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Two Morse functions on the circle have isomorphic Morse homology

Example

Assume the Axiom of Choice (The Axiom of Choice). On $S^{1}$ consider the height function $f_{0} (θ) = \cos θ$ with maximum $p$ at $θ = 0$ and minimum $q$ at $θ = π$ , and a Morse function $f_{1}$ with two maxima and two minima obtained from $f_{0}$ by a birth--death pair as in the local calculation below (Morse functions and excellent Morse functions, Nondegenerate critical points, nullity, index, and coindex, Morse--Smale pairs).

Then the Morse complex of $f_{0}$ is $Λ \cdot p \overset{\partial}{\to} Λ \cdot q$ with $\partial p = 0$ , since the two trajectories from $p$ to $q$ (the two arcs of $S^{1} ∖ {p, q}$ ) carry opposite signs in the conventions of Unstable orientations induce orientations of the trajectory moduli spaces: both arcs inherit the same orientation from the orientation of the one-dimensional unstable manifold $W^{u} (p) = S^{1} ∖ {q}$ , while the flow direction is opposite on the two arcs (The signed Morse differential over the integers, The mod-two Morse differential). Hence $H M_{0} (f_{0}) = Λ$ and $H M_{1} (f_{0}) = Λ$ (Morse homology of a Morse--Smale pair); the same computation with the extra acyclic summand gives $H M_{*} (f_{1}) ≅ H M_{*} (f_{0})$ , and the continuation isomorphism of Reverse continuation is an inverse on Morse homology realizes this identification. Both groups agree with $H_{*} (S^{1}; Λ)$ under Morse homology is naturally isomorphic to singular homology, so the canonical Morse homology Canonical Morse homology of a closed manifold is $H M_{0} (S^{1}; Λ) = H M_{1} (S^{1}; Λ) = Λ$ and $H M_{k} = 0$ for $k \neq 0, 1$ .

Facts & Assumptions

Given: The Axiom of Choice and the height function $f_{0} = \cos θ$ on $S^{1}$ with maximum $p$ and minimum $q$ , and the function $f_{1}$ obtained from it by a birth--death pair.

[F1]

On the circle both $f_{0}$ and $f_{1}$ are Morse, and both pairs are Morse--Smale with respect to the round metric: unstable and stable manifolds of the (at most one-dimensional) trajectory spaces meet transversally (Morse--Smale pairs, Morse functions and excellent Morse functions).

[F2]

The two trajectories from $p$ to $q$ are the two arcs of $S^{1} ∖ {p, q}$ ; both are contained in the one-dimensional unstable manifold $W^{u} (p)$ and inherit its orientation, while the flow direction is opposite on the two arcs; by comparison with the flow orientation their signs are opposite. The unparametrized moduli space here is zero-dimensional (Unstable orientations induce orientations of the trajectory moduli spaces).

[F3]

Because the only two critical points of $f_{0}$ have indices $1$ and $0$ , the Morse complex of $f_{0}$ has no other differentials; the signed count of [F2] makes $\partial p = 0$ over $Z$ and $2 \equiv 0$ over $Z / 2$ (The signed Morse differential over the integers, The mod-two Morse differential).

[F4]

After the explicit basis change below, the birth--death pair adds an acyclic two-term summand to the complex of $f_{0}$ , so $H M_{*} (f_{1}) ≅ H M_{*} (f_{0})$ ; the continuation map between the two Morse--Smale pairs is an isomorphism on homology (the local calculation below, Reverse continuation is an inverse on Morse homology).

[F5]

For a closed manifold the Morse homology of any Morse--Smale pair is isomorphic to singular homology, and the canonical Morse homology is well defined up to canonical isomorphism (Morse homology is naturally isomorphic to singular homology, Canonical Morse homology of a closed manifold).

Verification

technique · direct

1.1F2F3givenalgebra

For $f_{1}$ , enumerate the critical points in circular order as $p, q_{1}, c, q_{2}$ , with $p, c$ maxima and $q_{1}, q_{2}$ minima. Orient both unstable intervals in the direction of increasing angle. The two outgoing arcs at each maximum then give $\partial p = q_{1} - q_{2}$ and $\partial c = q_{2} - q_{1}$ (over $Z / 2$ replace minus by plus). In the integral bases $P = p + c$ , $C = c$ , $Q = q_{1}$ , $B = q_{2} - q_{1}$ , one has $\partial P = 0$ , $\partial C = B$ . These are invertible basis changes over $Z$ and over the stated coefficient rings. The pair $C \mapsto B$ has contracting homotopy $B \mapsto C$ , so the complex is the direct sum of the minimal zero-differential complex on $P, Q$ and this acyclic pair.

1.2F1F2F3given

By [F1] both pairs are Morse--Smale, so the Morse complexes are defined. By [F3] the Morse complex of $f_{0}$ is $Λ p \to Λ q$ with $\partial p = 0$ , because the only differential is the signed count of the two arcs from $p$ to $q$ and the two signs cancel.

2.1step 1.2algebra

Hence $H_{1} (f_{0}) = \ker \partial_{1} = Λ p$ and $H_{0} (f_{0}) = Λ q / im \partial_{1} = Λ q$ , so $H M_{0} (f_{0}) = Λ = H M_{1} (f_{0})$ and $H M_{k} (f_{0}) = 0$ for $k \neq 0, 1$ .

3.1F4step 2.1

By [F4] the birth--death pair adds an acyclic summand, so the homology of the complex of $f_{1}$ is the same: $H M_{*} (f_{1}) ≅ H M_{*} (f_{0})$ , and the continuation isomorphism realizes the identification.

4.1F5step 3.1∎

By [F5] both computations agree with the singular homology of the circle, $H_{0} (S^{1}; Λ) = H_{1} (S^{1}; Λ) = Λ$ and $H_{k} = 0$ otherwise, so the canonical Morse homology of $S^{1}$ is $Λ$ in degrees $0$ and $1$ ; this is the claimed comparison of the minimal and the stabilised function.

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Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (complete author PDF of the English book, 628 pp.) (standard reference, not scraped)
Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes, complete author PDF, 115 pp.) (standard reference, not scraped)