TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The mod-two Morse differential squares to zero

Statement

Assume the Axiom of Choice. Let $(f, X)$ be Morse--Smale on a closed manifold. Then $\partial_{k - 1} \circ \partial_{k} = 0$ for every $k$ (The mod-two Morse differential). Equivalently, $(C M_{*} (f, X; Z / 2), \partial)$ is a chain complex over $Z / 2$ (Chain complex in an abelian category) and its homology is the mod-two Morse homology of $(f, X)$ .

Facts & Assumptions

Given: A Morse--Smale pair $(f, X)$ on a closed manifold, the Axiom of Choice, and an integer $k$ .

[A1]

The Axiom of Choice; the boundary parity lemma [F3] and the finiteness underlying [F1] are supplied through it, together with ${A C}_{ω}$ via the bridge (The Axiom of Choice, AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

On a basis element $p \in {Crit}_{k} (f)$ the differential is $\partial_{k} p = \sum_{q} n_{2} (p, q) q$ with $n_{2} (p, q) = # M (p, q) m o d 2$ , and each $M (p, q)$ with index drop one is finite; coefficients are computed in $Z / 2$ (The mod-two Morse differential, The mod-two Morse chain group, The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[F2]

For $λ (p) - λ (q) = 2$ the compactification $\overline{M} (p, q)$ is a compact $1$ -manifold with boundary whose boundary is the disjoint union of the products $M (p, r) \times M (r, q)$ over critical points $r$ of index $λ (p) - 1$ ; in index drop two every broken trajectory of length at least two is once-broken (The index-two compactification is a compact one-manifold with boundary, Breaking length is bounded by the index drop).

[F3]

Under ${A C}_{ω}$ the boundary of a compact smooth $1$ -manifold has even cardinality (Boundary of a compact 1-manifold has even cardinality).

[F4]

There are no Morse--Smale trajectories with nonpositive index drop, so a product $M (p, r) \times M (r, q)$ is empty whenever one of the two index drops is nonpositive (No Morse--Smale trajectories for nonpositive index drop, Morse--Smale pairs).

[F5]

A chain complex in an abelian category is a graded family of objects with degree $- 1$ endomorphisms squaring to zero; for $Z / 2$ -modules this is the stated complex over $Z / 2$ (Chain complex in an abelian category).

Proof

technique · direct

1.1F1F4algebra

Let $p \in {Crit}_{k} (f)$ and $q \in {Crit}_{k - 2} (f)$ . Expanding the definition, the coefficient of $q$ in $\partial_{k - 1} (\partial_{k} p)$ is the sum over $r \in {Crit}_{k - 1} (f)$ of the products $n_{2} (p, r) n_{2} (r, q)$ in $Z / 2$ ; both index drops here are equal to one, so both factors are parities of finite cardinalities by [F1], and the product of the two parities is the parity of the cardinality of the product $M (p, r) \times M (r, q)$ . Hence the coefficient equals the parity of the cardinality of the finite disjoint union $⨆_{r \in {Crit}_{k - 1} (f)} M (p, r) \times M (r, q)$ .

2.1F2step 1.1

Here $λ (p) - λ (q) = 2$ , so by [F2] the disjoint union of step 1.1 is exactly the boundary of the compact $1$ -manifold with boundary $\overline{M} (p, q)$ ; hence the coefficient of $q$ in $\partial^{2} p$ is $# \partial \overline{M} (p, q) m o d 2$ . For any critical point $q^{'}$ that is not of index $k - 2$ , the coefficient of $q^{'}$ in $\partial^{2} p$ is zero because $\partial_{k - 1}$ takes values in $C M_{k - 2} (f, X; Z / 2)$ , whose basis is ${Crit}_{k - 2} (f)$ .

3.1A1F3step 2.1

By [F3] the boundary of the compact $1$ -manifold $\overline{M} (p, q)$ has even cardinality, so the coefficient of every $q$ of index $k - 2$ in $\partial^{2} p$ vanishes in $Z / 2$ ; by step 2.1 all other coefficients vanish as well. Hence $\partial_{k - 1} \circ \partial_{k} = 0$ on basis elements, and therefore on all of $C M_{k} (f, X; Z / 2)$ by linearity.

4.1F5step 3.1∎

Since this holds for every $k$ , the pair $(C M_{*} (f, X; Z / 2), \partial)$ satisfies the defining condition of a chain complex in the abelian category of $Z / 2$ -modules by [F5]; its homology is the mod-two Morse homology of $(f, X)$ by definition.

Depends on

Used by

Morse homology of a Morse--Smale pair Definition

Dependency tree · two levels

55 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)