Selected publications in manifold theory

Revision as of 17:33, 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
1926	Lefschetz	Intersections and transformations of complexes and manifolds
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}} |- |1908 || [http://www-history.mcs.st-and.ac.uk/Biographies/Tietze.html Tietze] || {{citeD2|Tietze1908|Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten}} |} </wikitex> ==1930 - 1960== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- |1926 || [http://www-history.mcs.st-and.ac.uk/Biographies/Lefschetz.html Lefschetz] || {{citeD2|Lefschetz1926|Intersections and transformations of complexes and manifolds}} |- |1931 || [http://www-history.mcs.st-and.ac.uk/Biographies/Hopf.html Hopf] || {{citeD2|Hopf1931|Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche}} |- |1931 || [http://www-history.mcs.st-and.ac.uk/Biographies/Morse.html Morse] || {{CiteD2|Morse1931|The critical points of a function of $n$ variables}} |- |1944 || [http://www-history.mcs.st-and.ac.uk/Biographies/Whitney.html Whitney] || {{CiteD2|Whitney1944|The self-intersections of a smooth $n$-manifold in n$-space}} |- |1947 || [http://www-history.mcs.st-and.ac.uk/Biographies/Chern.html Chern] || {{citeD2|Chern1947|On the characteristic classes of Riemannian manifolds}} |- |1953 || [http://www-history.mcs.st-and.ac.uk/Biographies/Hirzebruch.html Hirzebruch] || {{citeD2|Hirzebruch1953a|On Steenrod's reduced powers, the index of inertia and the Todd genus}} |- |1954 || [http://www-history.mcs.st-and.ac.uk/Biographies/Thom.html Thom] || {{citeD2|1954|Quelques propriétés globales des variétés différentiables}} |- |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 $-spheres.</ref> |- |1961 || [http://www-history.mcs.st-and.ac.uk/Biographies/Smale.html Smale] || {{citeD2|Smale1961|Generalized Poincaré's conjecture in dimensions greater than four}} Proved that every smooth homotopy sphere of dimension 6 or more is homeomorphic to the standard sphere |- |1963 || [http://en.wikipedia.org/wiki/Kervaire Kervaire] [http://www-history.mcs.st-and.ac.uk/Biographies/Milnor.html Milnor] || {{citeD2|Kervaire&Milnor1963|Groups of homotopy spheres I}} Applied surgery theory to the diffeomorphism classification of smooth structures the n-sphere |- |1965 || [http://www-history.mcs.st-and.ac.uk/Biographies/Novikov_Sergi.html Novikov] || {{citeD2|Novikov1965|Rational Pontrjagin classes. Homeomorphism and homotopy type of closed manifolds I}} |} </wikitex> ==1971 - 1980== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 19?? || ... || ... |} </wikitex> ==1981 - 1990== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- |1982 || [http://www-history.mcs.st-and.ac.uk/Biographies/Freedman.html Freedman] || {{citeD2|Freedman1982|The topology of four-dimensional manifolds}} Proves the Poincaré conjecture in dimension 4 |- |1983 || [http://www-history.mcs.st-and.ac.uk/Biographies/Donaldson.html Donaldson] || {{citeD2|Donaldson1983|An application of gauge theory to four-dimensional topology}} |} </wikitex> ==1991 - 2000== {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 19?? || ... || ... |} ==2001 - present== <wikitex>; {| class="wikitable sortable" width="100%" ! style="width:8%" | Year ! style="width:10%" | Authors ! Publication and remarks |- | 20?? || ... || ... |} </wikitex> == Discussion == ==Footnotes== <references/> == References == {{#RefList:}} [[Category:History]]n$ variables
1944	Whitney	The self-intersections of a smooth $n$-manifold in $2n$-space
1947	Chern	On the characteristic classes of Riemannian manifolds
1953	Hirzebruch	On Steenrod's reduced powers, the index of inertia and the Todd genus
1954	Thom	Quelques propriétés globales des variétés différentiables
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 |- |1965 || Novikov || Rational Pontrjagin classes. Homeomorphism and homotopy type of closed manifolds I |}

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