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The relative Morse complex of an adapted cobordism

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(W; M_{0}, M_{1})$ be a compact smooth cobordism triad with adapted excellent Morse function $f$ and adapted complete downward gradient-like field $X$ pointing outward along $M_{0}$ and inward along $M_{1}$ , with $(f, X)$ Morse--Smale in the sense that all unstable/stable intersections in $int W$ are transverse (Smooth cobordism triad for Morse theory, Morse function adapted to a cobordism, Morse--Smale pairs). Then:

all critical points of $f$ are interior and finite in number (A Morse function on a compact manifold has finitely many critical points, Nondegenerate critical points, nullity, index, and coindex); moreover every $X$ -trajectory joining two critical points is contained in the compact interior region ${x \in W : f (q) \leq f (x) \leq f (p)}$ determined by the endpoint values, hence meets neither $\partial W$ nor sufficiently small boundary collars whose $f$ -values lie below all interior critical values at $M_{0}$ and above all interior critical values at $M_{1}$ (Deformation lemma for a critical point free slab, Normalized gradient crosses a compact regular band in controlled time, Regular interval diffeomorphism);
the relative Morse chain groups $C M_{k} (f, X; Λ)$ , free on the interior critical points of index $k$ over $Λ = Z / 2$ and $Λ = Z$ (with a chosen positive orientation ray at each critical point for $Λ = Z$ ) (The mod-two Morse chain group, The signed Morse differential over the integers), with the trajectory differentials of The mod-two Morse differential and The signed Morse differential over the integers restricted to the interior trajectory moduli spaces, form chain complexes (by the relative cellular coefficient comparison below); their homology is denoted $H M_{*} (W, M_{0}; Λ)$ and called the relative Morse homology of the adapted data;
the compactified unstable manifolds of the interior critical points give the exact disk-attachment pair $(Z, M_{0})$ of Compactified unstable manifolds give the Morse--Smale CW decomposition. Its stagewise cellular approximation gives a finite CW model $(X, M_{0})$ , with one relative cell per critical point. The local coefficient computation of Cellular boundary coefficients are the signed trajectory counts identifies the relative Morse complex with the cellular complex of the relative handle decomposition of $(W, M_{0})$ (Handle decomposition relative to the incoming boundary, Morse functions and handle decompositions correspond, A handle decomposition gives a relative CW complex, One critical point handle attachment, Unstable disk is the handle core); hence there is a chain isomorphism between the relative Morse complex and the relative handle (cellular) chain complex, and consequently $H M_{*} (W, M_{0}; Λ) ≅ H_{*}^{c e l l} (W, M_{0}; Λ) ≅ H_{*} (W, M_{0}; Λ),$ the last isomorphism being the relative cellular comparison theorem applied with the constant local system (Cellular chains compute local homology, Homology and cohomology with local coefficients, Relative singular homology).

Here the relative handle cellular complex is the CW model $(X, M_{0})$ constructed by transporting and cellularly approximating the exact disk attaching maps. Those evaluation maps need not themselves extend the incoming CW structure. Its cellular homology is transported to the displayed $(W, M_{0})$ notation along the proved equivalence of pairs. The original value-ordered handle stages are not asserted to be the skeleta.

Facts & Assumptions

Given: The Axiom of Choice, the adapted excellent normalized Morse--Smale triad and the two coefficient rings.

[F1]

All critical points are interior and finite; boundary values zero and one lie strictly below and above their finite value range (Morse function adapted to a cobordism, A Morse function on a compact manifold has finitely many critical points).

[F2]

The compactified unstable disks give the exact disk-attachment pair $(Z, M_{0}) ≃ (W, M_{0})$ . With a supplied or constructed finite CW structure on $M_{0}$ , the supplier's Proof 9.1 constructs a CW model by cellular approximation and attachment comparison at each index stage, relative to $M_{0}$ ; its Proof 8.1 compares the exact attaching maps with value-ordered handle cores by homotopies below each critical value (Compactified unstable manifolds give the Morse--Smale CW decomposition).

[F3]

The coefficient supplier's Proof 1.1–3.1 computes the local degrees of the exact disk boundary projections from first-break sheets: they are the rigid trajectory signs with the precise ordered orientations, also for interval endpoints in degree one (Cellular boundary coefficients are the signed trajectory counts). The cellular boundary squares to zero (The cellular boundary squares to zero).

[F4]

Cellular chains of a CW pair with the constant local system compute ordinary relative singular homology (Cellular chains compute local homology, Homology and cohomology with local coefficients, Relative singular homology).

[F5]

The connecting map of a singular-homology triple is the pair connector followed by the relative quotient map. Its cycle formula sends $[c]$ to $[\partial c]$ , so it commutes with maps of triples (Long exact sequence of a triple in singular homology).

Proof

technique · direct, through the exact disk filtration and its CW comparison

1.1F1F3given

The finite critical set of [F1] gives the finite free relative Morse modules. Along a connecting orbit $f$ decreases, so its image lies in the compact interior slab between its endpoint values. Compactness of the boundary permits collars small enough that their values lie outside the entire interior critical-value range; that slab avoids these collars. The rigid counts are finite by the compact-slab argument of [F3]. This proves the confinement and finiteness assertions without using an arbitrary originally chosen collar.

2.1F2F3F5step 1.1construct

Write $Z_{k}$ for $M_{0}$ with all exact disks of index at most $k$ attached, and put $Z_{j} = M_{0}$ for $j < 0$ . Each disk attachment is a cofibration: its source boundary has a radial collar, which descends to the attached pair. Thus $Z_{k} / Z_{k - 1}$ is a wedge of $k$ -spheres, with the disjoint-point interpretation in degree zero, and $D_{k} : = H_{k} (Z_{k}, Z_{k - 1}; Λ)$ is free on the oriented critical disks. Define $d_{k}$ by the triple connector to $H_{k - 1} (Z_{k - 1}, Z_{k - 2}; Λ)$ , with $d_{0} = 0$ . By [F5], its coefficients are the boundary-map degrees after collapsing $M_{0}$ , disks of index at most $k - 2$ and the other $(k - 1)$ -disks. For $k \geq 2$ , the inverse image of the surviving open disk is exactly the first-break sheets ${γ} \times W^{u} (q)$ ; all exits are collapsed. The local degree computation of [F3] gives the trajectory count matrix in the stated critical rays. For $k = 1$ , the interval endpoints in $M_{0}$ vanish in the relative quotient and the remaining signed endpoints give the same count. This computes the exact filtered connector without declaring $Z$ a CW structure extending the base.

3.1F2F3F4F5step 2.1construct

Apply the stagewise construction of [F2], starting at $X_{- 1} = M_{0}$ . Transport each index- $k$ attaching map through the preceding homotopy inverse, cellularly approximate it into the ordinary $(k - 1)$ -skeleton and attach a $k$ -disk. The attachment comparison extends the preceding equivalence to $E_{k} : X_{k} \to Z_{k}$ , relative to $M_{0}$ and compatible with earlier stages. On each new disk it uses a boundary collar homotopy and preserves the oriented relative disk generator. Hence $E_{k}$ induces isomorphisms $H_{k} (X_{k}, X_{k - 1}) ≅ D_{k}$ by the pair sequences, and [F5] makes them commute with the triple connectors. Since $X_{k} = M_{0} \cup X^{(k)}$ , these are precisely the relative cellular modules and differential of [F4]. Step 2.1 therefore identifies that cellular complex with the relative Morse complex, and [F3] gives squared zero. The handle-core homotopies of [F2] identify this chosen CW model with a cellular model of the relative handle attachments; the model pair is equivalent to $(W, M_{0})$ .

4.1F4step 2.1step 3.1∎

Apply [F4] to the finite CW pair $(X, M_{0})$ with constant local system. Its cellular homology is its relative singular homology, which the pair equivalence of step 3.1 identifies with $H_{*} (W, M_{0}; Λ)$ . Combining this with the chain isomorphism of step 2.1 gives the displayed $H M_{*} (W, M_{0}; Λ) ≅ H_{*}^{c e l l} (W, M_{0}; Λ) ≅ H_{*} (W, M_{0}; Λ)$ . This also supplies the stated relative Morse homology notation.

Depends on

Used by

Relative Morse homology of a single-handle cobordism Example

Dependency tree · two levels

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