A generalization of calculus of variations which draws the relationship between the stationary points of a
smooth real-valued function on a manifold
and the global topology of the manifold.
For example, if a compact manifold
admits a function whose only stationary points are a maximum and a minimum, then
the manifold is a sphere. Technically
speaking, Morse theory applied to a function  on a manifold  with  and  shows that every cobordism can be realized as a finite sequence of surgeries. Conversely, a sequence of surgeries gives a cobordism.
There are a number of classical applications of Morse theory, including counting geodesics on a Riemann surface
and determination of the topology of a Lie
group (Bott 1960, Milnor 1963). Morse theory has received much attention in the
last two decades as a result of the paper by Witten (1982) which relates Morse theory
to quantum field theory and also directly connects the stationary points of a smooth
function to differential forms on the manifold.
Bott, R. Morse Theory and Its Applications to Homotopy Theory: Lectures
by R. Bott. Bonn, Germany: Universität Bonn, 1960.
Chang, K. C. Infinite Dimensional Morse Theory and Multiple Solution Problems.
Boston, MA: Birkhäuser, 1993.
Goresky, M. and MacPherson, R. Stratified Morse Theory. New York: Springer-Verlag, 1988.
Milnor, J. W. Morse Theory. Princeton, NJ: Princeton University Press,
1963.
Rassias, G. (Ed.). Morse Theory and Its Applications.
Veverka, J. F. The Morse Theory and Its Application to Solid State Physics.
Kingston, Ontario, Canada: Queen's University, 1966.
Witten, E. "Supersymmetry and Morse Theory." J. Diff. Geom. 17,
661-692, 1982.
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