[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86474-en":3,"doc-seo-86474-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86474,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Formalizing Abstract Simplicial Complexes And Stellar Subdivisions in Lean","Theory of simplicial complexes underpins topology as a computationally amenable framework for invariants. The work presents a Lean proof-assistant formalization of abstract simplicial complexes and stellar subdivisions using a purely combinatorial foundation. It includes formal morphisms, and key constructions such as links and joins. The study analyzes how stellar subdivisions interact with these operations, proving identities used for triangulated manifolds, including link equivalences across subdivisions.","Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean  \nGarett Cunningham  \nUniversity of Connecticut [garett.cunningham@uconn.edu](garett.cunningham@uconn.edu)  \nDaniel Zach Universit¨at Regensburg  \n[Daniel.Zach@stud.uni-regensburg.de](Daniel.Zach@stud.uni-regensburg.de)  \narXiv :2607 . 102 16v 1 [ cs .LO] 11 Jul 2026  \nStefan Friedl  \nUniversit¨at Regensburg  \n[stefan.friedl@mathematik.uni-regensburg.de](stefan.friedl@mathematik.uni-regensburg.de)  \nJuly 14, 2026  \nAbstract  \nThe theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex K and in a stellar subdivision σsK, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.  \n1 Introduction  \nThe theory of simplicial complexes finds its roots in some of the cornerstone results of premodern geometry, beginning with the study of polyhedra in 3-dimensional Euclidean geometry. Perhaps the first topological result is the Euler characteristic and its independence from the choice of triangulation for a  \nThe authors acknowledge support from the German-U.S. Fulbright Commission and CRC 1085 “Higher Invariants” at Universit¨at Regensburg, funded by the SFB.  \nsurface. The notion of a simplicial complex generalizes and subsumes this study, having grown to become an indispensable tool for algebraic topology, largely due to its quality of being especially amenable to computational techniques. The result is an array of methods for defining and computing topological invariants, such as simplicial homology and genus in the most elementary cases. Moreover, simplicial structure can be heavily leveraged to study the homotopy type of a space by gaining access to discretized versions of standard tools—for example, discrete Morse theory and simplicial collapse algorithms [21, 22] . Likewise, a specialization of spectral sequences computes higher homotopy groups in the simply connected case [35], as implemented in Kenzo [36] .  \nClassically, the field of combinatorial topology has made a rigorous program of extending and refining this toolkit. The resulting efforts have received wide applications, especially in the modern era of computers. Topological data analysis often leverages representations of data as simplicial complexes to compute useful metrics or invariants. For example, persistent homology [11] provides information regarding the structure of a dataset that is independent of certain biases in sampling. The software package Regina [6] is designed for explicit computations in low-dimensional topology based on a triangulation of a space (e.g. , a 3-manifold or knot), allowing preprocessing by simplifying a triangulation.  \nConcretely, this is done using bistellar and stellar subdivisions. These have the advantage of being especially pleasant for algorithms or combinatorial description. Stellar subdivisions have classically been a common choice for a “standard” subdivision [14, 20, 31, 34] . As such, they have become an essential tool in combinatorial topology—for example, in the classical proof of Newman’s theorem [9] . Theorems of Alexander [1] and Pachner [29] show that (tr","cbCaid5mCddqLW5D","https://ap.wps.com/l/cbCaid5mCddqLW5D","pdf",551239,4,1,28,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What does the document formalize in the Lean proof assistant?\",\"answer\":\"It formalizes abstract simplicial complexes and stellar subdivisions, together with related constructions and operations needed to study them.\"},{\"question\":\"What is the framework used for the formalization?\",\"answer\":\"A purely combinatorial framework is adopted to avoid reliance on ambient geometric embedding data.\"},{\"question\":\"What results are proved about stellar subdivisions?\",\"answer\":\"The work studies how stellar subdivisions interact with operations like links and joins, and proves identities including equivalences between links in a complex and in its stellar subdivision.\"}]",1784211941,71,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"formalizing-abstract-simplicial-complexes-and-stellar-subdivisions-in-lean","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/formalizing-abstract-simplicial-complexes-and-stellar-subdivisions-in-lean/86474/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","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},"What does the document formalize in the Lean proof assistant?","Question",{"text":75,"@type":76},"It formalizes abstract simplicial complexes and stellar subdivisions, together with related constructions and operations needed to study them.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the framework used for the formalization?",{"text":80,"@type":76},"A purely combinatorial framework is adopted to avoid reliance on ambient geometric embedding data.",{"name":82,"@type":73,"acceptedAnswer":83},"What results are proved about stellar subdivisions?",{"text":84,"@type":76},"The work studies how stellar subdivisions interact with operations like links and joins, and proves identities including equivalences between links in a complex and in its stellar subdivision.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]