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Morse and cellular boundaries for a surface handle presentation

Example

Assume the Axiom of Choice (The Axiom of Choice). Let $M = S^{2}$ with a height function $f$ having two maxima $p_{1}, p_{2}$ of index $2$ , one saddle $a$ of index $1$ and one minimum $r$ of index $0$ (the ``other sphere'' picture) (Nondegenerate critical points, nullity, index, and coindex).

Then: the trajectories from each maximum to the saddle form the finite set $M (p_{i}, a)$ with $# M (p_{i}, a) = 1$ , so the Morse differential is $\partial p_{i} = \pm a$ (The mod-two Morse differential, The signed Morse differential over the integers); the two trajectories from the saddle to the minimum give $\partial a = 0$ by the boundary-orientation cancellation (Unstable orientations induce orientations of the trajectory moduli spaces); and the cellular boundary of the Morse--Smale CW decomposition (Compactified unstable manifolds give the Morse--Smale CW decomposition) is, with the cell orientations of the unstable manifolds, $[e_{p_{i}} : e_{a}] = \pm 1$ and $[e_{a} : e_{r}] = 0$ (Incidence number of two CW cells, Cellular boundary is the incidence degree matrix, Cellular boundary from three consecutive skeleta). Thus the two boundary matrices agree up to the index normalization of Cellular boundary coefficients are the signed trajectory counts, realizing The Morse complex is chain isomorphic to the handle cellular complex in a case with nonzero coefficients: the cellular complex $Λ {p_{1}, p_{2}} \to Λ {a} \to Λ {r}$ has homology $Λ$ in degrees $0$ and $2$ , matching $H_{*} (S^{2}; Λ)$ .

Facts & Assumptions

Given: The Axiom of Choice, the "other sphere" Morse function $f$ on $S^{2}$ with maxima $p_{1}, p_{2}$ , saddle $a$ and minimum $r$ , and the Morse--Smale CW decomposition of its compactified unstable manifolds.

[F1]

Each maximum has exactly one steepest-descent trajectory to the saddle in this model, so $# M (p_{i}, a) = 1$ ; the Morse coefficient of $a$ in $\partial p_{i}$ is the trajectory sign, equal to $\pm 1$ (The mod-two Morse differential, The signed Morse differential over the integers, Nondegenerate critical points, nullity, index, and coindex).

[F2]

The unstable manifold of the saddle is one-dimensional; its compactification is a compact one-manifold with boundary whose boundary points are the two trajectories to the minimum, and the outward-normal-first orientation gives the two boundary signs opposite to each other, so the trajectory sign is the comparison of the oriented unstable interval with the outward flow direction: it is positive at one end and negative at the other. Thus the signed count of $M (a, r)$ is zero; over $Z / 2$ the count is $2 \equiv 0$ . Hence $\partial a = 0$ in both coefficient cases (Compactified unstable manifolds give the Morse--Smale CW decomposition, Unstable orientations induce orientations of the trajectory moduli spaces, The mod-two Morse differential).

[F3]

The cell attachments of the Morse--Smale CW decomposition have incidence numbers equal to the Morse coefficients up to the dimension-dependent sign of the coefficient comparison: $[e_{p_{i}} : e_{a}] = \pm 1$ and $[e_{a} : e_{r}] = 0$ (Compactified unstable manifolds give the Morse--Smale CW decomposition, Cellular boundary coefficients are the signed trajectory counts, Incidence number of two CW cells, Cellular boundary is the incidence degree matrix, Cellular boundary from three consecutive skeleta).

[F4]

The chain isomorphism of The Morse complex is chain isomorphic to the handle cellular complex identifies the homology of the Morse complex with the homology of the cellular complex, which for the presented cell structure is $Λ$ in degrees $0$ and $2$ , matching $H_{*} (S^{2}; Λ)$ .

Verification

technique · direct

1.1F1F2given

Apply the CW supplier of [F2] to $(- f, - X)$ , which is Morse--Smale because its stable and unstable manifolds are those of $X$ interchanged. Its dual CW structure has two zero-cells, one one-cell and one two-cell. The one-skeleton is connected: attaching a two-disk along its connected boundary circle cannot join distinct components, whereas the final sphere is connected. Hence the unique edge joins the two distinct vertices. Its two half-orbits, reversed back to $X$ , are exactly the two stable saddle branches, each tending to a different original maximum. Thus each maximum supplies exactly one orbit to the saddle, proving the geometric assertion of [F1]. By [F1] the only index-drop-one trajectory moduli with source a maximum are $M (p_{1}, a)$ and $M (p_{2}, a)$ , each a single point; thus $\partial p_{i} = \pm a$ in the integral case and $\partial p_{i} = a$ over $Z / 2$ .

1.2F2given

By [F2] the compactified unstable manifold of the saddle is an interval whose boundary consists of the two trajectories to the minimum, and the boundary orientation makes their signs cancel; hence $\partial a = 0$ over $Z$ , and over $Z / 2$ the count is $2 \equiv 0$ .

2.1F3step 1.1step 1.2

There are no other critical points, so the Morse complex is $Λ {p_{1}, p_{2}} \overset{\partial}{\to} Λ {a} \overset{\partial = 0}{\to} Λ {r}$ with $\partial p_{i} = \pm a$ ; by [F3] the cellular boundary matrix coincides with this one up to the index normalization, so $[e_{p_{i}} : e_{a}] = \pm 1$ and $[e_{a} : e_{r}] = 0$ , realizing the coefficient comparison in a case with a nonzero coefficient.

3.1F4step 2.1algebra∎

The homology of this complex is $Λ$ in degree $0$ (generated by $r$ ), $Λ$ in degree $2$ (generated by the class $p_{1} - p_{2}$ or $p_{1} + p_{2}$ according to the signs), and zero in degree $1$ , since the image of $\partial$ on $Λ {p_{1}, p_{2}}$ is the whole of $Λ {a}$ ; by [F4] this agrees with $H_{*} (S^{2}; Λ)$ and with the cellular complex of the decomposition.

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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)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (complete author PDF, 291 pp.) (standard reference, not scraped)