nLab Morse theory



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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Morse theory is the method of studying the topology of a smooth manifold MM by the study of Morse functions MM\to\mathbb{R} and their associated gradient flows.

Classical Morse theory centered around simple statements like Morse inequalities, concerning just the Betti numbers. It is useful not only for studying manifolds, but also for studying infinite CW-type spaces homotopically filtered in manifolds, as by Milnor and Bott (especially the stable homotopy of the classical groups) for spaces of paths in a smooth manifold.

Novikov–Morse theory is a variant using multivalued functions. There is also a discrete Morse theory for combinatorial cell complexes.

There are some infinite-dimensional generalizations like Floer instanton homology for 3-dimensional manifolds and also the Hamiltonian variant of Floer homology (and cohomology) for (finite dimensional) symplectic manifolds. An especially well-studied case is that of the cotangent bundle of a differentiable manifold with its standard symplectic structure; this is sometimes called Floer–Oh homology. Floer homology has been partly motivated by Arnold’s conjecture on periodic trajectories in classical mechanics. The symplectic variant of Floer cohomology is related to quantum cohomology.

The founders of Morse theory were Marston Morse, Raoul Bott and Albert Schwarz.


On a smooth manifold MM, a smooth function φ:M\varphi: M \to \mathbb{R} is said to be Morse (or a Morse function) if

  • the zero set of dφ d \varphi consists of isolated points, and
  • the Hessian of φ\varphi at these points is nondegenerate.

The Morse functions on MM are dense in most reasonable topologies you could put on C (M)C^{\infty}(M). A further condition which is useful, in case MM is not compact, is

  • if the (closed!) preimage of (,λ]( -\infty , \lambda ] under φ\varphi is compact for all λ\lambda, then φ\varphi is said to be coercive, whether or not it is Morse.

Together with a (smooth) Riemann structure g=,g =\langle \cdot,\cdot\rangle, any real function φ\varphi on MM defines a flow on MM by the equation

x˙,Y x=Y xφ=dφ(Y x). - \langle \dot x, Y_x \rangle = Y_x \varphi = d\varphi(Y_x).

The Morse functions are notable in that the flows they define have isolated fixed-points with trivially linearizable dynamics, and …[fixme: less vague?]… no other stable cyles.

When φ\varphi is Morse and coercive, the unstable manifolds of the fixpoints can be arranged into a CW complex C unstable(φ,g)C_{unstable} (\varphi,g), canonically homeomorphic to MM. When MM is compact, φ\varphi and φ-\varphi are automatically both coercive, and φ-\varphi induces a dual CW complexe C stable(φ,g) C_{stable} (\varphi,g) . Concretely, ….[details].

Sketch of a trivial application

Let XY X \to Y be a surjective submersion of compact smooth manifolds, and assume YY is connected. By suitable implicit function theorems, the preimage of any parametrized non-stationary curve γ:(0,1)Y\gamma :(0,1)\to Y is a submanifold of XX, and furthermore the parameter is a Morse function on this submanifold, having no critical points. (It is not coercive). By a little more analysis, the Morse gradient flow is therefore a smooth family of homotopy equivalences. A trivial adjustment of the Riemann structure further allows that the Morse flow sends fibers to fibers diffeomorphically, so that in fact the fibers over neighboring points of YY are diffeomorphic. But since YY is connected, this implies that all the fibers are diffeomorphic, so that XYX\to Y is a smooth fiber bundle over YY.

Slightly less-trivial example

The restriction to the unit sphere in n+1\mathbb{R}^{n+1} of a generic quadratic form is Morse with 2(n+1)2(n+1) critical points — two of each index; and furthermore this restriction clearly descends to ℝℙ n\mathbb{RP}^n as a Morse function with n+1n+1 critical points, one of each index. It can be shown that this is indeed the minimal collection of critical points supported by real projective space.



Morse complex and homology

  • John Milnor, Lectures on the h-cobordism theorem, Notes by L. Siebenmann & J. Sondow, Princeton Univ.

    Press, 1965.

  • Matthias Schwarz, Morse homology, Progress in Mathematics 111, 1993

There is also a variant due to Barannikov, and in a more abstract form due to Viterbo:

Relation to supersymmetric quantum mechanics

The relation to supersymmetric quantum mechanics is due to

  • Edward Witten. Supersymmetry and Morse theory. J. Diff Geom. 17(4): 661-692 (1982). (doi)

Reviews include

  • Gábor Pete, section 2 of Morse theory, lecture notes 1999-2001 (pdf)

Last revised on January 21, 2024 at 18:35:19. See the history of this page for a list of all contributions to it.