[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84427-en":3,"doc-seo-84427-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84427,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Persistent Stiefel–Whitney Classes of Tangent Bundles","Stiefel–Whitney classes provide topological invariants of vector bundles, and those of the tangent bundle encode key manifold information such as orientability and embeddability in Euclidean space. An algorithm is presented to compute these classes directly from a finite point-sample: it builds a filtration of simplicial complexes from the point cloud, computes persistent cohomology, then applies the Wu formula to recover Stiefel–Whitney classes using only cup products and Steenrod squares. The Wu-class step reduces to linear equations, yielding polynomial-time computation, with correctness and robustness proved even under barcode noise and spurious features.","arXiv :2503 . 15854v3 [math .AT] 11 Jul 2026  \nPersistent Stiefel–Whitney Classes of Tangent  \nBundles  \nDongwoo Gang 1*  \n1* Department of Mathematical Sciences, Seoul National University,  \nSeoul, South Korea.  \nCorresponding author(s). E-mail(s): [dongwoo.gang@snu.ac.kr](dongwoo.gang@snu.ac.kr) ;  \nAbstract  \nStiefel–Whitney classes are topological invariants of vector bundles, and those of the tangent bundle capture essential features of a manifold, such as whether it is orientable and how it can be embedded in Euclidean space. We present an algorithm that computes these classes for the tangent bundle directly from a finite sample of points. Starting from the point cloud, we build a filtration of simplicial complexes and compute its persistent cohomology, and then apply the Wu formula, which recovers the Stiefel–Whitney classes from the cup product and the Steenrod squares alone, without estimating tangent spaces or a smooth structure. The key step, finding the Wu classes, reduces to solving a system of linear equations, so the computation runs in polynomial time in the number of simplices. We prove that whenever the sample recovers the shape of a closed manifold, the computed classes agree with the true Stiefel–Whitney classes of its tangent bundle, and that this remains true even when the data carry spurious topological features on which the Steenrod squares vanish, so the classes can beidentified over a wide range of scales rather than only where the sample matches the manifold exactly. We illustrate the method on triangulated four-dimensional manifolds and on point clouds coming from image patches and from a molecular conformation space.  \nKeywords: Stiefel–Whitney class, Tangent bundle, Cohomology operations, Wu formula, Topological data analysis  \n1  \n1 Introduction  \nThe tangent bundle of a smooth manifold is the vector bundle whose fibers are its tangent spaces, and it plays a fundamental role in differential geometry and topology [1 , 2] . Stiefel–Whitney classes are the characteristic classes of real vector bundles with Z/2 coefficients. Those of the tangent bundle serve as powerful invariants of the base manifold [3–5] . For example, a smooth manifold is orientable if and only if the first Stiefel–Whitney class of its tangent bundle vanishes. More generally, higher Stiefel–Whitney classes impose additional obstructions, such as constraints on whether a smooth manifold can be embedded into Euclidean space of a given dimension.  \nHere is an intuitive example illustrating the significance of the Stiefel–Whitney classes. The torus and the Klein bottle cannot be distinguished by their cohomology groups with coefficients in Z/2, but the first Stiefel–Whitney class of their tangent bundles distinguishes them. More broadly, every closed surface can be completely classified by its Euler characteristic and the first Stiefel–Whitney class of its tangent bundle.  \nTopological Data Analysis (TDA) applies algebraic topology to extract topological features from point clouds embedded in high-dimensional spaces [6–8] . A central tool, persistent (co)homology, tracks connected components, loops, and higher-dimensional cycles across multiple scales in a filtration. Although recent work has introduced methods for computing Stiefel–Whitney classes of vector bundles in a TDA setting [9–11], many aspects remain unexplored. A fundamental challenge in computing characteristic classes in the persistent setting is that vector bundles are traditionally defined over a single CW complex, whereas persistent topology operates on filtrations of simplicial complexes. Recent studies have made progress in bridging this gap.  \nTinarrage [9] defined and computed persistent Stiefel–Whitney classes of line bundles, using a functorial classifying map into a finite projective space, a finitedimensional model of RP∞ = Gr 1 (R∞ ) . Rank-n bundles are instead classified by the Grassmannian Grn (R∞ ), whose finite-dimensional models are considerably harder to triangulate for n ≥ ","cbCait6Ck6LvO8Se","https://ap.wps.com/l/cbCait6Ck6LvO8Se","pdf",1659167,1,27,"English","en",105,"# Introduction\n## Background on tangent bundles and Stiefel–Whitney classes\n## Topological data analysis and persistent (co)homology\n## Related work on persistent Stiefel–Whitney classes\n## Wu formula and its extension to persistent cohomology","[{\"question\":\"How does the algorithm compute Stiefel–Whitney classes of a tangent bundle from point-cloud data?\",\"answer\":\"It constructs a simplicial filtration from the point cloud, computes persistent cohomology, and then applies the Wu formula to recover the Stiefel–Whitney classes using cup products and Steenrod squares on the persistent cohomology.\"},{\"question\":\"Why does the method avoid estimating tangent spaces or requiring a smooth structure?\",\"answer\":\"Wu’s approach expresses Stiefel–Whitney classes of the tangent bundle purely in terms of cup products and Steenrod squares on mod-2 cohomology, which can be extended to operations on persistent cohomology.\"},{\"question\":\"What guarantees the robustness of the computed persistent classes when the data contain spurious topological features?\",\"answer\":\"The paper proves agreement with the true tangent-bundle Stiefel–Whitney classes whenever the sample recovers a closed manifold’s shape, and remains valid even with spurious bars that are invisible to the Steenrod squares used in the Wu-based reconstruction.\"}]",1784195574,68,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"persistent-stiefelwhitney-classes-of-tangent-bundles","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/persistent-stiefelwhitney-classes-of-tangent-bundles/84427/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the algorithm compute Stiefel–Whitney classes of a tangent bundle from point-cloud data?","Question",{"text":75,"@type":76},"It constructs a simplicial filtration from the point cloud, computes persistent cohomology, and then applies the Wu formula to recover the Stiefel–Whitney classes using cup products and Steenrod squares on the persistent cohomology.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does the method avoid estimating tangent spaces or requiring a smooth structure?",{"text":80,"@type":76},"Wu’s approach expresses Stiefel–Whitney classes of the tangent bundle purely in terms of cup products and Steenrod squares on mod-2 cohomology, which can be extended to operations on persistent cohomology.",{"name":82,"@type":73,"acceptedAnswer":83},"What guarantees the robustness of the computed persistent classes when the data contain spurious topological features?",{"text":84,"@type":76},"The paper proves agreement with the true tangent-bundle Stiefel–Whitney classes whenever the sample recovers a closed manifold’s shape, and remains valid even with spurious bars that are invisible to the Steenrod squares used in the Wu-based reconstruction.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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