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The Morse complex of the circle

Example

Assume AC. Let $f : S^{1} \to R$ be the height function on the round circle and let $X = μ (θ) \sin θ \partial_{θ}$ be the normalized positive multiple of its downward round gradient constructed below, where $f (θ) = \cos θ$ . It has exactly the same two orbit arcs as the round gradient. Then $(f, X)$ is Morse--Smale with a single maximum $p$ of index $1$ , a single minimum $q$ of index $0$ and no other critical points (Morse--Smale pairs, Morse functions and excellent Morse functions, Nondegenerate critical points, nullity, index, and coindex). The two open arcs from $p$ to $q$ are exactly the two elements $γ_{1}, γ_{2}$ of $M (p, q)$ (Unparametrized Morse trajectory moduli space), so the space is finite (Index-one trajectory moduli spaces are finite). Modulo two, The mod-two Morse differential gives $\partial p = q + q = 0$ and $\partial q = 0$ . For the signed differential, choose orientations of $W^{u} (p)$ and of the zero-dimensional $W^{u} (q)$ ; the flow traverses the two components of $W^{u} (p) \cap W^{s} (q)$ in opposite directions relative to that orientation, so the comparison signs satisfy $ϵ (γ_{1}) = - ϵ (γ_{2})$ and The signed Morse differential over the integers gives $\partial p = ϵ (γ_{1}) q + ϵ (γ_{2}) q = 0$ . Thus both the mod-two and the integral Morse complexes of $(f, X)$ have homology $Z / 2$ in degrees $0, 1$ (respectively $Z$ in degrees $0, 1$ ).

Facts & Assumptions

Given: AC, the circle with $f (θ) = \cos θ$ , the normalized field $X$ constructed in step 1.1, and orientations of both unstable manifolds.

[F1]

The height function on the round circle is Morse with exactly two nondegenerate critical points: a maximum $p$ of index $1$ and a minimum $q$ of index $0$ (Morse functions and excellent Morse functions, Nondegenerate critical points, nullity, index, and coindex, Morse--Smale pairs).

[F2]

Smooth cutoffs exist, and a normalized downward gradient-like field has the prescribed linear local model (A smooth bump between concentric Euclidean balls, Downward gradient-like vector fields for a Morse function). Under AC, a smooth vector field on a compact manifold is complete (Every smooth vector field on a compact manifold is complete).

[F3]

Under the Axiom of Choice, for index drop one the unparametrized moduli space is finite and its cardinality may be reduced modulo two (Index-one trajectory moduli spaces are finite, The Axiom of Choice, AC implies DC implies countable choice).

[F4]

On a basis element of the mod-two chain group the differential counts the index-one moduli space modulo two, and the signed differential sums the comparison signs over the same finite sets (The mod-two Morse differential, The signed Morse differential over the integers).

[F5]

The orientation of $W^{u} (p)$ and the chosen normal-quotient orientation of $W^{s} (q)$ orient the two intersection arcs. The comparison sign is $+ 1$ or $- 1$ according to agreement with positive flow; reversing the orientation at $q$ reverses both arc signs together (Unstable orientations induce orientations of the trajectory moduli spaces, The orientation line of a Morse critical point).

Verification

technique · direct, by explicit enumeration

1.1F1F2F3givenconstructalgebra

Choose smooth positive periodic $μ$ equal to $4 / (1 + \cos θ)$ near $p = 0$ and $4 / (1 - \cos θ)$ near $q = π$ , using disjoint cutoff neighbourhoods and the positive constant $2$ elsewhere. In the signed Morse coordinates $\sqrt{1 - \cos θ}$ near $p$ and $\sqrt{1 + \cos θ}$ near $q$ , with signs chosen across each pole, $X = μ \sin θ \partial_{θ}$ is respectively $2 w \partial_{w}$ and $- 2 w \partial_{w}$ . Also $d f (X) = - μ \sin^{2} θ < 0$ off the poles, and the Hessians are $- 1$ and $+ 1$ . The smooth field is complete by [F2]. Stable and unstable sets are the poles and their complementary open intervals, whose nonempty intersections are transverse. On each of the two arcs $X$ never vanishes, so $\int d θ / (μ \sin θ)$ gives a time coordinate onto $R$ , with endpoints $p$ backward and $q$ forward. Thus each arc is exactly one orbit class, and $M (p, q) = {γ_{1}, γ_{2}}$ .

2.1F4step 1.1

Modulo two, [F4] gives $\partial p = n_{2} (p, q) q$ with $n_{2} (p, q) = # M (p, q) m o d 2 = 2 m o d 2 = 0$ , so $\partial p = 0$ ; and $\partial q = 0$ because there is no critical point of index $- 1$ .

2.2F4F5step 1.1

For the signed differential, [F5] says that the single orientation of $W^{u} (p) ≅ R$ orients both arcs, and the flow direction along the two arcs is opposite with respect to it: traversing $S^{1}$ from $p$ to $q$ along one arc and back along the other reverses the direction. Hence $ϵ (γ_{1}) = - ϵ (γ_{2})$ , and [F4] gives $\partial p = (ϵ (γ_{1}) + ϵ (γ_{2})) q = 0$ , while $\partial q = 0$ .

3.1F4step 2.1step 2.2∎

Both complexes therefore have zero differentials with one generator in degree $1$ and one in degree $0$ ; their homology is $Z / 2$ in degrees $0$ and $1$ for the mod-two complex and $Z$ in degrees $0$ and $1$ for the integral complex, with all other graded pieces zero. The integral differential is a chain complex differential by The integral Morse differential squares to zero, consistent with the computation $\partial p = 0$ .

Depends on

Used by

Changing an unstable orientation changes two sets of basis signs Example

Dependency tree · two levels

69 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology, Part I Ch. 3, complete author PDF (standard reference, not scraped)
Alexander F. Ritter, Part III Morse Homology (Cambridge lecture notes), Lectures 17-19, complete combined PDF (standard reference, not scraped)