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:
“We think of religion as the symbolic expression of our highest moral ideals; we think of magic as a crude aggregate of superstitions. Religious belief seems to become mere superstitious credulity if we admit any relationship with magic. On the other hand our anthropological and ethnographical material makes it extremely difficult to separate the two fields.”
—Ernst Cassirer (1874–1945)
“There is a relationship between cartooning and people like Miró and Picasso which may not be understood by the cartoonist, but it definitely is related even in the early Disney.”
—Roy Lichtenstein (b. 1923)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)
“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)
“That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.”
—Cornelia Otis Skinner (1901–1979)