|
|
|
|
Morse homology
|
(Definition)
|
|
|
Morse homology is a tool developed by Thom, Smale, and Milnor for homology theory.
Take  to be a smooth compact manifold. Throughout we assume that  is a suitable Morse function, that is, all critical points of  are nondegenerate. We must first make some definitions before defining the Morse homology. Choose a Riemannian metric on  so that the notion of a gradient vector field makes sense. The map
 such that
with
 , is called the negative gradient flow of  . Let  be a critical point of  , and define
 and   into its unstable manifolds and arrive at something that is homologically equivalent to its handle decomposition, but this decomposition was not a CW complex, hence it was hard to say anything about the
homotopy type of  . But Smale realized that if we impose more conditions on the metric itself, then we can make this into a CW complex.
The pair  , where  is a Morse function and  is the Riemannian metric, is called Morse-Smale pair, if for every pair  ,  of critical points of  ,  is transverse to  . This is known as the Morse-Smale condition. This condition actually holds for a generic Riemannian metric on M. With this restriction, this makes Thom's decomposition into a CW complex.
We can define a complex called the Morse complex as follows:
Let
 be the set of critical points of  of index  . We define the chain group ,  to be the formal linear combination with integer coefficients of elements of
 . We must also keep track of the signs of the flow lines. (However, it is true if you count mod 2, the Morse complex computes homology with coefficients in  .) To make this a chain complex we must define the differential map. The map
 applied to a critical point  is a formal sum of critical points with  given by this number. It is possible to prove that
 , making this into a chain complex.
The homology of this complex is called the Morse homology. It can be shown to be isomorphic to the singular homology of  .
Note: There is another way of realizing the Morse homology using Hodge theory, an idea pioneered by Edward Witten. His idea is essentially to conjugate the  operator by  and it can be shown that this conjugation again leads to another isomorphism between the set of harmonic forms and the De Rham cohomology. This parameter  is like a curve of chain complexes and Witten claimed that if  is large enough, then we can obtain a space whose dimension is the number of critical points of a given index and the boundary operator induced on  is the number of critical paths between critical points, as before. Witten did not prove this idea rigorously, but it was done later by Helffer and Sjostrand.
|
"Morse homology" is owned by PrimeFan. [ full author list (3) | owner history (5) ]
|
|
(view preamble)
Cross-references: paths, induced, boundary operator, dimension, curve, parameter, de Rham cohomology, harmonic, isomorphism, conjugation, operator, conjugate, Hodge theory, isomorphic, number, sum, chain complex, lines, coefficients, integer, linear combination, group, chain, index, Morse complex, complex, restriction, generic, transverse, metric, homotopy type, CW complex, decomposition, handle decomposition, equivalent, unstable manifolds, stable, flow, negative, map, vector field, gradient, Riemannian metric, definitions, nondegenerate, critical points, Morse function, manifold, compact, smooth, theory, homology
There is 1 reference to this entry.
This is version 14 of Morse homology, born on 2005-06-20, modified 2006-10-09.
Object id is 7174, canonical name is MorseHomology.
Accessed 2764 times total.
Classification:
| AMS MSC: | 58A05 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differentiable manifolds, foundations) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
