Related Spaces
More examples of open, contractible 3-manifolds may be constructed by proceeding in similar fashion and picking different embeddings of Ti+1 in Ti in the iterative process. Each embedding should be an unknotted solid torus in the 3-sphere. The essential properties are that the meridian of Ti should be null-homotopic in the complement of Ti+1, and in addition the longitude of Ti+1 should not be null-homotopic in Ti − Ti+1. Another variation is to pick several subtori at each stage instead of just one. The cones over some of these continua appear as the complements of Casson handles in a 4-ball.
The dogbone space is not a manifold but its product with R1 is homeomorphic to R4.
Read more about this topic: Whitehead Manifold
Famous quotes containing the words related and/or spaces:
“The near explains the far. The drop is a small ocean. A man is related to all nature. This perception of the worth of the vulgar is fruitful in discoveries. Goethe, in this very thing the most modern of the moderns, has shown us, as none ever did, the genius of the ancients.”
—Ralph Waldo Emerson (1803–1882)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)