Mapping Class Group
It is a theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of any closed, oriented genus- surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along explicit curves generate the mapping class group (this is called by the punning name "Lickorish twist theorem"); this number was later improved by Stephen P. Humphries to, for, which he showed was the minimal number.
Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "Y-homeomorphisms."
Read more about this topic: Dehn Twist
Famous quotes containing the words class and/or group:
“God of justice, save the people
From the clash of race and creed,
From the strife of class and faction,
Make our nation free indeed;”
—William Pierson Merrill (1867–1954)
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)