LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Properties of bounded-domain degree

Statement

Let $d \geq 1$ , let $U \subset R^{d}$ be bounded and open, and let $f : \overline{U} \to R^{d}$ be continuous. For $y \notin f (\partial U)$ , put $K = f^{- 1} ({y}) \cap U$ and define $D (f, U, y)$ by

$f_{*} [U]_{K} = D (f, U, y) [R^{d}]_{{y}} in H_{d} (R^{d}, R^{d} ∖ {y}; Z),$

where $[U]_{K}$ is the unique relative orientation class restricting to the standard local generator on $K$ , and $[R^{d}]_{{y}}$ is the positive local generator. This integer is well-defined. It is nonzero only if $f^{- 1} (y) \cap U \neq \emptyset$ , and $D (id, U, y) = 1$ whenever $y \in U$ . If $U_{1}, U_{2} \subset U$ are disjoint bounded open sets, $y \notin f (\partial U_{i})$ for $i = 1, 2$ , and every point of $f^{- 1} (y) \cap U$ lies in $U_{1} \cup U_{2}$ , then

$D (f, U, y) = D (f, U_{1}, y) + D (f, U_{2}, y) .$

If $H : [0, 1] \times \overline{U} \to R^{d}$ and $y : [0, 1] \to R^{d}$ are continuous and $y (t) \notin H_{t} (\partial U)$ for every $t$ , then $D (H_{0}, U, y (0)) = D (H_{1}, U, y (1))$ .

Statement references

Compatible orientation classes over compact subsets

Chizat and Bach, Appendix C.3.1

Hatcher, Algebraic Topology, §2.2

Facts & Assumptions

Given: The oriented open domains, maps, targets, and relative-homology construction in the Statement.

[F1]

Every compact subset $K$ of a boundaryless $d$ -manifold has a unique relative orientation class in $H_{d} (M, M ∖ K; Z)$ restricting to the chosen local generator at every point; these classes commute with restriction to smaller compact supports. Compatible orientation classes over compact subsets

[F2]

A continuous map of pairs induces a homomorphism on relative singular homology, and induced maps respect composition. Functoriality of relative homology

[F3]

Relative homology is the homology of the quotient singular-chain complex; excision identifies relative groups after removing a closed set lying in the interior of the relative subspace. Relative singular homology Excision for singular homology

[F4]

At an interior point of $R^{d}$ , the local relative homology group $H_{d} (R^{d}, R^{d} ∖ {y}; Z)$ is free of rank one. Local homology detects manifold dimension, interior, and boundary

Proof

Given: The objects in the Statement and the relative-homology degree definition.

1.1F1F2F3

If $f^{- 1} (y) \cap U = \emptyset$ , then $K = \emptyset$ and $H_{d} (U, U ∖ K) = H_{d} (U, U) = 0$ , so $D (f, U, y) = 0$ . For the identity and $y \in U$ , $K = {y}$ and the induced map sends the chosen local orientation generator to itself, so $D (id, U, y) = 1$ .

2.1F1F2F3step 1.1

Under the additivity hypotheses, $K$ is the disjoint union of $K_{i} = K \cap U_{i}$ . The closed set $Z = U ∖ (U_{1} \cup U_{2})$ has closure in $U$ disjoint from $K$ : the compact set $K$ lies in the open set $U_{1} \cup U_{2}$ . Excision [F3] identifies $H_{d} (U, U ∖ K)$ with $H_{d} (U_{1} \cup U_{2}, (U_{1} \cup U_{2}) ∖ K)$ . Since $U_{1}, U_{2}$ are disjoint open sets, every singular simplex in their union lies in exactly one of them, so this relative group and its chains split as the direct sum of the two relative groups. The unique orientation class [F1] therefore corresponds to $[U_{1}]_{K_{1}} + [U_{2}]_{K_{2}}$ . The restrictions of $f$ commute with the inclusions of these pairs, so functoriality [F2] maps this sum to the sum of the two target classes. Their integer coefficients consequently satisfy $D (f, U, y) = D (f, U_{1}, y) + D (f, U_{2}, y)$ .

3.1F1F2F3constructstep 2.1

For moving-target invariance set $g_{t} (x) = H_{t} (x) - y (t)$ ; translation identifies the target generator with the standard generator at $0$ . The zero set $S = {(t, x) : g_{t} (x) = 0}$ is a closed subset of compact $[0, 1] \times \overline{U}$ and is disjoint from $[0, 1] \times \partial U$ . Thus its projection $K_{*} \subset U$ is compact. Every $g_{t}$ defines a map of the fixed pair $(U, U ∖ K_{*})$ into $(R^{d}, R^{d} ∖ {0})$ . The singular prism operator for the homotopy satisfies $\partial P + P \partial = (g_{1})_{#} - (g_{0})_{#}$ ; each prism over a simplex in $U ∖ K_{*}$ avoids $0$ , so the chain homotopy descends to relative chains. Hence $(g_{0})_{*} = (g_{1})_{*}$ on $H_{d} (U, U ∖ K_{*})$ . Restriction compatibility in [F1] sends $[U]_{K_{*}}$ to $[U]_{K_{t}}$ for the actual time- $t$ zero set, so these two induced integers are exactly $D (H_{0}, U, y (0))$ and $D (H_{1}, U, y (1))$ . They are equal, proving moving-target invariance.

4.1F1F2F4step 1.1step 2.1step 3.1∎

The target group is $Z$ by [F4]; the source class is unique by [F1]; and the induced map exists by [F2]. Thus the integer in the Statement is well-defined. Steps 1.1–3.1 establish all asserted properties.

Degree construction diagram

For a compact preimage $K$ , the definition applies the map of relative homology groups induced by $f$ ; the diagram records that map and its local orientation generators. The proof above establishes the properties independently of the diagram.

Depends on

Used by

Dependency tree · two levels

23 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

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Chapter 2 (standard reference, not scraped)