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:
“I began to expand my personal service in the church, and to search more diligently for a closer relationship with God among my different business, professional and political interests.”
—Jimmy Carter (James Earl Carter, Jr.)
“When any relationship is characterized by difference, particularly a disparity in power, there remains a tendency to model it on the parent-child-relationship. Even protectiveness and benevolence toward the poor, toward minorities, and especially toward women have involved equating them with children.”
—Mary Catherine Bateson (20th century)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)
“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)
“The failure of women to produce genius of the first rank in most of the supreme forms of human effort has been used to block the way of all women of talent and ambition for intellectual achievement.”
—Anna Garlin Spencer (1851–1931)