A reasonable notion of dimension for singular intersection homology
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910835995049984 |
|---|---|
| author | Chataur, David Saralegi-Aranguren, Martintxo Tanré, Daniel |
| author_facet | Chataur, David Saralegi-Aranguren, Martintxo Tanré, Daniel |
| contents | M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces $S$ of a Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing~$S$. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing $S$. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex.
In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann's CS sets. In terms of King's paper, this means that polyhedral dimension is a ``reasonable'' dimension. The proof uses a Mayer-Vietoris argument which needs an adaptated subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06090 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A reasonable notion of dimension for singular intersection homology Chataur, David Saralegi-Aranguren, Martintxo Tanré, Daniel Algebraic Topology 55N33, 55M10, 55Q70, 14F43 M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces $S$ of a Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing~$S$. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing $S$. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann's CS sets. In terms of King's paper, this means that polyhedral dimension is a ``reasonable'' dimension. The proof uses a Mayer-Vietoris argument which needs an adaptated subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them. |
| title | A reasonable notion of dimension for singular intersection homology |
| topic | Algebraic Topology 55N33, 55M10, 55Q70, 14F43 |
| url | https://arxiv.org/abs/2211.06090 |