Relationship With Dimensions of Spaces of Differential Forms
In geometric situations when is a closed manifold, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms. The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality (when those apply), and the universal coefficient theorem of homology theory.
There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms. This requires also the use of some of the results of Hodge theory, about the Hodge Laplacian.
In this setting, Morse theory gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of critical points of a Morse function of a given index:
Witten gave an explanation of these inequalities by using the Morse function to modify the exterior derivative in the de Rham complex.
Read more about this topic: Betti Number
Famous quotes containing the words relationship with, relationship, dimensions, spaces, differential and/or forms:
“Guilty, guilty, guilty is the chant divorced parents repeat in their heads. This constant reminder remains just below our consciousness. Nevertheless, its presence clouds our judgment, inhibits our actions, and interferes in our relationship with our children. Guilt is a major roadblock to building a new life for yourself and to being an effective parent.”
—Stephanie Marston (20th century)
“When a mother quarrels with a daughter, she has a double dose of unhappiness—hers from the conflict, and empathy with her daughter’s from the conflict with her. Throughout her life a mother retains this special need to maintain a good relationship with her daughter.”
—Terri Apter (20th century)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)
“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
—Jean Baudrillard (b. 1929)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“Year chases year, decay pursues decay,
Still drops some joy from with’ring life away;
New forms arise, and diff’rent views engage,”
—Samuel Johnson (1709–1784)