TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The relevant third oriented bordism group is finite

Statement

Assume AC, as in the smooth Casson–Gordon bordism convention. For every prime-power integer $d$ and $G = Z / d \oplus Z$ , the oriented bordism group $Ω_{3}^{S O} (K (G, 1))$ is finite.

Statement references

The oriented bordism group of maps

The group structure

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, Theorem 8.4.7, pp. 170–171

Facts & Assumptions

Given: The standing AC convention of the smooth Casson–Gordon bordism construction, a prime power $d$ , and $G = C_{d} \times Z$ .

[F1]

These are geometric smooth oriented bordism groups of maps. Oriented bordism groups of maps to a space, Oriented bordism classes over a space form an abelian group

[F2]

The complete relative-source cutting, low-dimensional coefficient fillings, skeletal exact-couple and periodic cyclic-resolution calculation gives $Ω_{3}^{S O} (K (C_{d} \times Z, 1)) = Z / d$ , for every positive integer $d$ , under AC. Low-degree geometric bordism over a finite cyclic group times the circle

Proof

1.1F1F2givenalgebra

Apply [F2] to the specified prime-power integer $d$ . Its explicit cyclic resolution works for all $d$ without assuming $d$ prime, and its geometric skeletal comparison uses the same smooth bordism convention [F1]. It gives the isomorphism $Ω_{3}^{S O} (K (G, 1)) ≅ Z / d$ .

2.1F2step 1.1algebra∎

The group $Z / d$ has $d$ elements and is finite, including every exponent in a prime power. This proves the assertion under the stated AC convention, rather than inferring the prime-power calculation from its prime-order special case.

Depends on

Used by

Prime-power characters admit a multiple bounding bordism Lemma

Dependency tree · two levels

4 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

Charles Livingston and Swatee Naik, Introduction to Knot Concordance (standard reference, not scraped)
A. J. Casson and C. McA. Gordon, Cobordism of Classical Knots (1986), with appendix by P. M. Gilmer (standard reference, not scraped)