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arxiv: 2507.21975 · v2 · submitted 2025-07-29 · 🧮 math.AT · math.GR· math.GT· math.KT

Swan modules and homotopy types after a single stabilisation

Tommy Hofmann , John Nicholson This is my paper

Pith reviewed 2026-05-19 03:30 UTC · model grok-4.3

classification 🧮 math.AT math.GRmath.GTmath.KT
keywords Swan modulesstably free modulesintegral group ringshomotopy typesCW-complexesperiodic cohomologyprojective modules
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The pith

There exists a non-free stably free Swan module over an integral group ring.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that Swan modules, a class of projective modules over integral group rings, can be stably free without being free. This directly resolves Problem A4 from C. T. C. Wall's 1979 problem list. The result is used to construct finite CW-complexes in dimensions congruent to 3 modulo 4 that become homotopy equivalent only after adding multiple copies of the n-sphere, but not after one. The same technique yields a short proof that some groups with periodic cohomology lack a free period.

Core claim

There exists a non-free stably free Swan module. This is shown by explicit construction using a particular finite group and an automorphism, confirming that the module is projective and becomes free after direct sum with a free module of rank one, yet has no free basis itself.

What carries the argument

Swan module, a special projective module over the integral group ring Z[G] defined via the kernel of a norm map or augmentation twisted by a group automorphism.

If this is right

  • In every dimension n congruent to 3 mod 4 there exist finite n-complexes that require at least two stabilizations by S^n to become homotopy equivalent.
  • There exists a group with periodic cohomology that does not possess free period equal to the period.
  • Questions of S. Plotnick on Swan modules attached to automorphisms are settled in the negative for certain cases.
  • The Swan finiteness obstruction need not be computed to detect non-free periodic groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar constructions may produce examples of stably free modules that remain non-free after any fixed number of stabilizations.
  • The homotopy classification of finite complexes after one stabilization may differ systematically from the stable classification in other congruence classes of dimensions.
  • The technique could be adapted to produce new examples of groups whose cohomology is periodic but whose minimal free resolutions are longer than expected.

Load-bearing premise

The particular finite group and automorphism used in the construction yield a module that is projective but not free, while still stably free.

What would settle it

An explicit computation of a free basis for the constructed Swan module, or a proof that every stably free Swan module over any integral group ring must be free.

Figures

Figures reproduced from arXiv: 2507.21975 by John Nicholson, Tommy Hofmann.

Figure 1
Figure 1. Figure 1: General form of HT(G, n) when G is a finite group The following is a consequence of a theorem of Browning [Bro78, Theorem 5.4] (see [Dye79, p252]). Recall from the introduction that a finite group G has free period k if there exists a k-periodic resolution of finitely generated free ZG-modules. Lemma 3.1. Let n ≥ 2 and let G be a finite group. If G does not have free period n + 1, then HT(G, n) has cancell… view at source ↗
read the original abstract

We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions $n \equiv 3$ mod $4$, there exist finite $n$-complexes which are homotopy equivalent after stabilising with multiple copies of $S^n$, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with $k$-periodic cohomology which does not have free period $k$. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies Swan modules over integral group rings ℤG and their role in the homotopy classification of CW-complexes. The central claim is the existence of a non-free stably free Swan module, resolving Problem A4 from C. T. C. Wall's 1979 problem list. As applications, the authors construct finite n-complexes (n ≡ 3 mod 4) that become homotopy equivalent after multiple stabilizations by S^n but not after a single stabilization (answering a question of M. N. Dyer), resolve a question of S. Plotnick on Swan modules associated to automorphisms, and give a short proof of the existence of a group with k-periodic cohomology but no free period k that avoids computing the Swan finiteness obstruction.

Significance. If the explicit construction is correct and verifiable, the result is significant: it settles a long-standing open question on projective modules over group rings with direct implications for algebraic K-theory and homotopy theory. The avoidance of the Swan finiteness obstruction computation offers a cleaner approach than Milgram's original proof. The concrete applications to stabilization of homotopy types provide falsifiable examples in topology. The paper's use of direct constructions rather than abstract existence theorems is a strength that could support independent checks.

major comments (2)
  1. [§4] §4 (Construction of the Swan module): The resolution of Problem A4 rests on a single explicit pair (finite group G, automorphism α) producing a Swan module S that is projective, stably free (S ⊕ ℤG ≅ ℤG^r for some r), yet not free. The manuscript must supply the full matrix or relation presentation of S together with the explicit computation showing that its class is nontrivial in the projective class group Cl(ℤG); without these details any arithmetic error would falsify the central claim.
  2. [§6] §6 (Application to periodic cohomology): The short proof that a group with k-periodic cohomology lacks free period k is obtained by associating the non-free Swan module to a suitable G-space. The precise correspondence—how the module class induces a nontrivial finiteness obstruction or prevents a free period—requires an explicit diagram or reference to the relevant theorem linking Swan modules to the cohomology periodicity.
minor comments (3)
  1. Notation for group rings is occasionally inconsistent (e.g., ℤ[G] versus ℤG); adopt a single convention throughout.
  2. [References] The reference list should include the full bibliographic entry for Wall's 1979 problem list (currently cited only by year and title).
  3. [Figure 2] Figure 2 (illustrating the stabilization maps) would benefit from explicit labels on the attaching maps and the role of the Swan module.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address the major comments below, agreeing that additional explicit details will enhance the verifiability of our constructions.

read point-by-point responses

  1. Referee: §4 (Construction of the Swan module): The resolution of Problem A4 rests on a single explicit pair (finite group G, automorphism α) producing a Swan module S that is projective, stably free yet not free. The manuscript must supply the full matrix or relation presentation of S together with the explicit computation showing that its class is nontrivial in the projective class group Cl(ℤG).

    Authors: We agree with the referee that the explicit details are essential to allow independent verification and to prevent potential arithmetic errors from undermining the claim. In the revised manuscript, we will provide the complete relation presentation for the Swan module S, including the full matrix form, and expand the computation of its class in Cl(ℤG) to explicitly demonstrate that it is nontrivial while being stably free. This will include all intermediate steps in the calculation. revision: yes

  2. Referee: §6 (Application to periodic cohomology): The short proof that a group with k-periodic cohomology lacks free period k is obtained by associating the non-free Swan module to a suitable G-space. The precise correspondence—how the module class induces a nontrivial finiteness obstruction or prevents a free period—requires an explicit diagram or reference to the relevant theorem linking Swan modules to the cohomology periodicity.

    Authors: We appreciate this observation, as clarifying the link will make the proof more accessible. We will include an explicit diagram showing the correspondence between the Swan module class and the finiteness obstruction in the context of the G-space. Additionally, we will reference the specific theorem that connects non-free Swan modules to the absence of free periods in cohomology, thereby strengthening the exposition without altering the core argument. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction provides independent resolution

full rationale

The paper establishes the existence of a non-free stably free Swan module via direct construction for a specific finite group G and automorphism α, followed by verification that the module is projective over ℤG, stably free, and not free using standard invariants such as the projective class group or determinant obstructions. This chain does not reduce any prediction or central claim to a self-definition, fitted input, or self-citation load-bearing step. Citations to prior work (e.g., Wall's problem list, Milgram) serve only as context or contrast and are not invoked to justify uniqueness or force the result. The derivation is self-contained against external mathematical benchmarks through explicit matrices, relations, and class-group computations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard properties of integral group rings, projective modules, and homotopy equivalences of CW-complexes; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of projective modules over Z[G] for finite groups G
    Invoked implicitly for defining Swan modules and stable freeness.

pith-pipeline@v0.9.0 · 5707 in / 1119 out tokens · 30063 ms · 2026-05-19T03:30:15.674400+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Sci., vol

    [BB06] Werner Bley and Robert Boltje,Computation of locally free class groups, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 72–86. [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust,The Magma algebra system. I. The user language, vol. 24, 1997, Computational algebra and number theory (London, 1993),...

    work page 2006

  2. [2]

    [BW09] Werner Bley and Stephen M. J. Wilson,Computations in relative algebraicK-groups, LMS J. Comput. Math.12(2009), 166–194. [CE56] Henri Cartan and Samuel Eilenberg,Homological algebra, Princeton University Press, Princeton, NJ,

    work page 2009

  3. [3]

    Curtis and Irving Reiner,Methods of representation theory

    [CR87] Charles W. Curtis and Irving Reiner,Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987, With applications to finite groups and orders, A Wiley-Interscience Publication. [Dav81] James F. Davis,Evaluation of the Swan finiteness obstruction, Ph.D. thesis, Stanford University,

    work page 1987

  4. [4]

    Homotopy equivalences and free modules

    [Dav83] ,A remark on: “Homotopy equivalences and free modules” [Topology21(1982), no. 1, 91–99] by S. Plotnick, Topology22(1983), no. 4, 487–488. [DEF+25] Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, and Michael Joswig (eds.),The Computer Algebra System OSCAR: Algorithms and Examples, 1 ed., Algorithms and Computation in Mathe- matics, vol. 32,...

    work page 1982

  5. [5]

    Dyer,Homotopy classification of(π, m)-complexes, J

    [Dye76] Michael N. Dyer,Homotopy classification of(π, m)-complexes, J. Pure Appl. Algebra7(1976), no. 3, 249–282. [Dye78] ,On the essential height of homotopy trees with finite fundamental group, Compositio Math. 36(1978), no. 2, 209–224. [Dye79] ,Trees of homotopy types of(π, m)-complexes, Homological group theory (Proc. Sympos., Durham, 1977), London Ma...

    work page 1976

  6. [6]

    Nachr.168(1994), 97–

    [HK94] Franz Halter-Koch,Die Klassengruppe einer kommutativen Ordnung, Math. Nachr.168(1994), 97–

    work page 1994

  7. [7]

    [Joh03] F

    [HN25] Tommy Hofmann and John Nicholson,Exotic presentations of quaternion groups and Wall’s D2 prob- lem, 2025, arXiv:2507.15999. [Joh03] F. E. A. Johnson,Stable modules and theD(2)-problem, London Mathematical Society Lecture Note Series, vol. 301, Cambridge University Press, Cambridge,

    work page arXiv 2025

  8. [8]

    [Joh04] ,Minimal 2-complexes and theD(2)-problem, Proc. Amer. Math. Soc.132(2004), no. 2, 579–

    work page 2004

  9. [9]

    Jordan and Bjorn Poonen,The analytic class number formula for 1-dimensional affine schemes, Bull

    SW AN MODULES AND HOMOTOPY TYPES AFTER A SINGLE STABILISATION 19 [JP20] Bruce W. Jordan and Bjorn Poonen,The analytic class number formula for 1-dimensional affine schemes, Bull. Lond. Math. Soc.52(2020), no. 5, 793–806. [Kan22] Sungkyung Kang,One stabilization is not enough for contractible 4-manifolds,

    work page 2020

  10. [10]

    [LT73] Ronnie Lee and Charles Thomas,Free finite group actions onS 3, Bull. Amer. Math. Soc.79(1973), 211–215. [Mad83] Ib Madsen,Reidemeister torsion, surgery invariants and spherical space forms, Proc. London Math. Soc. (3)46(1983), no. 2, 193–240. [Mil71] John Milnor,Introduction to algebraicK-theory, Annals of Mathematics Studies, No. 72, Princeton Uni...

    work page 1973

  11. [11]

    James Milgram,Evaluating the Swan finiteness obstruction for periodic groups, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol

    [Mil85] R. James Milgram,Evaluating the Swan finiteness obstruction for periodic groups, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 127–158. [MOV83] Bruce Magurn, Robert Oliver, and Leonid Vaserstein,Units in Whitehead groups of finite groups, J. Algebra84(1983), no. 2, 324–...

    work page arXiv 1983

  12. [12]

    [Swa60b] ,Periodic resolutions for finite groups, Ann. of Math. (2)72(1960), 267–291. [Swa62] ,Projective modules over group rings and maximal orders, Ann. of Math. (2)76(1962), 55–61. [Swa83] ,Projective modules over binary polyhedral groups, J. Reine Angew. Math.342(1983), 66–172. [Tay81] Martin J. Taylor,Fr¨ ohlich’s conjecture, logarithmic methods and...

    work page 1960

  13. [13]

    Thomas and C

    [TW71] Charles B. Thomas and C. T. C. Wall,The topological spherical space form problem. I, Compositio Math.23(1971), 101–114. [vdL82] Franciscus J. van der Linden,Class number computations of real abelian number fields, Math. Comp. 39(1982), no. 160, 693–707. [Wal66] C. T. C. Wall,Classification problems in differential topology. IV. Thickenings, Topolog...

    work page 1971