbb {R}), \partial _{*})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>;</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> whose generators consist of the regular singularities of $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> and the representatives of the de Rham cohomology of the links of the n -sheet cone singularities. The differential $$\partial _{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> </mml:math> carries a dynamical aspect, tracking the flow trajectories between consecutive singularities. The relevance of this chain complex becomes clear when we prove that its homology is isomorphic to the intersection homology of the underlying pseudomanifold.