TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Smooth cobordism is an equivalence relation

Statement

For every $n \geq 0$ , unoriented cobordism of closed smooth $n$ -manifolds (Unoriented smooth cobordism of closed manifolds) and oriented cobordism of closed oriented smooth $n$ -manifolds (Oriented smooth cobordism) are reflexive by cylinders, symmetric by dual bordisms (orientation-reversed in the oriented case), and transitive by collar gluing. Restricted to any set $S$ of such manifolds in either theory, cobordism is therefore an equivalence relation (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ) and partitions $S$ into classes $[M]_{S}$ . The collection of manifolds on arbitrary underlying sets is not itself a set; global bordism sets and the notation $[M]$ use the bounded models introduced in the subsequent bordism-group definition. No choice principle is used, and no claim about diffeomorphism classification is made.

Facts & Assumptions

Given: An integer $n \geq 0$ , closed smooth $n$ -manifolds, and in the oriented theory closed oriented smooth $n$ -manifolds. A closed manifold is compact without boundary (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Smooth manifolds and their smooth charts).

[F1]

For every closed smooth $n$ -manifold $M$ the cylinder $M \times [0, 1]$ with its product collars is a bordism from $M$ to $M$ , and in the oriented case it carries an orientation making it an oriented bordism from $(M, o)$ to $(M, o)$ ; hence $M$ is cobordant to itself in both theories (Cylinders give reflexivity of cobordism).

[F2]

Swapping the boundary parts and collar parametrisations of a bordism from $M_{0}$ to $M_{1}$ yields a bordism from $M_{1}$ to $M_{0}$ ; in the oriented case the dual with the opposite orientation on the same manifold is an oriented bordism from $M_{1}$ to $M_{0}$ (Reversing a cobordism gives symmetry).

[F3]

If $W_{1}$ is a bordism from $M_{0}$ to $M_{1}$ and $W_{2}$ a bordism from $M_{1}$ to $M_{2}$ , the collar gluing produces a bordism $W_{1} \cup_{M_{1}} W_{2}$ from $M_{0}$ to $M_{2}$ , and in the oriented case the orientations glue to an orientation making it an oriented bordism from $M_{0}$ to $M_{2}$ (Collar gluing and seam smoothing give transitivity).

[F4]

A binary relation on a set is an equivalence relation when it is reflexive, symmetric and transitive; for an equivalence relation the equivalence classes form a partition of the set and $[a]$ denotes the class of $a$ (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

1.1F1

(Reflexivity.) Let $M$ be a closed smooth $n$ -manifold, oriented by $o$ in the oriented theory. By [F1] the cylinder $M \times [0, 1]$ with its product collars is a bordism from $M$ to $M$ , and with the orientation $(- 1)^{n} (o \otimes d t)$ it is an oriented bordism from $(M, o)$ to $(M, o)$ . Thus $M$ is cobordant to itself in both theories.

1.2F2

(Symmetry.) If $M_{0}$ is cobordant to $M_{1}$ , choose a bordism $(W, θ_{0}, θ_{1})$ ; the dual data of [F2] give a bordism from $M_{1}$ to $M_{0}$ . If $(W, θ_{0}, θ_{1})$ is an oriented bordism from $M_{0}$ to $M_{1}$ , the same manifold with the opposite orientation and the swapped collars is an oriented bordism from $M_{1}$ to $M_{0}$ , with induced orientations $- M_{1}$ and $M_{0}$ on the incoming and outgoing faces.

1.3F3

(Transitivity.) If $M_{0}$ is cobordant to $M_{1}$ through $W_{1}$ and $M_{1}$ is cobordant to $M_{2}$ through $W_{2}$ , the collar gluing of [F3] gives a bordism from $M_{0}$ to $M_{2}$ ; if both bordisms are oriented, the glued orientation makes the result an oriented bordism from $M_{0}$ to $M_{2}$ .

2.1F4step 1.1step 1.2step 1.3∎

(Equivalence relation on a set.) Let $S$ be any set of closed smooth $n$ -manifolds, with supplied orientations in the oriented theory. Restricted to $S$ , cobordism is reflexive by step 1.1, symmetric by step 1.2 and transitive by step 1.3. Thus [F4] gives an equivalence relation on $S$ and the quotient classes $[M]_{S}$ . The three properties hold for arbitrary manifolds, but the set-theoretic quotient is asserted only on a set of models; the subsequent definition constructs global bordism sets this way. The argument uses only supplied collar data and no choice principle.

Depends on

Used by

Null-cobordant closed manifolds Definition
Unoriented and oriented bordism groups Definition
The pair of pants is a cobordism realizing addition of circles Example
Zero-dimensional bordism groups Proposition
Cartesian product makes bordism a graded ring Theorem
Disjoint union makes bordism classes abelian groups Theorem

Dependency tree · two levels

45 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

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
John Milnor and James Stasheff, Characteristic Classes (original pagination) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002) (standard reference, not scraped)