Martintxo Saralegi-Aranguren
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View article: Intersection homotopy, refinements and coarsenings
Intersection homotopy, refinements and coarsenings Open
In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to loca…
View article: The Gysin braid for S3-actions onmanifolds
The Gysin braid for S3-actions onmanifolds Open
Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic …
View article: Relation between intersection homology and homotopy groups
Relation between intersection homology and homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which Gajer associates to a…
View article: Smith-Gysin Sequence
Smith-Gysin Sequence Open
Starting with a manifold $M$ and a semi-free action of $S^3$ on it, we have the Smith-Gysin sequence: $$ \cdots \to H^{*}( M) \to H^{*-3}(M/S^3, M^{S^3}) \oplus H^{*} (M^{S^3}) \to H^{*+1}(M/S^3, M^{S^3}) \to H^{*+1}(M) \to \cdots $$ In th…
View article: Hard Lefschetz property for $\mathbb{S}^3$-actions
Hard Lefschetz property for $\mathbb{S}^3$-actions Open
The Hard Lefschetz Property (HLP) has recently been formulated in the context of isometric flows without singularities on manifolds. In this category, two versions of the HLP (transverse and not) have been proven to be equivalent, thus gen…
View article: Homotopy truncations of homotopically stratified spaces
Homotopy truncations of homotopically stratified spaces Open
Intersection homology of Goresky and MacPherson can be defined from the\nDeligne sheaf, obtained from truncations of complexes of sheaves. As\nintersection homology is not the homology of a particular space, the search for\na family of spa…
View article: Homotopy truncations of homotopically stratified spaces
Homotopy truncations of homotopically stratified spaces Open
Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces…
View article: The Gysin braid for $S^3$-actions on manifolds
The Gysin braid for $S^3$-actions on manifolds Open
Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic …
View article: Perverse homotopy groups
Perverse homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates t…
View article: Relation between intersection homology and homotopy groups
Relation between intersection homology and homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates t…
View article: Hard Lefschetz Property for Isometric Flows
Hard Lefschetz Property for Isometric Flows Open
The hard Lefschetz property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), b…
View article: SIMPLICIAL INTERSECTION HOMOLOGY REVISITED
SIMPLICIAL INTERSECTION HOMOLOGY REVISITED Open
Intersection homology is defined for simplicial, singular and PL chains. In the case of a filtered simplicial complex, it is well known that the three versions are isomorphic. This isomorphism is established by using the PL case as an inte…
View article: Lefschetz Duality for Isometric Flows
Lefschetz Duality for Isometric Flows Open
The Hard Lefschetz Property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), b…
View article: Hard Lefschetz Property for Isometric Flows
Hard Lefschetz Property for Isometric Flows Open
The Hard Lefschetz Property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), b…
View article: POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY
POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY Open
Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseu…
View article: Variations on Poincaré duality for intersection homology
Variations on Poincaré duality for intersection homology Open
Intersection homology with coefficients in a field restores Poincaré duality for some spaces with singularities, as stratified pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and stratified pseudomanifolds ar…
View article: Blown-up intersection cochains and Deligne's sheaves
Blown-up intersection cochains and Deligne's sheaves Open
In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to…
View article: Blown-up intersection cohomology
Blown-up intersection cohomology Open
In previous works, we have introduced the blown-up intersection cohomology\nand used it to extend Sullivan's minimal models theory to the framework of\npseudomanifolds, and to give a positive answer to a conjecture of M. Goresky\nand W. Pa…
View article: Singular decompositions of a cap product
Singular decompositions of a cap product Open
In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that thi…
View article: Intersection Homology. General perversities and topological invariance
Intersection Homology. General perversities and topological invariance Open
Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent pervers…
View article: Intersection Homology. General perversities and topological invariance
Intersection Homology. General perversities and topological invariance Open
Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent pervers…