[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-151787-en":3,"doc-seo-151787-105":29,"detail-sidebar-cat-0-en-105":93},{"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":11},151787,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Lecture 3 - MU and complex cobordism","Lecture 3 introduces complex cobordism MU and the construction of MU^*(X) from almost complex manifolds modulo cobordism. It explains graded ring operations via disjoint union and Cartesian product, then states the Thom–Pontryagin/Milnor result that MU^*(pt) is a polynomial ring generated by classes of complex projective spaces CP^n. The lecture connects orientations of MU to formal group laws using Quillen’s theorem and discusses the logarithm, logarithmic additivity, and a proof strategy involving Milnor hypersurfaces and Chern numbers.","Lecture 3: MU and complex cobordism  \nPhilip Egger  \nTranscribed by Paul VanKoughnett  \nDe􀀌nition 1 . Let X be a space and M1 , M2 two almost complex n-manifolds. (This means that we have chosen a reduction of the structure group of the stable tangent bundle to the unitary group.) Let f1 : M1 ! X , f2 : M2 ! X be two continuous functions. Then f1 and f2 are bordant if there exists an almost complex (n + 1)-manifold W with @W = M1 tM2 and a map H : W ! X such that Hj @W = f1 t f2 .  \nWe de􀀌ne MUn (X) to be the set of maps from n-manifolds into X modulo bordism. [Evidently this is an equivalence relation: a cylinder gives re􀀍exivity, symmetry is clear, and gluing bordisms gives transitivity.]  \nMU􀀃 (X) has a graded ring structure, where addition is given by disjoint union (or equivalently, connect sum), and multiplication is given by Cartesian product.  \nThe most obvious case to consider is when X = 􀀃 . Then we can ignore the maps, and two manifolds are bordant i􀀋 they are the boundary of a manifold, which is just the ordinary bordism relation. We have calculated MU􀀃 (􀀃) = MU􀀃 .  \nTheorem 2 (Thom-Pontryagin, Milnor) . MU􀀃 􀀊 Q is a polynomial ring generated by the classes of CPn for n 􀀕 1. In fact, MU􀀃 = Z [tn : jtnj = 2n] .  \nWe can choose tn such that (n + 1)tn = [CPn ] . Note that this ring is concentrated in even degrees: everything odd is nilbordant!  \nNow, recall that he Lazard ring L is isomorphic to Z [xn : jxnj = 2n] . There's no canonical isomorphism L  MU􀀃 , but choosing an orientation for M U􀀃 gives you one. Let's go into this in more detail.  \nDe􀀌nition 3 . A multiplicative cohomology theory E 􀀃 is complex orientable if for i􀀃 : CP1 ,! CP1 , i􀀃 : E2 CP1 ! E2 CP1 is surjective. Equivalently, for every complex n-plane bundle V , we can choose a Thom class u 2 E2n(V; V0 ) such that the image of u in 􀀙􀀃 E under the Thom isomorphism is a generator (where V0 is the zero section) . A choice of such a generator and a Thom class for all V getting sent to that generator is a complex orientation.  \nWe brie􀀍y de􀀌ne the cohomology theory MU 􀀃 on k-manifolds X . For Z a (k 􀀀 n)-manifold, a complexoriented map Z ! X is a proper map Z ! X with an almost complex structure on its stable normal bundle. MUn (X) is the set of complex-oriented maps from (k 􀀀 n)-manifolds to X modulo the cobordism relation: two maps Z1 ! X , Z2 ! X are cobordant if there is an (k 􀀀 n + 1)-manifold W and a map W ! X 􀀂 R such that Z1 ! X and Z2 ! X are respectively the 􀀌bers over 0 and 1 .  \nIt is now easy to see that MU 􀀃 is complex orientable. [Indeed, a generator 􀀒 of MU 2 CP1 is given by any map 􀀃 ! CP1 , which is equivalently a complex line in C2 ; extending this successively to a hyperplane in each CPn by throwing in the new basis vectors gives an element of MU 2 CP1 that evidently pulls back to 􀀒 .]  \nTheorem 4 (Quillen, [2]) . Every choice of orientation of MU􀀃 corresponds to a formal group law over MU􀀃 , which in turn corresponds to an isomorphism 􀀞 : L  MU􀀃 .  \nDe􀀌nition 5 . For F = F (x; y) a formal group law over a torsion-free ring R, the logarithm of F is  \nlogF (x) = Z0 x ~~ @~~@Fyd(tt;~~ ~~0) 2 Q 􀀊 R[[x]]:  \n2  \nProposition 6 . logF (x +F y) = logF (x) + logF (y) .  \nThis can be proved by messing around with power series.  \nProposition 7 . There is an orientation on MU such that  \nlogF (x) =X [Cnn1] xn+1:  \nn􀀕0  \nProof of Quillen's theorem. Let Fu be the universal formal group law over L, F the above formal group law over MU􀀃 , 􀀞 : L ! MU􀀃 the map classifying this FGL. We want to show that 􀀞 is an isomorphism. (It is a lemma that there is an isomorphism of FGLs between any two FGL over MU􀀃 , so it su􀀎ces to prove Quillen's theorem for the single case where F is the orientation given by Proposition 7.)  \nFirst, we show 􀀞 􀀊 Q is an isomorphism. Fu 􀀊 Q has logarithm Pn􀀕0 np1 xn+1 for some pn , and F 􀀊 Q has logarithm Pn􀀕0 [Cnn1] xn+1 . Thus the map must send pn to [CPn], meaning that the degree of pn is 2n, and it is then clear that the pn are g","cbCaijddc27dsIzC","https://ap.wps.com/l/cbCaijddc27dsIzC","pdf",172862,1,3,"English","en",105,"# Definition and basic properties of MU\n## Bordism definition and graded ring structure\n# Computing MU^*(pt)\n## Thom–Pontryagin/Milnor theorem\n# Complex orientations and formal group laws\n## Complex orientability and Thom classes\n# Quillen’s theorem and the logarithm\n## Logarithm of a formal group law and MU orientation\n# Proof sketch via Milnor hypersurfaces\n## Pushforward, image, and generation by [H_ij]\n# Construction of the spectrum MU\n## Thom spaces and stabilization","[{\"question\":\"How does the lecture define MU^n(X) using cobordism?\",\"answer\":\"MU^n(X) is defined as maps from n-manifolds into X modulo the bordism relation. Two maps are cobordant if there exists an (n+1)-manifold W mapping to X whose fibers over two points correspond to the original maps.\"},{\"question\":\"What is the Thom–Pontryagin/Milnor description of MU^*(pt)?\",\"answer\":\"MU^*(pt) is a polynomial ring generated by classes of CP^n for n≥1. Concretely, MU^* ⊗ Q can be expressed as Z[t_n : |t_n|=2n], and the ring is concentrated in even degrees because odd elements are nilbordant.\"},{\"question\":\"How do orientations of MU relate to formal group laws?\",\"answer\":\"Quillen’s theorem states that every choice of orientation of MU corresponds to a formal group law over MU^*, which in turn yields an isomorphism from the Lazard ring L to MU^*. The lecture also introduces the logarithm of the formal group law and the additivity property log_F(x+_F y)=log_F(x)+log_F(y).\"},{\"question\":\"What is the role of Milnor hypersurfaces in the proof?\",\"answer\":\"Milnor hypersurfaces H_ij ⊂ CP^i × CP^j are used to show that the map from L to MU^* is surjective. The classes [H_ij] are shown to lie in the image and to generate MU^*, using pushforward computations and a developed theory of Chern numbers.\"}]","Lecture 3 - MU and complex cobordism | PDF",1787849323,{"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":88,"head_meta":90,"extra_data":92,"updated_unix":28},"lecture-3-mu-and-complex-cobordism","",{"@graph":35,"@context":87},[36,52,66],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,49],{"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":21},"https://docshare.wps.com/document/research-report/",{"item":50,"name":13,"@type":42,"position":51},"https://docshare.wps.com/document/lecture-3-mu-and-complex-cobordism/151787/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-08-27",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79,83],{"name":70,"@type":71,"acceptedAnswer":72},"How does the lecture define MU^n(X) using cobordism?","Question",{"text":73,"@type":74},"MU^n(X) is defined as maps from n-manifolds into X modulo the bordism relation. Two maps are cobordant if there exists an (n+1)-manifold W mapping to X whose fibers over two points correspond to the original maps.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"What is the Thom–Pontryagin/Milnor description of MU^*(pt)?",{"text":78,"@type":74},"MU^*(pt) is a polynomial ring generated by classes of CP^n for n≥1. Concretely, MU^* ⊗ Q can be expressed as Z[t_n : |t_n|=2n], and the ring is concentrated in even degrees because odd elements are nilbordant.",{"name":80,"@type":71,"acceptedAnswer":81},"How do orientations of MU relate to formal group laws?",{"text":82,"@type":74},"Quillen’s theorem states that every choice of orientation of MU corresponds to a formal group law over MU^*, which in turn yields an isomorphism from the Lazard ring L to MU^*. The lecture also introduces the logarithm of the formal group law and the additivity property log_F(x+_F y)=log_F(x)+log_F(y).",{"name":84,"@type":71,"acceptedAnswer":85},"What is the role of Milnor hypersurfaces in the proof?",{"text":86,"@type":74},"Milnor hypersurfaces H_ij ⊂ CP^i × CP^j are used to show that the map from L to MU^* is surjective. The classes [H_ij] are shown to lie in the image and to generate MU^*, using pushforward computations and a developed theory of Chern numbers.","https://schema.org",{"og:url":50,"og:type":89,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":91,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,122,125,130,133,137],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":45,"category_name":114,"show_sort_weight":115,"slug":116},6,"Technology",50,"technology",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":119,"show_sort_weight":120,"slug":121},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":123,"slug":124},30,"research-report",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":128,"slug":129},9,"Religion & Spirituality",20,"religion-spirituality",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":128,"slug":132},"World Cup","world-cup",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":134,"slug":136},10,"Lifestyle","lifestyle",{"id":138,"doc_module":4,"doc_module_name":45,"category_name":139,"show_sort_weight":108,"slug":140},19,"General","general"]