Martintxo Saralegi-Aranguren ✉️

Most significant publications

Abstracts

Perverse homotopy

• Intersection homotopy, refinements and coarsenings
  (With D. Tanré). Arxiv
• Perverse homotopy groups
  Accepted for publication in Journal of the European Mathematical Society.
  (With D. Chataur and D. Tanré). Arxiv
• Homotopy truncations of homotopically stratified spaces
  Proceedings of the A.M.S. 152 (2024), 1319-1332.
  (With D. Chataur and D. Tanré). Arxiv

Intersection (co)homology

• Simplicial intersection homology revisited
  Accepted for publication in Topology and Applications.
  (With D. Chataur and D. Tanré). Arxiv
• A reasonable notion of dimension for singular intersection homology
  Journal of Homotopy and Related Structures 19(2024), 121–150.
  (With D. Chataur and D. Tanré). Arxiv
• Addendum to "Refinement invariance of intersection (Co)homologies"
  Homology, Homotopy and Applications 26(2024), 259-274. Arxiv
• Refinement invariance of intersection (co)homologies
  Homology, Homotopy and Applications 23 (2021), 311-340. Arxiv
• Poincaré duality, cap products and Borel-Moore intersection Homology
  Quarterly Journal of Mathematics 71(2020), 943-958.
  (With D. Tanré). Arxiv
• Blown-up intersection cochains and Deligne's sheaves
  Geometriæ Dedicata 204(2020), 315-337.
  (With D. Chataur and D. Tanré). Arxiv
• Variations on Poincaré duality for intersection homology
  Enseignement Mathématique 65(2020), 117-154.
  (With D. Tanré). Arxiv
• Intersection homology. General perversities and topological invariance
  Illinois Journal of Mathematics 63(2019), 127-163.
  (With D. Chataur and D. Tanré). Arxiv
• Lefschetz duality for intersection (co)homology
  Mathematische Zeitschrift 291(2019), 1-16.
  Arxiv
• Poincaré duality with cap products in intersection homology
  Advances in Mathematics 326(2018), 314-351.
  (With D. Chataur and D. Tanré). Arxiv
• Blown-up intersection cohomology
  An alpine bouquet of algebraic topology. Contemporary Mathematics, 708(2018), 45-102.
  (With D. Chataur and D. Tanré). Arxiv
• Intersection Cohomology. Simplicial Blow-up and Rational Homotopy.
  Memoirs AMS 254(2018), 1-108.
  (With D. Chataur and D. Tanré). Arxiv
• Singular decompositions of a cap-product
  Proceedings of the AMS 145(2017), 3645-3656.
  (With D. Chataur and D. Tanré). Arxiv
• Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon.
  Algebraic & Geometric Topology 16 (2016), 1851-1904.
  (With D. Chataur and D. Tanré). Arxiv
• De Rham intersection cohomology for general perversities.
  Illinois Journal of Mathematics 49 (2005), 737-758.
• Cohomologie d'intersection modérée. Un Théorème de deRham.
  Pacific Journal of Math. 169 (1995), 235-289.
  (With B. Cenkl et G. Hector)
• Homological properties of stratified spaces.
  Illinois Journal of Mathematics 38 (1994), 47-70.
• Homologie d'intersection and variétés homologiques.
  C.R.A.S. 314 (1992), 847-851.
  (With J.P. Brasselet)
• L²-cohomologie des espaces stratifiés.
  Manuscripta Math. 76 (1992), 21-32.
  (With G. Hector)
• Théorème de de Rham pour les variétés stratifiées.
  Annals of Global Anal. and Geo. 9 (1991), 211-243.
  (With J.P. Brasselet and G. Hector)
• Homología de intersección: comparación para perversidades diferentes.
  Actas XII Jornadas Hispano-Lusas de Matemáticas, Universidade do Minho vol. II (1987), 735-758.

Singular Riemannian Foliations

• Hard Lefschetz Property for S3-actions
  Proceedings of the A.M.S. 153 (2025), 1263–1273.
  (Avec J. I. Royo Prieto et R. Wolak) Arxiv
• Hard Lefschetz Property for Isometric Flows
  Transformation Groups 29, 409–423 (2024)
  (With J.I. Royo Prieto and R. Wolak) DOI
• Cohomological tautness for singular Riemannian foliations
  Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales 143 (2019), 4263-428.
  (With J. I. Royo Prieto et R. Wolak) Arxiv
• Poincaré Duality of the basic intersection cohomology of a Killing foliation
  Monatsh Math. 180 (2016), 145-166
  (With R. Wolak) Arxiv
• Finiteness of the basic intersection cohomology of a Killing foliation
  Mathematische Zeitschrift 272 (2012), 443-457
  (With R. Wolak) Arxiv
• Cohomological tautness for Riemannian foliations
  Russ. Journal of Math. Physics 16 (2009), 450-46.
  (With J. I. Royo Prieto et R. Wolak) Arxiv
• Tautness for Riemannian foliations on non-compact manifolds
  Manuscripta Math. 126 (2008), 177-200
  (With J. I. Royo Prieto and R. Wolak) Arxiv
• Top dimensional group of the basic intersection cohomology for singular Riemannian foliations
  Bull. Polish Ac Sc. 53 (2005), 429-440
  (With J. I. Royo Prieto and R. Wolak) Arxiv
• The BIC of a singular foliation defined by an abelian group of isometries
  Ann. Polon. Math. 89 (2006), 203-246
  (With R. Wolak) Arxiv
• The BIC of a conical foliations
  Mat. Zametki 77 (2005), 235-257 / Translation in Math. Notes 77 (2005), 213-231
  (With R. Wolak) Arxiv

Compact Lie group actions

• The Gysin Braid for S³-actions on manifolds
  Accepted for publication in Pacific Journal of Mathematics
  (With J.I. Royo Prieto). Arxiv
• Smith-Gysin Sequence
  Differential Geometric Structures and Applications, IWDG 2023. Springer Proceedings in Mathematics & Statistics, 440 (2024), 239–248.
  (With J.I. Royo Prieto and R. Wolak). Arxiv
• Equivariant intersection cohomology of the circle actions
  Real Academia de Ciencias Exactas, Físicas y Naturales 108 (2014), 49-62.
  (With J.I. Royo Prieto) Arxiv
• The Gysin sequence for S³-actions on manifolds
  Publicationes Mathematicae Debrecen 3 (2013), 275-289.
  (With J.I. Royo Prieto) Arxiv
• Intersection cohomology of circle actions
  Topology and its Applications 254 (2007), 2764-2770.
  (With G. Padilla) Arxiv
• Minimal models for non-free circle actions
  Illinois Journal of Mathematics 44 (2000), 784-820.
  (With A. Roig) PDF
• Cohomologie d'intersection des actions toriques simples
  Indagationes Math. 7 (1996), 389-417. PDF
• Gysin sequences
  Analysis and geometry in foliated manifolds, Santiago de Compostela, 1994, 207–222. PDF

• A Gysin sequence for semifree actions of S³
  Proceedings of the A.M.S. 118 (1993), 1335–1345. PDF
• Intersection cohomology of S¹-actions
  Transactions of the A.M.S. 338 (1993), 263–288.
  (With G. Hector). PDF
• The Euler class for flows of isometries
  Research Notes in Math. 131 (1985), 220–227. PDF

Others

• Euler y un balón de fútbol
  SIGMA, Servicio Central de Publicaciones del Gobierno Vasco 36 (2011), 125-135.
  (With J.I. Royo Prieto) PDF
• A six dimensional compact symplectic solvmanifold without Kähler structures
  Osaka J. Math. 33 (1996), 19–35.
  (With M. Fernández and M. de León). PDF
• Cosymplectic reduction for singular momentum maps-actions
  J. Phys. A: Math. Gen. 26 (1993), 5032–5043.
  (With M. de León). PDF
• Fuzzy filters
  Journal of Math. Anal. and Appl. 129 (1988), 560–566.
  (With M.A. de Prada). PDF
• Una nota sobre convergencia en espacios topológicos fuzzy
  Actas IX Jornadas Hispano-Lusas, vol. II (1982), 763–766.
  (With M.A. de Prada). PDF

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Slides and Videos

• Dos cohomologías de intersección y una conjetura de Goresky y Pardon
  Seminario Topología Algebraica, UNAM (México), Agosto 2019. Slides · Video
  In collaboration with D. Chataur & D. Tanré.
• Intersection cohomologies
  MPS Conference on Singularities, Simons Foundation (New York, USA), Agosto 2018. Slides · Video
  In collaboration with D. Chataur & D. Tanré.
• Filtered (co)-intersection Poincaré duality
  Workshop on Stratified Spaces, Fields Institute (Toronto, Canadá), Agosto 2016. Slides · Video
  In collaboration with D. Chataur & D. Tanré.
• Gysin sequence for smooth S³-actions
  Knots, Manifolds, and Group Actions, Słubice (Polonia), Septiembre 2013. Slides
  In collaboration with J.I. Royo Prieto.
• Qu'est-ce qu'un mathématicien?
  Le Tour de France des Déchiffreurs, Voyage en mathématiques, Enero 2012. Video 1 · Video 2
  Interview with É. Mathéron by V. Vassallo.
• The basic intersection cohomology of a singular Riemannian foliation
  XVIII Encuentro de Topología, Sevilla (España), Octubre 2011. Slides
  In collaboration with R. Wolak.
• Sobre la Conjetura de Poincaré
  Paseo en la geometría, Bilbao (España), Abril 2008. Slides
• Minimality and singular Riemannian foliations
  Primer Congreso Hispano-Francés de Matemáticas, Zaragoza (España), Julio 2007. Slides
  In collaboration with J.I. Royo Prieto & R. Wolak.
• The basic intersection cohomology of a singular Riemannian foliation
  6th Conference on Geometry and Topology, Krynica (Polonia), Mayo 2004. Slides
  In collaboration with R. Wolak.

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Errare humanum est, perseverare diabolicum

• Homological properties of stratified spaces
  There is an unforgivable oblivion in Proposition 2.2.5: the perversity p must lie between the 0 perversity and the top perversity. One of the main results (Theorem 4.15) also needs this hypothesis. Usual Goresky-MacPherson perversities satisfy this.
  This error is corrected in De Rham intersection cohomology for general perversities, where tame intersection homology appears in a hidden way (see Blown-up intersection cohomology for an explicit definition).
• Blown-up intersection cohomology
  There is a gap in the proof of Theorem E (b) (p. 91): we should have proved that H(ω) is a p-allowable cochain in order to get that H(ω) is an integrating cochain of ω. But this is not true: another integrating cochain of ω is needed. This is done here, where the corrections are in red.
• Refinement invariance of intersection (co)homologies
  Error on page 24, last formula (13th line from the bottom). The correct version is here, corrections in red.
• Mayer-Vietoris We often find in the literature on intersection homology that the Mayer–Vietoris property (or the equivalent Excision Formula) is stated to hold for intersection homology, using the following argument:

  Mutatis mutandis, the classical proof for singular homology applies.

  This reasoning appears, for example, in:
  
  
  • Morse Theory and Intersection Homology Theory (page 145),
  • An Introduction to Intersection Homology Theory (page 58),
  • Topological invariance of intersection homology without sheaves (page 152),
  • Théorème de De Rham pour les Varietes Stratifies (page 222), …
  In fine, the assertion is true, although the situation is so different between the singular case and the intersection case that some additional explanation is not out of place. This has been done, for example, in:
  
  
  • Singular intersection homology (Section 4.4),
  • Homological properties of stratified spaces (Section 2.3),
  • Intersection cohomology,simplicial blow-up and rational homotopy (Pages 87-89), …
  
  The proof of the Mayer–Vietoris technique in the context of singular homology, associated to an open cover 𝒰 ={U,V} of X, essentially uses these two tools:
  
  MV1 – For any singular chain η of X there exists an integer k tal que Sd^k(η) is a 𝒰-singular chain.
  MV2 – Any 𝒰-singular chain ξ possesses a decomposition ξ = ξ_U + ξ_V where ξ_U is a singular chain of U and ξ_V is a singular chain of V.
  Notice that this decomposition is not unique.
  
  In the context of intersection chains, property MV-1 still holds, replacing singular chains with intersection chains. The second property also remains valid when singular chains are replaced by allowable chains. However, we need to work specifically with intersection chains. This is more delicate, since it does not hold in full generality. It may happen that ξ_U and ξ_V are not intersection chains, while ξ is. In other words, ∂ξ may be allowable, while ∂ξ_U and ∂ξ_V are not.
  
  In fact, property MV-2 still applies, but we need to provide an explicit method to construct such a decomposition.
  
  The method developed in the last two references mentioned above involves a precise description of the allowable simplices that are not intersection simplices. Each such simplex σ contains a bad face τ that detects this phenomenon. Let us now consider an intersection chain ξ, and let τ₁ ,…,τ_m be the bad faces of the simplices in ξ. The chain ξ₁ consists of all simplices in ξ that contain τ₁ , and the chains ξ₂ ,…,ξ_m are defined in the same way. They are intersection chains. The chain made up of intersection simplices is denoted by ξ₀ , it is also an intersection chain. Applying property MV-1, we may assume that each of the chains ξ₀ , ξ₁ ,…,ξ_m is entirely contained in either U or V. Since ξ = ξ₀ + ξ₁ + ··· +ξ_m we obtain property MV-2.
  
  
  In the context of singular homology, we can replace “singular chains” with “singular simplices” in properties MV-1 and MV-2. However, we cannot do the same in the context of intersection homology: we cannot replace “intersection chain” with “intersection simplex,” since an intersection chain is not necessarily a sum of intersection simplices.

  
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