Related Exotic Structures
Casson handles are homeomorphic to D2×R2 by Freedman's theorem (where D2 is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to D2×R2. In other words, some Casson handles are exotic D2×R2s.
It is not known (as of 2009) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.
Read more about this topic: Exotic R4
Famous quotes containing the words related, exotic and/or structures:
“Generally there is no consistent evidence of significant differences in school achievement between children of working and nonworking mothers, but differences that do appear are often related to maternal satisfaction with her chosen role, and the quality of substitute care.”
—Ruth E. Zambrana, U.S. researcher, M. Hurst, and R.L. Hite. “The Working Mother in Contemporary Perspectives: A Review of Literature,” Pediatrics (December 1979)
“I do not approve of anything that tampers with natural ignorance. Ignorance is like a delicate exotic fruit; touch it and the bloom is gone.”
—Oscar Wilde (1854–1900)
“It is clear that all verbal structures with meaning are verbal imitations of that elusive psychological and physiological process known as thought, a process stumbling through emotional entanglements, sudden irrational convictions, involuntary gleams of insight, rationalized prejudices, and blocks of panic and inertia, finally to reach a completely incommunicable intuition.”
—Northrop Frye (b. 1912)