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Homotopy groups of spheres
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==References== ===Notes=== {{reflist}} ===Sources=== {{sfn whitelist |CITEREFFuks2001 |CITEREFMahowald2001}} {{refbegin|2}} * {{citation |last1= Adams |first1= J. Frank |author1-link= Frank Adams |title= On the groups J(X) IV |journal= [[Topology (journal)|Topology]] |volume= 5 |year= 1966 |issue= 1 |doi= 10.1016/0040-9383(66)90004-8 |pages= 21–71 |doi-access= free }}. See also {{citation |title= Correction |journal= Topology |volume= 7 |year= 1968 |doi= 10.1016/0040-9383(68)90010-4 |issue= 3 |author= Adams, J |pages= 331 |doi-access= free }}. * {{citation |last1= Barratt |first1= Michael G. |last2= Jones |first2= John D. S. |last3= Mahowald |first3= Mark E. |author3-link=Mark Mahowald |title= Relations amongst Toda brackets and the Kervaire invariant in dimension 62 |journal= [[Journal of the London Mathematical Society]] |volume= 30 |year= 1984 |issue= 3 |pages= 533–550 |doi= 10.1112/jlms/s2-30.3.533 |mr= 0810962 |citeseerx= 10.1.1.212.1163 }}. * {{citation |last1= Berrick |first1= A. J. |last2= Cohen |first2= Frederick R. |author2-link= Frederick R. Cohen |last3= Wong |first3= Yan Loi |last4= Wu |first4= Jie |title= Configurations, braids, and homotopy groups |journal= [[Journal of the American Mathematical Society]] |volume= 19 |year= 2006 |pages= 265–326 |url= http://www.math.nus.edu.sg/~matwujie/publications.html |doi= 10.1090/S0894-0347-05-00507-2 |mr= 2188127 |issue= 2 |doi-access= free }}. * {{citation |last1= Cartan |first1= Henri |author1-link= Henri Cartan |last2= Serre |first2= Jean-Pierre |author2-link= Jean-Pierre Serre |title= Espaces fibrés et groupes d'homotopie. I. 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Cohen |last2= Moore |first2= John C. |author2-link= John Coleman Moore |last3= Neisendorfer |first3= Joseph A. |title= The double suspension and exponents of the homotopy groups of spheres |journal= [[Annals of Mathematics]] |series=Second Series |volume= 110 |date=November 1979 |pages= 549–565 |doi= 10.2307/1971238 |mr= 554384 |issue= 3 |jstor= 1971238 }}. * {{citation |last= Cohen |first= Joel M. |title= The decomposition of stable homotopy |journal= Annals of Mathematics |series=Second Series |volume= 87 |year= 1968 |pages= 305–320 |doi= 10.2307/1970586 |mr= 0231377 |jstor= 1970586 |issue= 2 |pmid= 16591550 |pmc= 224450 }}. * {{Citation | doi=10.3792/pjaa.82.141 | last1=Deitmar | first1=Anton | title=Remarks on zeta functions and ''K''-theory over '''F'''<sub>1</sub> | url=http://projecteuclid.org/getRecord?id=euclid.pja/1162820095 | mr=2279281 | year=2006 | journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences | issn=0386-2194 | volume=82 | issue=8 | pages=141–146| arxiv=math/0605429 }}. *{{citation | last1 = Fine | first1 = Benjamin | last2 = Rosenberger | first2 = Gerhard | contribution = 8.1 Winding Number and Proof Five | doi = 10.1007/978-1-4612-1928-6 | isbn = 0-387-94657-8 | mr = 1454356 | pages = 134–136 | publisher = Springer-Verlag, New York | series = Undergraduate Texts in Mathematics | title = The Fundamental Theorem of Algebra | year = 1997}} * {{springer|id=s/s086650|title=Spheres, homotopy groups of the|first=Dmitry B.|last=Fuks|author-link=Dmitry Fuchs}}. *{{citation | last = Hu | first = Sze-tsen | author-link = Sze-Tsen Hu | location = New York & London | mr = 0106454 | publisher = Academic Press | series = Pure and Applied Mathematics | title = Homotopy theory | volume = 8 | year = 1959}} * {{citation |last= Isaksen |first= Daniel C. |title= Stable Stems |journal = Memoirs of the American Mathematical Society |year= 2019 |volume= 262 |issue= 1269 |doi= 10.1090/memo/1269 |isbn= 978-1-4704-3788-6 |mr= 4046815 |doi-access= free }}. * {{citation |last1= Isaksen |first1= Daniel C. |last2= Wang |first2= Guozhen |last3= Xu |first3= Zhouli |author3-link= Zhouli Xu |title= Stable homotopy groups of spheres: from dimension 0 to 90 |year= 2023 |doi= 10.1007/s10240-023-00139-1 |journal= Publications mathématiques de l'IHÉS |volume=137 |pages=107–243 |eprint= 2001.04511 |mode=cs2 }}. * {{citation |last1= Kervaire |first1= Michel A. |author1-link= Michel Kervaire |last2= Milnor |first2= John W. |author2-link=John Milnor |title= Groups of homotopy spheres: I |journal= [[Annals of Mathematics]] |volume= 77 |year= 1963 |issue= 3 |pages= 504–537 |mr= 0148075 |doi= 10.2307/1970128 |jstor= 1970128 }}. * {{citation |last= Kochman |first= Stanley O. |title= Stable homotopy groups of spheres. A computer-assisted approach |series= Lecture Notes in Mathematics |volume= 1423 |publisher= [[Springer-Verlag]] |location= Berlin |year= 1990 |isbn= 978-3-540-52468-7 |doi= 10.1007/BFb0083795 |mr= 1052407 }} Also see the corrections in {{harv|Kochman|Mahowald|1995}} *{{Citation | last1=Kochman | first1=Stanley O. | last2=Mahowald | first2=Mark E. | author2-link=Mark Mahowald| title=The Čech centennial (Boston, MA, 1993) | chapter-url=https://books.google.com/books?id=nEaGVNx2MsoC&pg=PA299 | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Contemporary Mathematics | mr=1320997 | year=1995 | volume=181 | chapter=On the computation of stable stems | pages=299–316 | isbn=978-0-8218-0296-0}} * {{citation |last= Mahowald |first= Mark |author-link=Mark Mahowald |chapter= Toward a global understanding of π<sub>∗</sub>(''S''<sup>''n''</sup>) |title= Proceedings of the International Congress of Mathematicians (Berlin, 1998) |volume= II |series= Documenta Mathematica, Extra Volume |year= 1998 |pages= 465–472 |chapter-url= http://www.emis.de/journals/DMJDMV/xvol-icm/06/Mahowald.MAN.html |mr= 1648096 }} * {{springer|id=E/e110020|title=EHP spectral sequence|first=Mark|last=Mahowald}}. *{{citation|first=John W.|last= Milnor|author-link=John Milnor |title=Differential topology forty-six years later|journal= [[Notices of the American Mathematical Society]] |volume=58|year=2011|issue= 6 |pages=804–809|url=https://www.ams.org/notices/201106/rtx110600804p.pdf}} *{{citation | last = Miranda | first = Rick | doi = 10.1090/gsm/005 | isbn = 0-8218-0268-2 | mr = 1326604 | publisher = American Mathematical Society | location = Providence, Rhode Island | series = Graduate Studies in Mathematics | title = Algebraic curves and Riemann surfaces | url = https://books.google.com/books?id=aN4bfzgHvvkC&pg=PA123 | volume = 5 | year = 1995}} * {{citation |doi= 10.2969/jmsj/02540707 |last= Nishida | first= Goro | author-link = Goro Nishida |title= The nilpotency of elements of the stable homotopy groups of spheres |journal= Journal of the Mathematical Society of Japan |volume= 25 |issue= 4 |year= 1973 |pages= 707–732 |issn= 0025-5645 |mr= 0341485 |doi-access= free |hdl= 2433/220059 |hdl-access= free }}. *[[Lev Pontryagin|Pontrjagin, Lev]], ''Smooth manifolds and their applications in homotopy theory'' American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959) * {{citation |last= Ravenel |first= Douglas C. |author-link = Douglas Ravenel |title= Complex cobordism and stable homotopy groups of spheres |edition= 2nd |url= http://www.math.rochester.edu/people/faculty/doug/mu.html |publisher= AMS Chelsea |year= 2003 |isbn= 978-0-8218-2967-7 |mr= 0860042 }}. * {{citation |last= Scorpan |first= Alexandru |year= 2005 |title= The wild world of 4-manifolds |publisher= [[American Mathematical Society]] |isbn= 978-0-8218-3749-8 |mr= 2136212 }}. * {{citation |last= Serre |first= Jean-Pierre |title= Homologie singulière des espaces fibrés. Applications |journal= Annals of Mathematics |series=Second Series |volume= 54 |year= 1951 |pages= 425–505 |mr= 0045386 |doi= 10.2307/1969485 |jstor= 1969485 |issue= 3 }}. * {{citation |last= Serre |first= Jean-Pierre |title= Sur la suspension de Freudenthal |journal= Comptes Rendus de l'Académie des Sciences, Série I |volume= 234 |year= 1952 |place= Paris |issn= 0764-4442 |pages= 1340–1342 |mr= 0046048 }}. * {{citation |last= Toda |first= Hirosi |author-link=Hiroshi Toda |title= Composition methods in homotopy groups of spheres |publisher= [[Princeton University Press]] |year= 1962 |isbn= 978-0-691-09586-8 |mr= 0143217 |series= Annals of Mathematics Studies |volume= 49 }}. *{{citation | last = Walschap | first = Gerard | contribution = Chapter 3: Homotopy groups and bundles over spheres | doi = 10.1007/978-0-387-21826-7 | isbn = 0-387-20430-X | mr = 2045823 | publisher = Springer-Verlag, New York | series = Graduate Texts in Mathematics | title = Metric structures in differential geometry | volume = 224 | year = 2004}} * {{citation |last1= Wang |first1= Guozhen |last2= Xu |first2= Zhouli |author2-link= Zhouli Xu |title= The triviality of the 61-stem in the stable homotopy groups of spheres |journal= [[Annals of Mathematics]] |volume= 186 |year= 2017 |issue= 2 |pages= 501–580 |doi=10.4007/annals.2017.186.2.3 |mr= 3702672 |arxiv= 1601.02184 |s2cid= 119147703 }}. * {{citation |last1= Gheorghe |first1= Bogdan |last2= Wang |first2= Guozhen |last3= Xu |first3= Zhouli |author3-link= Zhouli Xu |title= The special fiber of the motivic deformation of the stable homotopy category is algebraic |journal= [[Acta Mathematica]] |volume= 226 |year= 2021 |issue= 2 |pages= 319–407 |doi=10.4310/ACTA.2021.v226.n2.a2 |doi-access=free |s2cid= 119303902 |arxiv= 1809.09290 }}. *{{citation | mr = 3204653 | publisher = The Univalent Foundations Program and Institute for Advanced Study | ref = {{harvid|Homotopy type theory|2013}} | title = Homotopy type theory—univalent foundations of mathematics | url = https://homotopytypetheory.org/book/ | year = 2013}} {{refend}} ===General algebraic topology references=== * {{citation |first= Allen |last= Hatcher |author-link= Allen Hatcher |title= Algebraic Topology |url= http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher= [[Cambridge University Press]] |isbn= 978-0-521-79540-1 |mr= 1867354 }}. * {{citation |last= May |first= J. Peter | author-link = J. Peter May |title= A Concise Course in Algebraic Topology |edition= revised |publisher= [[University of Chicago Press]] |year= 1999b |isbn= 978-0-226-51183-2 |url= http://www.math.uchicago.edu/~may/CONCISE |series= Chicago lectures in mathematics |mr= 1702278 }}. ===Historical papers=== * {{citation |last= Čech |first= Eduard |author-link= Eduard Čech |title= Höherdimensionale Homotopiegruppen |journal= Verhandlungen des Internationalen Mathematikerkongress, Zürich |year= 1932 }}. * {{citation |last= Hopf |first= Heinz |author-link= Heinz Hopf |title= Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche |journal= [[Mathematische Annalen]] |volume= 104 |issue= 1 |pages= 637–665 |year= 1931 |url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002274760 |doi= 10.1007/BF01457962 |s2cid= 123533891 }}. * {{citation |last= May |first= J. Peter | author-link = J. Peter May |chapter= Stable Algebraic Topology 1945–1966 |title= History of Topology |chapter-url= http://hopf.math.purdue.edu/cgi-bin/generate?/May/history |editor= I. M. James |pages= 665–723 |publisher= [[Elsevier|Elsevier Science]] |isbn= 978-0-444-82375-5 |year= 1999a }}.
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