Math 2.0

Manifolds and cell complexes Definition

Morse homology

Builds on Smooth manifoldMorse functionRiemannian metricTransversality

Definition

Let MM be a compact smooth manifold without boundary, let f ⁣:M→ℝf \colon M \to \R be a Morse function, and let gg be a Riemannian metric on MM. Let pp be a critical point, and consider the flow lines of the vector field −∇f-\nabla f. The unstable manifold Wu(p)W^u(p) is the set of points whose flow line tends to pp as time goes to −∞-\infty. The stable manifold Ws(p)W^s(p) is the set of points whose flow line tends to pp as time goes to +∞+\infty. Assume that the pair (f,g)(f, g) is Morse–Smale: Wu(p)W^u(p) and Ws(q)W^s(q) are transverse for all critical points p,qp, q. For k≥0k \ge 0, let CkC_k be the ℤ/2\Z/2-vector space with basis the critical points of ff of index kk. Let pp be a critical point of index kk and qq one of index k−1k - 1. Let n(p,q)∈ℤ/2n(p, q) \in \Z/2 be the number, modulo 22, of flow lines of −∇f-\nabla f from pp to qq, counted up to time shift; it is finite. Let ∂ ⁣:Ck→Ck−1\partial \colon C_k \to C_{k-1} be the linear map with ∂p=∑qn(p,q) q\partial p = \sum_q n(p, q) \, q. Then ∂∘∂=0\partial \circ \partial = 0, and the Morse homology of (f,g)(f, g) is the homology of the chain complex (C∗,∂)(C_*, \partial).

Remarks

The Morse homology is isomorphic to H∗(M;ℤ/2)H_*(M; \Z/2), so it does not depend on ff or gg. With a choice of orientations of the unstable manifolds, one can count flow lines with signs and get a chain complex over ℤ\Z whose homology is H∗(M;ℤ)H_*(M; \Z). The identity ∂2=0\partial^2 = 0 comes from broken flow lines p→q→rp \to q \to r with index drop 22. They are exactly the ends of the one-dimensional spaces of flow lines from pp to rr, and these ends come in pairs. Example: the height function on the round sphere SnS^n, n≥1n \ge 1, has one critical point of index 00 and one of index nn. For n≥2n \ge 2 the indices differ by more than one, so ∂=0\partial = 0. For n=1n = 1 there are two flow lines from the maximum to the minimum, so again ∂=0\partial = 0 modulo 22. In both cases the homology is ℤ/2\Z/2 in degrees 00 and nn. Non-example: take a metric on the upright torus in which the flow line leaving the upper saddle runs straight into the lower saddle. Then WuW^u of the upper saddle and WsW^s of the lower saddle meet along a curve, so they are not transverse. The pair is not Morse–Smale and the definition does not apply; a small change of the metric removes the problem.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Morse function, concept, no evidence levelMorse functionconceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTransversality, concept, no evidence levelTransversalityconceptMorse homology, definition, no evidence levelMorse homologydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Morse function concept
  2. Riemannian metric concept
  3. Smooth manifold concept
  4. Transversality concept
  5. Morse homology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Morse_homology
About this page
Tag7MJSQ7MT23
URIm2://7MJSQ7MT23
Kinddefinition
WritingW0 stub
MSC57R70, 57R58, 58E05
DefinesMorse homology
UsesSmooth manifold, Morse function, Riemannian metric, Transversality
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.