LemmaStatement: 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.

The homotopy effect of a surgery killing a relative class below the middle

Statement

Assume ${A C}_{ω}$ . Let $(f, b) : M^{m} \to X$ be a degree-one normal map with $M$ connected, let $2 p + 2 \leq m$ , and perform the $p$ -surgery on valid framed data representing $x \in π_{p + 1} (f)$ , with result $(f^{'}, b^{'})$ . For $p \geq 2$ , $π_{i} (f^{'}) ≅ π_{i} (f)$ for $1 \leq i \leq p$ , and $π_{p + 1} (f^{'})$ is the quotient of $π_{p + 1} (f)$ by the subgroup generated by the $π_{1} (M)$ -translates of $x$ . When $f$ induces a fundamental-group isomorphism, in particular when $f$ is $p$ -connected, this subgroup is the $Z [π_{1} (X)]$ -submodule generated by $x$ . For $p = 1$ , the corresponding quotient uses the normal subgroup generated by those translates in the possibly nonabelian relative group $π_{2} (f)$ ; when $f$ induces a fundamental-group isomorphism the relative group is abelian and has the usual $π_{1} (X)$ -module formulation. For $p = 0$ , the relative fundamental pointed set is the coset set for the source-image subgroup enlarged by the loop represented by the new core. No relative degree-zero group is asserted. The trace realizes these comparisons. In particular the usual $p$ -connected high-degree surgery step preserves lower relative groups and kills precisely the indicated generated class.

Facts & Assumptions

[F1]

The represented sphere has an actual framing compatible with its prescribed stable normal data and gives a normal trace extension over the finite CW target. Stable normal data supplies framings of the surgery spheres below the middle dimension

[F2]

A relative map cell gives the precise orbit-generated quotient, normal closure in degree two, and image-subgroup cosets in degree one. A relative map cell kills its class with the correct fundamental-group action

[F5]

Lifting criterion for maps from path-connected locally path-connected spaces. Lifting criterion for maps from path-connected locally path-connected spaces

[F6]

Long exact sequence of relative homotopy groups. Long exact sequence of relative homotopy groups

Proof

Given: Countable choice, valid normal-map surgery data, its trace $F : T \to X$ , and $2 p + 2 \leq m$ .

1.1givenconstructF1F2

Read the trace from $M$ : it is, relative to its incoming face, homotopy equivalent to $M$ with one $(p + 1)$ -cell attached, and the map on its core is the specified nullhomotopy representing $x$ . This is the handle-core description in the normal-map surgery supplier. Apply the local relative-map-cell lemma. For $p \geq 2$ it gives the orbit-generated quotient and the lower isomorphisms; for $p = 1$ it gives the normal/action closure in the relative degree-two group, and for $p = 0$ the enlarged-image coset description. The natural action is by $π_{1} (M)$ unless the source and target fundamental groups have been identified.

2.1step 1.1constructF1F2

Read the trace from $M^{'}$ : its single dual relative cell has dimension $q = m - p \geq p + 2$ . Apply [F2] to this dual cell and the same trace extension, with source the connected smooth $M^{'}$ . It gives $π_{i} (f^{'}) ≅ π_{i} (F)$ for $1 \leq i < q$ , including every $i \leq p + 1$ , with the degree-one comparison understood as a pointed-set comparison. The outgoing face is connected: its inclusion into the trace, whose incoming face is connected, has the homotopy type of an attachment of a cell of dimension $q \geq 2$ , which cannot join distinct components. The dual-cell argument of [F2] smooths only inside that open cell and requires no supplied CW structure on $M^{'}$ . This compares relative map groups; the absolute inclusion $M^{'} \to T$ need not be an isomorphism in degree $q - 1$ , and no such stronger assertion is used.

3.1step 1.1step 2.1constructF5F6∎

Combine the two trace computations. If $f_{*} : π_{1} (M) \to π_{1} (X)$ is an isomorphism, identify the orbit action with the target group-ring action. For $p = 1$ in this case all relative degree-two boundaries lift to the universal covers; both covering spaces are simply connected, and the relative group is a quotient of the abelian absolute second homotopy group by the pair sequence, so it is abelian. Otherwise retain the normal-closure formulation. The $p$ -connected application for $p \geq 2$ has the fundamental-group isomorphism automatically. These give every stated degree convention and the corrected quotient, with the exact borderline inequality $q \geq p + 2$ .

Depends on

Used by

Surgery below the middle dimension improves connectivity Proposition

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

Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, Oxford University Press 2002; complete electronic copy) (standard reference, not scraped)
Wolfgang Lück, A Basic Introduction to Surgery Theory (complete lecture notes, ICTP/Münster) (standard reference, not scraped)