Editing Exotic sphere (section)

==4-dimensional exotic spheres and Gluck twists==
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by {{harvs|txt | last1=Freedman | first1=Michael | authorlink1=Michael Freedman| last2=Gompf | first2=Robert | authorlink2=Robert Gompf| last3=Morrison | first3=Scott | last4=Walker | first4=Kevin | year=2010}} who say that it is believed to be false.

Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres ({{harvs|txt|last1=Cappell|first1=Sylvain|authorlink1=Sylvain Cappell|last2=Shaneson|first2=Julius|authorlink2=Julius Shaneson|year=1976}}) and those derived by '''Gluck twists''' {{harv|Gluck|1962}}. Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere ''S'' in ''S''<sup>4</sup> and gluing it back in using a diffeomorphism of its boundary ''S''<sup>2</sup>×''S''<sup>1</sup>. The result is always homeomorphic to ''S''<sup>4</sup>. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, {{harvs|txt|last=Gordon|first=Cameron|authorlink=Cameron Gordon (mathematician)| year=1976}}, {{harvs|txt|last=Montesinos|first=José|year=1983}}, {{harvs|txt|last=Plotnick|first=Steven P. |year=1984}}, {{harvtxt|Gompf|1991}}, {{harvtxt|Habiro|Marumoto|Yamada|2000}}, {{harvs|txt|last=Akbulut|first=Selman|authorlink=Selman Akbulut| year=2010}}, {{harvtxt|Gompf|2010}}, {{harvtxt|Kim | Yamada|2017}}.