Morse Homology — Mathematics Knowledge System

Overview

Morse homology recovers the singular homology of a smooth manifold from the critical points and gradient flow lines of a Morse function. Given a Morse function f on a compact manifold M with a Riemannian metric, one forms a chain complex: the chains are generated by critical points graded by their Morse index, and the boundary operator counts (with sign) rigid gradient flow lines between critical points of adjacent index. The resulting homology is independent of the choice of Morse function and metric, and equals the singular homology of M.

Intuition

On a surface, imagine water flowing downhill on a Morse function's gradient field. Critical points are where the gradient vanishes: minima, maxima, and saddles. The chain complex counts gradient flow trajectories from saddle points to minima (boundary map). Amazingly, this combinatorial count of flow lines reproduces the full topology of the surface. The Euler characteristic equals (number of minima) - (number of saddles) + (number of maxima), and the homology ranks satisfy the Morse inequalities.

Formal Definition

Definition

Let f: M -> R be a Morse function on a compact smooth manifold, g a generic Riemannian metric making (f, g) Morse-Smale. Define the Morse chain complex: C_k = free abelian group on critical points of index k. The boundary map partial_k: C_k -> C_{k-1} sends each index-k critical point p to the signed count of gradient flow lines from p to index-(k-1) critical points q. The Morse homology HM_*(M, f, g) is independent of (f, g) and equals H_*(M; Z).

C_k(f) = \bigoplus_{p \in \mathrm{Crit}_k(f)} \mathbb{Z} \cdot p

Morse chains

\partial_k(p) = \sum_{q \in \mathrm{Crit}_{k-1}(f)} n(p,q)\, q

Boundary operator (n(p,q) = signed count of flow lines)

\partial_{k-1} \circ \partial_k = 0,\quad H_k^{\mathrm{Morse}}(M) = \ker \partial_k / \mathrm{im}\, \partial_{k+1}

Homology

\chi(M) = \sum_k (-1)^k |\mathrm{Crit}_k(f)|

Euler characteristic via Morse function

Notation

Notation	Meaning
$\mathrm{Crit}_k(f)$	Critical points of f with Morse index k
$n(p,q)$	Signed count of rigid gradient flow lines from p to q
$W^u(p)$	Unstable manifold of critical point p
$HM_k(M)$	k-th Morse homology group of M

Theorems

Theorem 1: Morse Homology Theorem

For any Morse function f and generic metric g on a compact manifold M, the Morse homology HM_*(M, f, g) is isomorphic to the singular homology H_*(M; Z). In particular it is independent of the choices of f and g.

Theorem 2: Strong Morse Inequalities

Let c_k = |Crit_k(f)| and b_k = rank H_k(M; Z). Then for each n: c_n - c_{n-1} + ... +/- c_0 >= b_n - b_{n-1} + ... +/- b_0. The weak Morse inequalities state c_k >= b_k for all k.

Theorem 3: Poincare Duality via Morse

If M is a closed oriented n-manifold, replacing f by -f exchanges index-k critical points with index-(n-k) critical points, giving a chain-level proof that HM_k(M) is isomorphic to HM_{n-k}(M), recovering Poincare duality.

Worked Examples

1
Critical points: minimum at south pole p (index 0), maximum at north pole q (index 2). No index-1 critical points (no saddles).
2
Chain complex: C_2 = Z*q, C_1 = 0, C_0 = Z*p. All boundary maps are zero (no flow lines between non-adjacent index).
3
Homology: HM_0 = Z, HM_1 = 0, HM_2 = Z. This matches H_*(S^2; Z).
$H_0(S^2) = \mathbb{Z},\quad H_1(S^2) = 0,\quad H_2(S^2) = \mathbb{Z}$

✓ Answer

Morse homology of S^2 is Z in degrees 0 and 2, zero elsewhere -- matching singular homology.

Practice Problems

Hardproof writing

Prove that partial_{k-1} o partial_k = 0 in the Morse chain complex by showing that the boundary of a 1-dimensional moduli space of gradient flow lines contributes zero.

Mediumfree response

Describe the Morse chain complex for the torus T^2 with the standard height function, identifying all critical points and flow lines.

Common Mistakes

Common Mistake

Thinking all gradient flow lines contribute to the boundary map.

Only rigid (isolated) flow lines -- those in 0-dimensional moduli spaces (index difference exactly 1) -- contribute to the boundary map. Higher-dimensional families are not counted.

Common Mistake

Confusing the Morse index with the dimension of the stable manifold.

The Morse index of a critical point p equals the dimension of the UNSTABLE manifold W^u(p), not the stable manifold. The stable manifold has dimension (n - index).

Quiz

The boundary operator in the Morse chain complex counts:

Critical points in each connected componentSigned gradient flow trajectories between critical points of adjacent indexHomotopy classes of paths between critical pointsHomology classes of the unstable manifolds

Morse homology of a compact manifold M is independent of:

The dimension of MThe coefficient ringThe choice of Morse function and Riemannian metricThe smooth structure of M

Historical Background

Morse theory originated with Marston Morse's work in the 1920s-30s, connecting the topology of manifolds to the critical points of smooth functions. The modern chain complex formulation -- now called Morse homology or Morse-Smale-Witten complex -- was developed by Thom (1949), Smale (1961), Witten (1982), and rigourously justified by Floer, Salamon, and Schwarz in the 1990s. Floer's infinite-dimensional generalisation (Floer homology) revolutionised symplectic topology and gauge theory.

1934
Morse publishes 'The Calculus of Variations in the Large'
Marston Morse
1961
Smale proves the h-cobordism theorem using Morse theory and gradient flows
Stephen Smale
1982
Witten gives a physics derivation of the Morse inequalities via supersymmetric quantum mechanics
Edward Witten
1995
Schwarz publishes a rigorous treatment of Morse homology as a chain complex
Matthias Schwarz

Summary

Morse homology builds a chain complex from critical points of a Morse function, with boundary map counting gradient flow lines.
The key theorem: Morse homology equals singular homology, independent of (f, g).
The Morse inequalities bound Betti numbers below by critical point counts: b_k <= c_k.
Morse homology is the prototype for Floer homology in infinite-dimensional settings.

References

BookMilnor, J. Morse Theory. Princeton University Press, 1963.
BookSchwarz, M. Morse Homology. Birkhauser, 1993.
WebsiteWikipedia -- Morse homology