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The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. <a href="https://mathscinet.ams.org/mathscinet/search/journal/profile?groupId=3285" target="new">Subscribers to MathSciNet</a> may click through for more detailed information. </div> </p> </section> </div> </header> <main> <section class="row"> <nav class="col-md-3">  <nav class="journals-list-group"> <script type="text/javascript" src="/journals/javascript/jams_nav.js"></script> <a href="/journals/journals"><i class="fa-sharp fa-solid fa-house"></i> Journals Home</a> <a href="/epubsearch">eContent Search</a> <a href="/aboutjams" class="">About Journal of the American Mathematical Society</a> <a href="/jamsedit" class="">Editorial Board</a> <a href="/jams/jamssubmit.html" class="">Author and Submission Information</a> <a href="/policies/policies" class="" target="_blank">Journal Policies</a> <a href="/subscriberinfo" class="" target="_blank">Subscription Information</a> <a href="/jemail" class="" target="_blank"> Journal Email Notifications</a> <hr> <a href="/pairing/pair_my_device.html"><i class="fa fa-globe" aria-hidden="true"></i> Remote Access</a> </nav> </nav> <article class="col-md-9"> <section>   <div class="row">  <ul class="nav nav-tabs"> <li role="presentation" class=""> <script type="text/javascript" src="/journals/javascript/jams_nav.js"></script> <a href="javascript: goto_most_recent_jams()" aria-controls="currissue" role="tab"> Current volume </a> </li> <li role="presentation" class="">  <a href="/cgi-bin/mstrack/accepted_papers/jams?active=press" aria-controls="press" role="tab">Articles in press </a> </li> <li role="presentation" class=""> <a href="/journals/jams/0000-000-00?active=recent"> Recently published</a> </li> <li role="presentation" class=""> <a href="/journals/jams/all_issues.html?active=allissues"> All issues : 1988 – Present </a> </li> </ul> <script src="/js/abstracts/ajaxcall.js"></script> </div> <p> </p> <style> #navCell { display: none;} </style> <script src="/journals/javascript/references.js" type="text/javascript"></script> <script async src="/js/mj.js"></script>  <script src="https://platform-api.sharethis.com/js/sharethis.js#property=598c93d1e2587a001253a011&product=unknown" async="async"></script>  <div class="sharethis-inline-share-buttons"></div>   <article class="col-md-12"> <header> <h2>Generalized group characters and complex oriented cohomology theories<br> <small>HTML articles powered by AMS MathViewer <a href="/AMSMathViewer"><i class="fa fa-info-circle" aria-label="about MathViewer"></i></a></small></h2> <dl> <dd>by <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Hopkins%2C%20Michael%20J.">Michael J. Hopkins</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Kuhn%2C%20Nicholas%20J.">Nicholas J. Kuhn</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a>;  </dd> <dd>J. Amer. Math. Soc. <strong>13</strong> (2000), 553-594 </dd> <dd> DOI: https://doi.org/10.1090/S0894-0347-00-00332-5 </dd> <dd> Published electronically: April 26, 2000 </dd> <dd class="mathviewerPDFLink"> <a href="S0894-0347-00-00332-5.pdf"><small>PDF</small></a> | <a href="https://www.copyright.com/openurl.do?issn=1088-6834&WT.mc.it=American%20Mathematical%20Society" target="_blank">Request permission</a> </dd> </dl> </header> <section id="Abstract"> <h2>Abstract:</h2> Let $BG$ be the classifying space of a finite group $G$. Given a multiplicative cohomology theory $E^{*}$, the assignment \[ G \longmapsto E^{*}(BG) \] is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for <em>complex oriented</em> cohomology theories $E^{*}$, using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin’s Theorem is proved for all complex oriented $E^*$: the abelian subgroups of $G$ serve as a detecting family for $E^*(BG)$, modulo torsion dividing the order of $G$. When $E^*$ is a complete local ring, with residue field of characteristic $p$ and associated formal group of height $n$, we construct a character ring of class functions that computes $\frac {1}{p}E^*(BG)$. The domain of the characters is $G_{n,p}$, the set of $n$–tuples of elements in $G$ each of which has order a power of $p$. A formula for induction is also found. The ideas we use are related to the Lubin–Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, $E_n^*$–theory, etc. The $n$th Morava K–theory Euler characteristic for $BG$ is computed to be the number of $G$–orbits in $G_{n,p}$. For various groups $G$, including all symmetric groups, we prove that $K(n)^*(BG)$ is concentrated in even degrees. Our results about $E^*(BG)$ extend to theorems about $E^*(EG\times _G X)$, where $X$ is a finite $G$–CW complex. </section> <section id="references"> <details> <summary>References</summary> <ul><li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Adams%2C%20J.%20F.">J. F. Adams</a>, <em>Stable homotopy and generalised homology</em>, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=402720">MR <strong>402720</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Adams%2C%20John%20Frank">John Frank Adams</a>, <em>Infinite loop spaces</em>, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1978. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=505692">MR <strong>505692</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Adams%2C%20J.%20F.">J. F. Adams</a>, <em>Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture</em>, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 483–532. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=764596">MR <strong>764596</strong></a>, DOI <a href="https://doi.org/10.1007/BFb0075584">10.1007/BFb0075584</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Araki%2C%20Shoro">Shôrô Araki</a>, <em>Equivariant stable homotopy theory and idempotents of Burnside rings</em>, Publ. Res. Inst. Math. Sci. <strong>18</strong> (1982), no. 3, 1193–1212. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=688954">MR <strong>688954</strong></a>, DOI <a href="https://doi.org/10.2977/prims/1195183305">10.2977/prims/1195183305</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Atiyah%2C%20M.%20F.">M. F. Atiyah</a>, <em>Characters and cohomology of finite groups</em>, Inst. Hautes Études Sci. Publ. Math. <strong>9</strong> (1961), 23–64. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=148722">MR <strong>148722</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Atiyah%2C%20M.%20F.">M. F. Atiyah</a>, <em>$K$-theory</em>, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=224083">MR <strong>224083</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Atiyah%2C%20Michael">Michael Atiyah</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Segal%2C%20Graeme">Graeme Segal</a>, <em>On equivariant Euler characteristics</em>, J. Geom. Phys. <strong>6</strong> (1989), no. 4, 671–677. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1076708">MR <strong>1076708</strong></a>, DOI <a href="https://doi.org/10.1016/0393-0440%2889%2990032-6">10.1016/0393-0440(89)90032-6</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Baker%2C%20Andrew">Andrew Baker</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Wurgler%2C%20Urs">Urs Würgler</a>, <em>Liftings of formal groups and the Artinian completion of $v_n^{-1}\textrm {BP}$</em>, Math. Proc. Cambridge Philos. Soc. <strong>106</strong> (1989), no. 3, 511–530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1010375">MR <strong>1010375</strong></a>, DOI <a href="https://doi.org/10.1017/S0305004100068249">10.1017/S0305004100068249</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Baker%2C%20Andrew">Andrew Baker</a>, <em>Hecke algebras acting on elliptic cohomology</em>, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 17–26. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1642886">MR <strong>1642886</strong></a>, DOI <a href="https://doi.org/10.1090/conm/220/03091">10.1090/conm/220/03091</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Bott%2C%20Raoul">Raoul Bott</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Taubes%2C%20Clifford">Clifford Taubes</a>, <a href="/jams/1989-02-01/S0894-0347-1989-0954493-5/" class="amsJournalReferenceTitle"><em>On the rigidity theorems of Witten</em></a>, <a href="/jams/">J. Amer. Math. Soc.</a> <strong>2</strong> (1989), <a href="/jams/1989-02-01/">no. 1</a>, 137–186. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=954493">MR <strong>954493</strong></a>, DOI <a href="/jams/1989-02-01/S0894-0347-1989-0954493-5/">10.1090/S0894-0347-1989-0954493-5</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=tom%20Dieck%2C%20Tammo">Tammo tom Dieck</a>, <em>Kobordismentheorie klassifizierender Räume und Transformationsgruppen</em>, Math. Z. <strong>126</strong> (1972), 31–39 (German). <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=298695">MR <strong>298695</strong></a>, DOI <a href="https://doi.org/10.1007/BF01580352">10.1007/BF01580352</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=tom%20Dieck%2C%20Tammo">Tammo tom Dieck</a>, <em>Transformation groups and representation theory</em>, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=551743">MR <strong>551743</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=tom%20Dieck%2C%20Tammo">Tammo tom Dieck</a>, <em>Transformation groups</em>, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=889050">MR <strong>889050</strong></a>, DOI <a href="https://doi.org/10.1515/9783110858372.312">10.1515/9783110858372.312</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Hazewinkel%2C%20Michiel">Michiel Hazewinkel</a>, <em>Formal groups and applications</em>, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=506881">MR <strong>506881</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Hopkins%2C%20Michael%20J.">Michael J. Hopkins</a>, <em>Characters and elliptic cohomology</em>, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 87–104. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1055870">MR <strong>1055870</strong></a>, DOI <a href="https://doi.org/10.1017/CBO9780511662614.010">10.1017/CBO9780511662614.010</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Hopkins%2C%20Michael%20J.">Michael J. Hopkins</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Kuhn%2C%20Nicholas%20J.">Nicholas J. Kuhn</a>, and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a>, <em>Morava $K$-theories of classifying spaces and generalized characters for finite groups</em>, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 186–209. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1185970">MR <strong>1185970</strong></a>, DOI <a href="https://doi.org/10.1007/BFb0087510">10.1007/BFb0087510</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Hunton%2C%20John">John Hunton</a>, <em>The Morava $K$-theories of wreath products</em>, Math. Proc. Cambridge Philos. Soc. <strong>107</strong> (1990), no. 2, 309–318. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1027783">MR <strong>1027783</strong></a>, DOI <a href="https://doi.org/10.1017/S0305004100068572">10.1017/S0305004100068572</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Osadcii%2C%20N.%20M.">N. M. Osadčiĭ</a>, <em>The algebras $L^{2}_{n}(\Gamma )$, and the lattice of closed ideals of these algebras</em>, Ukrain. Mat. Ž. <strong>26</strong> (1974), 669–670, 717 (Russian). <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=365154">MR <strong>365154</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Kriz%2C%20Igor">Igor Kriz</a>, <em>Morava $K$-theory of classifying spaces: some calculations</em>, Topology <strong>36</strong> (1997), no. 6, 1247–1273. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1452850">MR <strong>1452850</strong></a>, DOI <a href="https://doi.org/10.1016/S0040-9383%2896%2900049-3">10.1016/S0040-9383(96)00049-3</a></li> [KL98]krizlee I. Kriz and K. P. Lee. Odd degree elements in the Morava $K(n)$ cohomology of finite groups. Preprint, 1998. To appear in <em>Topology and its applications</em>. <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Kuhn%2C%20Nicholas%20J.">Nicholas J. Kuhn</a>, <a href="/tran/1987-304-01/S0002-9947-1987-0906812-8/" class="amsJournalReferenceTitle"><em>The Morava $K$-theories of some classifying spaces</em></a>, <a href="/tran/">Trans. Amer. Math. Soc.</a> <strong>304</strong> (1987), <a href="/tran/1987-304-01/">no. 1</a>, 193–205. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=906812">MR <strong>906812</strong></a>, DOI <a href="/tran/1987-304-01/S0002-9947-1987-0906812-8/">10.1090/S0002-9947-1987-0906812-8</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Kuhn%2C%20Nicholas%20J.">Nicholas J. Kuhn</a>, <em>Character rings in algebraic topology</em>, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 111–126. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1055872">MR <strong>1055872</strong></a>, DOI <a href="https://doi.org/10.1017/CBO9780511662614.012">10.1017/CBO9780511662614.012</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Laitinen%2C%20Erkki">Erkki Laitinen</a>, <em>On the Burnside ring and stable cohomotopy of a finite group</em>, Math. Scand. <strong>44</strong> (1979), no. 1, 37–72. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=544579">MR <strong>544579</strong></a>, DOI <a href="https://doi.org/10.7146/math.scand.a-11795">10.7146/math.scand.a-11795</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Landweber%2C%20Peter%20S.">Peter S. Landweber</a>, <a href="/proc/1971-027-01/S0002-9939-1971-0268885-1/" class="amsJournalReferenceTitle"><em>Complex bordism of classifying spaces</em></a>, <a href="/proc/">Proc. Amer. Math. Soc.</a> <a href="/proc/1971-027-01/"><strong>27</strong></a> (1971), 175–179. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=268885">MR <strong>268885</strong></a>, DOI <a href="/proc/1971-027-01/S0002-9939-1971-0268885-1/">10.1090/S0002-9939-1971-0268885-1</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Landweber%2C%20Peter%20S.">Peter S. Landweber</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a>, and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Stong%2C%20Robert%20E.">Robert E. Stong</a>, <em>Periodic cohomology theories defined by elliptic curves</em>, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 317–337. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1320998">MR <strong>1320998</strong></a>, DOI <a href="https://doi.org/10.1090/conm/181/02040">10.1090/conm/181/02040</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Lang%2C%20Serge">Serge Lang</a>, <em>Cyclotomic fields</em>, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=485768">MR <strong>485768</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Lewis%2C%20L.%20G.%2C%20Jr.">L. G. Lewis Jr.</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=May%2C%20J.%20P.">J. P. May</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Steinberger%2C%20M.">M. Steinberger</a>, and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=McClure%2C%20J.%20E.">J. E. McClure</a>, <em>Equivariant stable homotopy theory</em>, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=866482">MR <strong>866482</strong></a>, DOI <a href="https://doi.org/10.1007/BFb0075778">10.1007/BFb0075778</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Lubin%2C%20Jonathan">Jonathan Lubin</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Tate%2C%20John">John Tate</a>, <em>Formal complex multiplication in local fields</em>, Ann. of Math. (2) <strong>81</strong> (1965), 380–387. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=172878">MR <strong>172878</strong></a>, DOI <a href="https://doi.org/10.2307/1970622">10.2307/1970622</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Morava%2C%20Jack">Jack Morava</a>, <em>Completions of complex cobordism</em>, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin-New York, 1978, pp. 349–361. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=513583">MR <strong>513583</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Landweber%2C%20P.%20S.">P. S. Landweber</a> (ed.), <em>Elliptic curves and modular forms in algebraic topology</em>, Lecture Notes in Mathematics, vol. 1326, Springer-Verlag, Berlin, 1988. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=970278">MR <strong>970278</strong></a>, DOI <a href="https://doi.org/10.1007/BFb0078035">10.1007/BFb0078035</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Quillen%2C%20Daniel">Daniel Quillen</a>, <em>The spectrum of an equivariant cohomology ring. I, II</em>, Ann. of Math. (2) <strong>94</strong> (1971), 549–572; ibid. (2) 94 (1971), 573–602. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=298694">MR <strong>298694</strong></a>, DOI <a href="https://doi.org/10.2307/1970770">10.2307/1970770</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a>, <em>Morava $K$-theories and finite groups</em>, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981) Contemp. Math., vol. 12, Amer. Math. Soc., Providence, RI, 1982, pp. 289–292. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=676336">MR <strong>676336</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a>, <em>Complex cobordism and stable homotopy groups of spheres</em>, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=860042">MR <strong>860042</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ravenel%2C%20Douglas%20C.">Douglas C. Ravenel</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Wilson%2C%20W.%20Stephen">W. Stephen Wilson</a>, <em>The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture</em>, Amer. J. Math. <strong>102</strong> (1980), no. 4, 691–748. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=584466">MR <strong>584466</strong></a>, DOI <a href="https://doi.org/10.2307/2374093">10.2307/2374093</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Segal%2C%20Graeme">Graeme Segal</a>, <em>Classifying spaces and spectral sequences</em>, Inst. Hautes Études Sci. Publ. Math. <strong>34</strong> (1968), 105–112. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=232393">MR <strong>232393</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Segal%2C%20Graeme">Graeme Segal</a>, <em>Equivariant $K$-theory</em>, Inst. Hautes Études Sci. Publ. Math. <strong>34</strong> (1968), 129–151. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=234452">MR <strong>234452</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Segal%2C%20G.%20B.">G. B. Segal</a>, <em>Equivariant stable homotopy theory</em>, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 59–63. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=423340">MR <strong>423340</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Serre%2C%20Jean-Pierre">Jean-Pierre Serre</a>, <em>Représentations linéaires des groupes finis</em>, Hermann, Paris, 1967 (French). <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=232867">MR <strong>232867</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Stretch%2C%20C.%20T.">C. T. Stretch</a>, <em>Stable cohomotopy and cobordism of abelian groups</em>, Math. Proc. Cambridge Philos. Soc. <strong>90</strong> (1981), no. 2, 273–278. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=620737">MR <strong>620737</strong></a>, DOI <a href="https://doi.org/10.1017/S0305004100058734">10.1017/S0305004100058734</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Graves%2C%20Lawrence%20M.">Lawrence M. Graves</a>, <em>The Weierstrass condition for multiple integral variation problems</em>, Duke Math. J. <strong>5</strong> (1939), 656–660. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=99">MR <strong>99</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Tanabe%2C%20Michimasa">Michimasa Tanabe</a>, <em>On Morava $K$-theories of Chevalley groups</em>, Amer. J. Math. <strong>117</strong> (1995), no. 1, 263–278. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1314467">MR <strong>1314467</strong></a>, DOI <a href="https://doi.org/10.2307/2375045">10.2307/2375045</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Green%2C%20John%20W.">John W. Green</a>, <em>Harmonic functions in domains with multiple boundary points</em>, Amer. J. Math. <strong>61</strong> (1939), 609–632. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=90">MR <strong>90</strong></a>, DOI <a href="https://doi.org/10.2307/2371316">10.2307/2371316</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Tezuka%2C%20M.">M. Tezuka</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Yagita%2C%20N.">N. Yagita</a>, <em>Cohomology of finite groups and Brown-Peterson cohomology</em>, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 396–408. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1000392">MR <strong>1000392</strong></a>, DOI <a href="https://doi.org/10.1007/BFb0085243">10.1007/BFb0085243</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Wurgler%2C%20Urs">Urs Würgler</a>, <em>Commutative ring-spectra of characteristic $2$</em>, Comment. Math. Helv. <strong>61</strong> (1986), no. 1, 33–45. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=847518">MR <strong>847518</strong></a>, DOI <a href="https://doi.org/10.1007/BF02621900">10.1007/BF02621900</a></li> </ul> </details> </section> <section id="similar"> <details> <summary>Similar Articles</summary> <ul> <li> Retrieve articles in <em>Journal of the American Mathematical Society</em> with MSC (2000): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N22&jrnl=one&onejrnl=jams">55N22</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N34&jrnl=one&onejrnl=jams">55N34</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N91&jrnl=one&onejrnl=jams">55N91</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55R35&jrnl=one&onejrnl=jams">55R35</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=57R85&jrnl=one&onejrnl=jams">57R85</a> </li> <li> Retrieve articles in all journals with MSC (2000): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N22">55N22</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N34">55N34</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55N91">55N91</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=55R35">55R35</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=57R85">57R85</a> </ul> </details> </section> <section id="additionalinformation"> <details open> <summary>Bibliographic Information</summary> <ul> <li><strong>Michael J. 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