[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-272162-105":59,"doc-detail-272162-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","spectral-sequences-chapter-5-preliminary-and-incomplete-introduction","Spectral Sequences - Chapter 5 - Preliminary and incomplete introduction","","Spectral sequences are introduced as a powerful algebraic topology tool for organizing complex relationships among homotopy, homology, and cohomology groups that are otherwise captured by exact sequences. The focus begins with the Serre spectral sequence for fibrations, explaining how its pages, differentials, and limiting terms encode information about the homology or cohomology of fiber, base, and total spaces. The chapter then outlines key applications and motivates the Adams spectral sequence for stable homotopy groups, especially of spheres.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/spectral-sequences-chapter-5-preliminary-and-incomplete-introduction/272162/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/spectral-sequences-chapter-5-preliminary-and-incomplete-introduction/272162.png","ImageObject",300,407,{"name":92,"@type":93},"Ezra","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem do spectral sequences help solve in algebraic topology?","Question",{"text":113,"@type":114},"They provide a systematic algebraic tool for cases where relationships between homotopy, homology, or cohomology groups are more complicated than those described by a single exact sequence.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the Serre spectral sequence relate fiber, base, and total space?",{"text":118,"@type":114},"For fibrations F → X → B, it relates the homology or cohomology of F, X, and B through a structured sequence of pages with differentials under technical hypotheses.",{"name":120,"@type":111,"acceptedAnswer":121},"Why is the second page E2 central in the Serre spectral sequence?",{"text":122,"@type":114},"The first page can be ignored in many expositions, while the main information starts at the second page, where entries E2^{p,q} are expressed using the homology of the fiber and base.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},272162,1789977603,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099514068035,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",".  \nSpectral Sequences  \nAllen Hatcher  \nThis is a preliminary and incomplete version of an extra 􀀌fth chapter for my Algebraic Topology textbook. Its aim is to give an introduction to spectral sequences as they arise in algebraic topology. The rather lengthy 􀀌rst section of the chapter is devoted to the Serre spectral sequence and some of its main applications. After this, the second section gives a short introduction to the more specialized Adams spectral sequence, which is geared toward computing stable homotopy groups, especially stable homotopy groups of spheres. The chapter then concludes with several independent sections on related topics and other spectral sequences. Here is a Table of Contents for the chapter:  \n5.1. The Serre Spectral Sequence . . . . . . . . . . . . . . . . 520  \nExact Couples 521 .  \nThe Serre Spectral Sequence for Homology 526 .  \nSerre Classes 532 .  \nGeneralizations and Further Properties 536 .  \nThe Serre Spectral Sequence for Cohomology 542 .  \nRational Homotopy Groups 551 .  \nLocalization of Spaces 553 .  \nCohomology of Eilenberg-MacLane Spaces 562 .  \nComputing Homotopy Groups of Spheres 573 .  \n5.2. The Adams Spectral Sequence . . . . . . . . . . . . . . . 580  \nSpectra 584 .  \nConstructing the Adams Spectral Sequence 594 .  \nComputing a Few Stable Homotopy Groups of Spheres 599 .  \nAdditional Topics  \n5.A. Whitehead’s Exact Sequence 605 .  \n5.B. The Bockstein Spectral Sequence [Not yet written] .  \n5.C. The Mayer-Vietoris Spectral Sequence [Not yet written] .  \n5.D. The EHP Sequence 611 .  \n5.E. Eilenberg-Moore Spectral Sequences 619 .  \nIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631  \nThere are many situations in algebraic topology where the relationship between certain homotopy, homology, or cohomology groups is expressed perfectly by an exact sequence. In other cases, however, the relationship may be more complicated anda more powerful algebraic tool is needed. In a wide variety of situations spectral sequences provide such a tool. For example, instead of considering just a pair (X, A) and the associated long exact sequences of homology and cohomology groups, one could consider an arbitrary increasing sequence of subspaces X0 ⊂ X1 ⊂ · · · ⊂ X with X = SiXi , and then there are associated homology and cohomology spectral sequences. Similarly, the Mayer-Vietoris sequence for a decomposition X = A ∪ B generalizes to a spectral sequence associated to a cover of X by any number of sets.  \nWith this great increase in generality comes, not surprisingly, a corresponding increase in complexity. This can be a serious obstacle to understanding spectral sequences on 􀀌rst exposure. But once the initial hurdle of ‘believing in’ spectral sequences is surmounted, one cannot help but be amazed at their power.  \nThe 􀀌rst spectral sequence that appeared in algebraic topology, and still the most important one, is the Serre spectral sequence which relates the homology or cohomology groups of the 􀀌ber, base, and total space of a 􀀌bration. The homotopy groups of these three spaces 􀀌t into a long exact sequence, but for homology or cohomology the relationship is much more complicated, as expressed in the spectral sequence. This increase in complexity can be seen already for a product 􀀌bration, where the homotopy groups of the product are just the products of the homotopy groups of the two factors, whereas for homology one has the more complicated K¨unneth formula.  \nThe 􀀌rst section of this chapter is devoted to the Serre spectral sequence and some of its main applications both to general theory and speci􀀌c calculations. After this we give a brief introduction to the Adams spectral sequence and its application to computing stable homotopy groups of spheres.  \nOne can think of a spectral sequence as a book consisting of a sequence of pages, each of which is a two-dimensional array of abelian groups. On each page there are maps between the groups, and these maps form chain complexes.","cbCaig94IUR0I15T","https://ap.wps.com/l/cbCaig94IUR0I15T","pdf",2769766,117,"English","# 5.1. The Serre Spectral Sequence\n## Exact Couples\n## The Serre Spectral Sequence for Homology\n## Serre Classes\n## Generalizations and Further Properties\n## The Serre Spectral Sequence for Cohomology\n## Rational Homotopy Groups\n## Localization of Spaces\n## Cohomology of Eilenberg-Moore Spaces\n## Computing Homotopy Groups of Spheres\n# 5.2. The Adams Spectral Sequence\n## Spectra\n## Constructing the Adams Spectral Sequence\n## Computing a Few Stable Homotopy Groups of Spheres","[{\"question\":\"What problem do spectral sequences help solve in algebraic topology?\",\"answer\":\"They provide a systematic algebraic tool for cases where relationships between homotopy, homology, or cohomology groups are more complicated than those described by a single exact sequence.\"},{\"question\":\"How does the Serre spectral sequence relate fiber, base, and total space?\",\"answer\":\"For fibrations F → X → B, it relates the homology or cohomology of F, X, and B through a structured sequence of pages with differentials under technical hypotheses.\"},{\"question\":\"Why is the second page E2 central in the Serre spectral sequence?\",\"answer\":\"The first page can be ignored in many expositions, while the main information starts at the second page, where entries E2^{p,q} are expressed using the homology of the fiber and base.\"}]","Spectral Sequences - Chapter 5 - Preliminary and incomplete introduction | PDF",1789463608,295]