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Fiber bundle
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== History == In [[topology]], the terms '''''fiber''''' (German: ''Faser'') and '''''fiber space''''' (''gefaserter Raum'') appeared for the first time in a paper by [[Herbert Seifert]] in 1933,<ref>{{cite journal|title=Topologie dreidimensionaler gefaserter Räume|first=Herbert|last= Seifert|author-link=Herbert Seifert| journal=[[Acta Mathematica]]| volume=60|year=1933|pages=147–238|doi=10.1007/bf02398271|doi-access=free}}</ref><ref>[https://projecteuclid.org/euclid.acta/1485887992 "Topologie Dreidimensionaler Gefaserter Räume"] on [[Project Euclid]].</ref><ref>{{Cite book |last=Seifert |first=H. |url=https://www.worldcat.org/oclc/5831391 |title=Seifert and Threlfall, A textbook of topology |date=1980 |publisher=Academic Press |others=W. Threlfall, Joan S. Birman, Julian Eisner |isbn=0-12-634850-2 |location=New York |oclc=5831391}}</ref> but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the '''base space''' (topological space) of a fiber (topological) space ''E'' was not part of the structure, but derived from it as a quotient space of ''E''. The first definition of '''fiber space''' was given by [[Hassler Whitney]] in 1935<ref>{{cite journal|title=Sphere spaces|first=Hassler|last= Whitney|author-link=Hassler Whitney| journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=21|issue=7|year=1935|pages=464–468|doi=10.1073/pnas.21.7.464|doi-access=free|pmid=16588001|pmc=1076627|bibcode=1935PNAS...21..464W}}</ref> under the name '''sphere space''', but in 1940 Whitney changed the name to '''sphere bundle'''.<ref>{{cite journal|title=On the theory of sphere bundles|first=Hassler|last= Whitney|author-link=Hassler Whitney| journal= [[Proceedings of the National Academy of Sciences of the United States of America]] |volume=26|issue=2|year=1940|pages=148–153|doi=10.1073/pnas.26.2.148|pmid=16588328|pmc=1078023|bibcode=1940PNAS...26..148W|doi-access=free}}</ref> The theory of fibered spaces, of which [[vector bundle]]s, [[principal bundle]]s, topological [[fibration]]s and [[fibered manifold]]s are a special case, is attributed to [[Herbert Seifert]], [[Heinz Hopf]], [[Jacques Feldbau]],<ref>{{cite journal|title=Sur la classification des espaces fibrés|first=Jacques| last=Feldbau|author-link=Jacques Feldbau|journal=[[Comptes rendus de l'Académie des Sciences]]|volume=208|year=1939|pages=1621–1623}}</ref> Whitney, [[Norman Steenrod]], [[Charles Ehresmann]],<ref>{{cite journal|title=Sur la théorie des espaces fibrés|first=Charles|last= Ehresmann|author-link=Charles Ehresmann |journal=Coll. Top. Alg. Paris|volume=C.N.R.S.|year=1947|pages=3–15}}</ref><ref>{{cite journal|title=Sur les espaces fibrés différentiables|first=Charles|last= Ehresmann|author-link=Charles Ehresmann | journal=[[Comptes rendus de l'Académie des Sciences]] |volume=224|year=1947|pages=1611–1612}}</ref><ref>{{cite journal|title=Les prolongements d'un espace fibré différentiable|first=Charles|last= Ehresmann|author-link=Charles Ehresmann |journal=[[Comptes rendus de l'Académie des Sciences]]|volume=240|year=1955|pages=1755–1757}}</ref> [[Jean-Pierre Serre]],<ref>{{cite journal|title=Homologie singulière des espaces fibrés. Applications|first=Jean-Pierre|last= Serre|author-link=Jean-Pierre Serre|journal=[[Annals of Mathematics]] |volume=54|issue=3|year=1951|pages=425–505|doi=10.2307/1969485|jstor=1969485}}</ref> and others. Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.<ref>See {{harvtxt|Steenrod|1951|loc=Preface}}</ref> Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a [[sphere bundle]],<ref>In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example: * {{cite journal | last = Whitney | first = Hassler | author-link = Hassler Whitney | title = Sphere spaces | journal = Proc. Natl. Acad. Sci. | volume = 21 | issue = 7 | pages = 462–468 | date = 1935 | doi = 10.1073/pnas.21.7.464 | pmid = 16588001 | pmc = 1076627 | bibcode = 1935PNAS...21..464W | doi-access = free }} * {{cite journal | last = Whitney | first = Hassler | author-link = Hassler Whitney | title = Topological properties of differentiable manifolds | journal = Bull. Amer. Math. Soc. | volume = 43 | issue = 12 | pages = 785–805 | date = 1937 | doi=10.1090/s0002-9904-1937-06642-0 | doi-access = free | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-43/issue-12/Topological-properties-of-differentiable-manifolds/bams/1183500133.pdf }} </ref> that is a fiber bundle whose fiber is a sphere of arbitrary [[dimension]].<ref> {{cite journal | last = Whitney | first = Hassler | author-link = Hassler Whitney | title = On the theory of sphere bundles | journal = Proc. Natl. Acad. Sci. | volume = 26 | issue = 2 | pages = 148–153 | date = 1940 | doi = 10.1073/pnas.26.2.148 | pmid = 16588328 | pmc = 1078023 | bibcode = 1940PNAS...26..148W | url = http://www.pnas.org/content/pnas/26/2/148.full.pdf | doi-access = free }}</ref>
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