Spheres
The Euler characteristic of the n-sphere is:
Thus, there is no non-vanishing section of the tangent bundle of even spheres, so the tangent bundle is not trivial -- i.e., is not a parallelizable manifold, and in particular does not admit a Lie group structure.
For odd spheres, a nowhere vanishing section is given by
which shows that the Euler class vanishes; this is just n copies of the usual section over the circle.
As the Euler class for an even sphere corresponds to, we can use the fact that the Euler class of a Whitney sum of two bundles is just the cup product of the Euler class of the two bundles to see that there are no non-trivial subbundles of the tangent bundle of an even sphere.
Since the tangent bundle of the sphere is stably trivial but not trivial, all other characteristic classes vanish on it, and the Euler class is the only ordinary cohomology class that detects non-triviality of the tangent bundle of spheres: to prove further results, one must use secondary cohomology operations or K-theory.
Read more about this topic: Euler Class, Examples
Famous quotes containing the word spheres:
“The world has already learned that woman has other virtues than meekness, patience, humility and endurance. She possesses courage above all fear, and a will that knows no obstacles; and when these are called forth by some great emergency, false modesty is trampled in the dust, and spheres are scattered to the winds.”
—A. Holley, U.S. women’s magazine contributor. The Lily, p. 38 (May 1852)
“Perhaps a modern society can remain stable only by eliminating adolescence, by giving its young, from the age of ten, the skills, responsibilities, and rewards of grownups, and opportunities for action in all spheres of life. Adolescence should be a time of useful action, while book learning and scholarship should be a preoccupation of adults.”
—Eric Hoffer (1902–1983)
“It launch’d forth filament, filament, filament, out of itself,
Ever unreeling them, ever tirelessly speeding them.
And you O my soul where you stand,
Surrounded, detached, in measureless oceans of space,
Ceaselessly musing, venturing, throwing, seeking the spheres to connect them,
Till the bridge you will need be form’d, till the ductile anchor hold,
Till the gossamer thread you fling catch somewhere, O, my soul.”
—Walt Whitman (1819–1892)