Smith conjecture
{{Short description|The fixed point set of a finite-order 3-sphere diffeomorphism can't be a non-trivial knot}}{{No footnotes|date=November 2021}}
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.
{{harvs|txt|last=Smith|first=Paul A.|authorlink=Paul Althaus Smith|year=1939|loc=remark after theorem 4}} showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in {{harv|Eilenberg|1949|loc=Problem 36}} if the fixed point set could be knotted. {{harvs|txt|authorlink=Friedhelm Waldhausen|last=Waldhausen|first=Friedhelm|year=1969}} proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by {{harvs|txt| last1=Morgan | first1=John| author1-link=John Morgan (mathematician)|last2=Bass | first2=Hyman|author2-link=Hyman Bass|year=1984}} and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland.
{{harvs|txt| last1=Montgomery | first1=Deane |author1-link=Deane Montgomery| last2=Zippin | first2=Leo|author2-link=Leo Zippin |year=1954}} gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category. {{harvs|txt|last=Giffen|first=Charles|year=1966}} showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.
See also
References
- {{Citation | last1=Eilenberg | first1=Samuel | author1-link=Samuel Eilenberg | title=On the Problems of Topology | jstor=1969448 | series=Second Series |mr=0030189 | year=1949 | journal=Annals of Mathematics | issn=0003-486X | volume=50 | issue=2 | pages=247–260 | doi=10.2307/1969448}}
- {{Citation | last1=Giffen | first1=Charles H. | title=The generalized Smith conjecture | jstor=2373054 |mr=0198462 | year=1966 | journal=American Journal of Mathematics | issn=0002-9327 | volume=88 | issue=1 | pages=187–198 | doi=10.2307/2373054}}
- {{Citation | last1=Montgomery | first1=Deane |author1-link=Deane Montgomery| last2=Zippin | first2=Leo|author2-link=Leo Zippin | title=Examples of transformation groups | jstor=2031959 |mr=0062436 | year=1954 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=5 | issue=3 | pages=460–465 | doi=10.2307/2031959| doi-access=free }}
- {{Citation | editor1-last=Morgan | editor1-first=John W. | editor2-last=Bass | editor1-link=John Morgan (mathematician)| editor2-first=Hyman | editor2-link=Hyman Bass| title=The Smith conjecture | url=https://books.google.com/books?id=sXcwg4zi1_EC | publisher=Academic Press | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-506980-9 |mr=758459 | year=1984 | volume=112}}
- {{Citation | last1=Smith | first1=Paul A. | author-link=Paul Althaus Smith| title=Transformations of finite period. II | jstor=1968950 |mr=0000177 | year=1939 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=40 | issue=3 | pages=690–711 | doi=10.2307/1968950| bibcode=1939AnMat..40..690S }}
- {{Citation | last1=Waldhausen | first1=Friedhelm | author1-link=Friedhelm Waldhausen | title=Über Involutionen der 3-Sphäre | doi=10.1016/0040-9383(69)90033-0 |mr=0236916 | year=1969 | journal=Topology | issn=0040-9383 | volume=8 | pages=81–91| url=https://pub.uni-bielefeld.de/record/1782176 | doi-access=free }}
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