Selected publications in manifold theory

Revision as of 17:06, 14 December 2010

This is a timeline of important publications in the theory of manifolds, including

• smooth manifolds,
• piecewise-linear manifolds;
• topological manifolds;
• homology manifolds.

See also:

• Chronology at MacTutor
• Geometry and topology at MacTutor

1 Up to 1900

2 1900 - 1930

3 1930 - 1960

Year	Authors	Publication and remarks
1931	Hopf	Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche
1931	[ Morse]	The critical points of a function of $This is a '''timeline of important publications in the theory of [[Wikipedia:manifold|manifold]]s''', including * [[Wikipedia:differentiable|smooth]] manifolds, * [[Wikipedia:piecewise-linear|piecewise-linear]] manifolds; * [[topological manifolds]]; * [[homology manifolds]]. See also: * [http://www-history.mcs.st-and.ac.uk/Chronology/index.html Chronology] at MacTutor * [http://www-history.mcs.st-and.ac.uk/Indexes/Geometry_Topology.html Geometry and topology] at MacTutor ==Up to 1900== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 1868 || [http://www-history.mcs.st-and.ac.uk/Biographies/Riemann.html Riemann] || {{citeD2|Riemann1868|Über die Hypothesen, welche der Geometrie zugrunde liegen}} |} </wikitex> ==1900 - 1930== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 1904 || [http://www-history.mcs.st-and.ac.uk/Biographies/Poincare.html Poincaré] || {{citeD2|Riemann1868|Cinquieme}} |} </wikitex> ==1930 - 1960== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- |1931 || [ Hopf] || {{citeD2|Hopf1931|Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche}} |- | 1944 || [http://www-history.mcs.st-and.ac.uk/Biographies/Whitney.html Whitney] || {{CiteD2|Whitney1944| ...}} |- | 1956 || [http://www-history.mcs.st-and.ac.uk/Biographies/Milnor.html Milnor]|| {{CiteD2|Milnor1956|On manifolds homeomorphic to the $-sphere}} Showed the existence of several smooth structures on a fixed topological manifold: in this case the $-sphere. |} </wikitex> ==1961 - 1970== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- |1960 || [http://en.wikipedia.org/wiki/Kervaire Kervaire] || {{citeD2|Kervaire1960a|A manifold which does not admit any differentiable structure}} Used surgery to find a PL manifold in dimension 10 which admits no smooth structure <ref>An interesting fact is that Milnor could easily have made this observation in his 1956 paper {{citeD|Milnor1956}} on $-spheres.</ref> |- |1961 || [http://www-history.mcs.st-and.ac.uk/Biographies/Smale.html Smale] || {{citeD2|Smale1961|Generalized Poincaré's conjecture in dimensions greater than four}} Proved that every smooth homotopy sphere of dimension 6 or more is homeomorphic to the standard sphere |- |1963 || [http://en.wikipedia.org/wiki/Kervaire Kervaire] [http://www-history.mcs.st-and.ac.uk/Biographies/Milnor.html Milnor] || {{citeD2|Kervaire&Milnor1963|Groups of homotopy spheres I}} Applied surgery theory to the diffeomorphism classification of smooth structures the n-sphere |} </wikitex> ==1971 - 1980== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 19?? || ... || ... |} </wikitex> ==1981 - 1990== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 1982 || [http://www-history.mcs.st-and.ac.uk/Biographies/Freedman.html Freedman] || {{citeD2|Freedman1982|The topology of four-dimensional manifolds}} Proves the Poincaré conjecture in dimension 4 |} </wikitex> ==1991 - 2000== {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 19?? || ... || ... |} ==2001 - present== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 20?? || ... || ... |} </wikitex> == Discussion == ==Footnotes== <references/> == References == {{#RefList:}} [[Category:History]]n$ variables
1944	Whitney	...
1956	Milnor	On manifolds homeomorphic to the $7$-sphere Showed the existence of several smooth structures on a fixed topological manifold: in this case the $7$-sphere.

4 1961 - 1970

|- |1961 || Smale || Generalized Poincaré's conjecture in dimensions greater than four Proved that every smooth homotopy sphere of dimension 6 or more is homeomorphic to the standard sphere |- |1963 || Kervaire Milnor || Groups of homotopy spheres I Applied surgery theory to the diffeomorphism classification of smooth structures the n-sphere |}

5 1971 - 1980

6 1981 - 1990

7 1991 - 2000

8 2001 - present

9 Discussion

10 Footnotes

1. ↑ An interesting fact is that Milnor could easily have made this observation in his 1956 paper [Milnor1956] on $7$-spheres.

11 References