DescriptionOne of the most basic classes of spaces in topology is the class of manifolds. These spaces, wich can be thought of as higher dimensional analogues of surfaces, appear in many branches of mathematics, for example, as solutions to algebraic equations or as configuration spaces of mechanical systems. Morse theory is an important theory to study these manifolds by looking at real valued functions on them, and it links topology with analysis and algebra.Assuming this function satisfies certain properties, crucial information on the global shape of the manifold can be derived from the critical points of the function. Critical points are simply the points on the manifold where the derivative of the function vanishes. This information can be described in algebraic terms and includes the Euler characteristic. Morse theory can also be applied to infinite dimensional manifolds with important applications to Riemannian Geometry. In 1998, Robin Forman introduced a combinatorial Morse theory for cell complexes which allowed him to use many of the techniques from the smooth theory to study the topology of cell complexes. These cell complexes are a generalization of simplicial complexes where one looks at cells instead of simplices. Morse functions on a cell complex simply assign to every cell a real number subject to a certain condition. One can then introduce the concept of a critical cell (as opposed to a critical point) and it is possible to recover the homotopy type of the cell complex from the critical cells. One of the nice features of this approach is that everything is discrete so that one does not have to worry about continuity or smoothness of functions defined on the cell complex. Both the classical and the combinatorial theory give rise to various applications, such as understanding configuration spaces, homology and cohomology. There are also generalizations to path spaces, or Novikov theory, where the reals as range are replaced by the circle. This is a rich field, with many possible directions to explore. PrerequisitesTopology III.ResourcesThe standard reference book seems to be Morse Theory by John Milnor, but another book is An introduction to Morse theory by Yukio Matsumoto.Another useful, but more advanced book, is Lectures on the h-cobordism theorem by John Milnor. For the combinatorial theory, the standard main article is Morse theory for Cell Complexes, Advances in Mathematics 134 (1998), 90-145, by R. Forman. Other articles, including introductory expositions can be obtained from his website. |
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