The aim of this paper is to construct a chain complex for a singular 2-manifold X with n-sheet cone singularities, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, equipped with a Gutierrez-Sotomayor flow <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>, that computes the intersection homology of X. By combining Morse theory and de Rham cohomology, we introduce a chain complex <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(C_{}(X,\varphi ; \mathbb {R}), \partial _{})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>φ</mi> <mo>;</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>∂</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> whose generators consist of the regular singularities of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> and the representatives of the de Rham cohomology of the links of the n-sheet cone singularities. The differential <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\partial _{*}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>∂</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> 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.

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

期刊论文
出版模式：

开放获取
发表日期：

2026年4月16日

查看全文

Dahisy V. S. Lima,
Denilson Tenório,
Nivaldo G. Grulha Jr.

摘要

The aim of this paper is to construct a chain complex for a singular 2-manifold X with n-sheet cone singularities, where $n \in \mathbb {N}$ $n \in N$ with $n \ge 2$ $n \geq 2$ , equipped with a Gutierrez-Sotomayor flow $\varphi $ $φ$ , that computes the intersection homology of X. By combining Morse theory and de Rham cohomology, we introduce a chain complex $(C_{*}(X,\varphi ; \mathbb {R}), \partial _{*})$ $(C_{*} (X, φ; R), \partial_{*})$ whose generators consist of the regular singularities of $\varphi $ $φ$ and the representatives of the de Rham cohomology of the links of the n-sheet cone singularities. The differential $\partial _{*}$ $\partial_{*}$ 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.