| _version_ | 1866901783592304640 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper investigates the homotopy type of stratified spaces, with a particular focus on the complex analytic and algebraic singularities they model. Stratified spaces provide a powerful framework for studying objects that are not manifolds but possess a regular, layered structure. The central goal of this work is to establish a systematic relationship between the local geometry of the strata and the global homotopy invariants of the total space. We develop a methodological approach that combines classical tools from algebraic topology, such as spectral sequences and obstruction theory, with modern techniques rooted in intersection homology and the theory of perverse sheaves. Our main results provide a decomposition of the homotopy type of a stratified space in terms of the homotopy types of the links of its strata, weighted by their codimensions. We demonstrate that the failure of this decomposition to be a simple product is controlled by a series of higher-order invariants related to the attaching maps of the strata. We apply this framework to compute the homotopy groups of several well-known singular spaces, including complex projective varieties with isolated singularities and the Whitney umbrella. The findings illustrate how the intricate weaving of strata dictates the global topology and provides a more refined understanding than that offered by classical homology theories, which often fail to capture the nuances of singular spaces. This work contributes to the broader program of extending the tools of differential topology to the realm of singular analytic and algebraic geometry. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17681141 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Homotopy Type of Stratified Singularities SÉRGIO DE ANDRADE, PAULO This paper investigates the homotopy type of stratified spaces, with a particular focus on the complex analytic and algebraic singularities they model. Stratified spaces provide a powerful framework for studying objects that are not manifolds but possess a regular, layered structure. The central goal of this work is to establish a systematic relationship between the local geometry of the strata and the global homotopy invariants of the total space. We develop a methodological approach that combines classical tools from algebraic topology, such as spectral sequences and obstruction theory, with modern techniques rooted in intersection homology and the theory of perverse sheaves. Our main results provide a decomposition of the homotopy type of a stratified space in terms of the homotopy types of the links of its strata, weighted by their codimensions. We demonstrate that the failure of this decomposition to be a simple product is controlled by a series of higher-order invariants related to the attaching maps of the strata. We apply this framework to compute the homotopy groups of several well-known singular spaces, including complex projective varieties with isolated singularities and the Whitney umbrella. The findings illustrate how the intricate weaving of strata dictates the global topology and provides a more refined understanding than that offered by classical homology theories, which often fail to capture the nuances of singular spaces. This work contributes to the broader program of extending the tools of differential topology to the realm of singular analytic and algebraic geometry. |
| title | Homotopy Type of Stratified Singularities |
| url | https://doi.org/10.5281/zenodo.17681141 |