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Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7

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Pith's one-line read The paper computes three Spin-G bordism groups through dimension seven.

desk verdict New Spin-Sp(4) computation, but the proof's central Adams calculation is absent — blank E2 tables and an invalid truncation argument leave the theorem uncheckable. read the letter →

arxiv 2504.15014 v1 pith:NRJKGIQZ submitted 2025-04-21 math.AT

Naoki Kuroda This is my paper · ORCID
classification math.AT MSC 57R90
keywords Spin-GbordismgroupsAdamsspectralsequenceSp(4)SU(8)Spin(16)Thomspectramanifoldgenerators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper determines the Spin-G bordism groups in dimensions up to 7 for three symmetry groups that arise as maximal compact subgroups of hidden symmetries in supergravity: $Sp(4)$, $SU(8)$, and $Spin(16)$. The main result is a complete list of groups, with explicit manifolds ($HP^1$, $CP^2$, $SU(3)/SO(3)$, and products with circles) generating each nonzero summand. These groups classify the possible global anomalies carried by spin manifolds with those twistings, so the computation is a step toward deciding which string-theory backgrounds are anomaly-free in low dimensions.

What carries the argument

The load-bearing device is the Adams spectral sequence for connective $ko$-homology of the Madsen-Tillmann spectra $MT(\mathrm{Spin}\text{-}G)$, arrived at through the Anderson-Brown-Peterson splitting. Because the relevant Thom spectra are not vector-bundle Thom spectra, the paper invokes a cited theorem that lets the standard $M\mathrm{ko}_{f_0,x_2}$ model still be used. The computation then rests on $A_1$-module presentations ($A_1$ is the subalgebra of the Steenrod algebra generated by $Sq^1$ and $Sq^2$) for $H^*_{ko}(M\mathrm{ko}_{f_0,x_2})$ up to degree 8, obtained from the $\mathbb{Z}_2$-cohomology rings of the classifying spaces; on the resulting $\mathrm{Ext}_{A_1}(-, \mathbb{Z}_2)$ $E_2$-terms; and on the claim that all Adams differentials vanish because they commute with the $h_0$-action. The Leray-Serre spectral sequence supplies the cohomology rings, and a theorem on Eilenberg-Mac Lane spectra is used to split off extensions at the end.

What would settle it

Recompute the $E_2$-term $\mathrm{Ext}_{A_1}(-, \mathbb{Z}_2)$ for the three $A_1$-modules in Figures 1-3 in total degrees up to 7 and check each possible differential $d_r$ against the $h_0$-action; since the paper's $E_2$-tables are blank, this independent calculation would settle Theorem 1.1. A second check would be to compute the three homomorphisms in (18) in degree 6 and verify that $CP^1\times CP^1\times CP^1$ indeed evaluates to $-2$ on the integral class $z_6$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for $k=0,\dots,7$, the three groups are $\Omega_k^{\mathrm{Spin}\text{-}\mathrm{Sp}(4)} = (\mathbb{Z},0,0,0,\mathbb{Z}\oplus\mathbb{Z},\mathbb{Z}_2\oplus\mathbb{Z}_2,\mathbb{Z}_2\oplus\mathbb{Z}_2,0)$, $\Omega_k^{\mathrm{Spin}\text{-}\mathrm{SU}(8)} = (\mathbb{Z},0,0,0,\mathbb{Z}\oplus\mathbb{Z},\mathbb{Z}_2,\mathbb{Z}\oplus\mathbb{Z}_2,0)$, and $\Omega_k^{\mathrm{Spin}\text{-}\mathrm{Spin}(16)} = (\mathbb{Z},0,0,0,\mathbb{Z}\oplus\mathbb{Z},\mathbb{Z}_2,\mathbb{Z}_2,0)$. The free part in degree 4 is generated by $HP^1$ and $CP^2$ for all three; in degree 5, $SU(3)/SO(3)$ generates a $\mathbb{Z}_2$ in all three cases and $HP^1\times S^1$ gives a second $\mathbb{Z}_2$ for $\mathrm{Spin}\text{-}\mathrm{Sp}(4)$; in degree 6, $HP^1\times S^1\times S^1$ and $CP^2\times CP^1$ generate $\mathbb{Z}_2\oplus\mathbb{Z}_2$ for $\mathrm{Spin}\text{-}\mathrm{Sp}(4)$, while $\mathrm{Spin}\text{-}\mathrm{SU}(8)$ has a free summand generated by $CP^1\times CP^1\times CP^1$ plus a $\mathbb{Z}_2$ generated by $CP^2\times CP^1$, and $\mathrm{Spin}\text{-}\mathrm{Spin}(16)$ has only the $\mathbb{Z}_2$ generated by $CP^2\times CP^1$. The paper also proves that the three theories are isomorphic in degrees up to 4 and have no odd-prime torsion.

Load-bearing premise

The computation rests on the unshown algebraic claim that the $A_1$-module structures in Figures 1-3 are exactly correct up to degree 8 and that every Adams differential vanishes; if either part fails, the listed groups change.

Editorial extensions

If this is right

  • In dimensions 0-7 the three Spin-G bordism groups are completely known, so any Spin-Sp(4), Spin-SU(8), or Spin-Spin(16) manifold in these dimensions is either bordant to one of the listed generators or to a boundary.
  • Because the three theories are isomorphic through degree 4, the first genuinely different torsion appears in degree 5.
  • The explicit generators give concrete representatives: $HP^1$ and $CP^2$ generate the degree-4 free part, $SU(3)/SO(3)$ generates a $\mathbb{Z}_2$ in degree 5 in every case, and circle products generate the extra torsion for $\mathrm{Spin}\text{-}\mathrm{Sp}(4)$.
  • The vanishing in degrees 1, 2, 3, and 7 means no nontrivial bordism obstructions exist for these twistings in those dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Pushing the same computation beyond degree 7 would require including the 13-dimensional relation noted in Remark 3.6, so the higher groups are not determined by this paper.
  • The same fiberwise arguments may give a uniform computation for $\mathrm{Spin}\text{-}\mathrm{Sp}(2n)$, $\mathrm{Spin}\text{-}\mathrm{SU}(2n)$, and $\mathrm{Spin}\text{-}\mathrm{Spin}(4n)$ for larger $n$, where the cohomology rings are known but the $A_1$-module complexity grows.
  • If these bordism classes are realized in string theory, the nonzero $\mathbb{Z}_2$ classes in degrees 5 and 6 would imply that certain five- and six-dimensional backgrounds cannot be made anomaly-free by adding local counterterms alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper claims to compute the Spin-G bordism groups Omega_k^{Spin-Sp(4)}, Omega_k^{Spin-SU(8)}, and Omega_k^{Spin-Spin(16)} for k=0,...,7, together with explicit manifold generators. The method combines the Spin-G bordism isomorphism with Madsen-Tillmann spectra, the Adams spectral sequence for connective ko-homology, and the Debray-Yu theorem for non-vector-bundle Thom spectra. The main results are stated in Theorem 1.1 and proved in Theorem 3.7, with additional geometric arguments in Section 4 identifying generators such as HP^1, CP^2, SU(3)/SO(3), and products of CP^1 and CP^2 factors.

Significance. If the computations were correct and fully justified, the paper would be a useful addition to the program of computing physics-motivated Spin-G bordism groups, and the explicit generator descriptions are valuable. The paper also demonstrates awareness of the modern non-vector-bundle Thom spectrum technology. However, the central Adams spectral sequence computation is not actually presented: the E2-term tables are blank, the truncation argument for the A1-modules is not sufficient, and the differential collapse is asserted rather than proved. As written, the main theorem is unsupported, so the significance cannot be assessed.

major comments (5)
  1. [Section 3.4, Theorem 3.7 and Tables 5-7] The Adams E2-terms are not displayed: Tables 5, 6, and 7 are blank. The proof merely says that 'using the concrete example from Section 4 of [3]' the desired E2-term is obtained. This is the central computation of the paper, and without the actual Ext groups, the reader cannot verify the claimed bordism groups.
  2. [Section 3.4, A1-module replacement] The proof replaces Mko f0,x2 by an A1-module that is required only to agree with the true module in degrees at most 7 as a Z2-vector space and to have matching A1-action in degrees at most 8. This is insufficient to determine Ext_{A1}^{s,t}(M,Z2) for t-s at most 7: for a fixed small t-s, t can be as large as s+7, and entries with s>0 can depend on module data in degrees above 8. The paper does not explain why the truncated module determines the relevant Ext groups.
  3. [Section 3.4, differential collapse] The proof asserts that all differentials vanish because they commute with the Ext_{A1}(Z2,Z2)-action and multiplication by h0. From d_r(h0 x)=h0 d_r(x) one cannot conclude d_r=0 without further information about the h0-towers in the relevant E2-range. The paper provides no E2 chart and no analysis of h0-towers, so the collapse is not justified.
  4. [Section 3.1, Proposition 3.1] The proof contains a concrete false statement: it claims pi_0(Sp(4)) is isomorphic to Z (and similarly for SU(8) and Spin(16)), but pi_0 of a connected Lie group is trivial. In addition, the proof that f and g are 4-equivalences lists pi_4(SU(8)) and pi_4(Spin(16)) but omits pi_4(Sp(4)) and the induced map on pi_4; the cohomological argument about H^4 is therefore not a complete proof of the claimed 4-equivalence.
  5. [Sections 2.3 and 3.4, use of Debray-Yu Theorem] The paper relies on [6, Theorem 2.28(3)] for non-vector-bundle Thom spectra, but does not verify that the hypotheses of that theorem hold for the three cases (G,H) = (Sp(4),Sp(4)/Z2), (SU(8),SU(8)/Z2), and (Spin(16),Ss(16)). This verification is load-bearing because the entire computational framework depends on it.
minor comments (6)
  1. [Section 3.1] The word 'Hurewitz' should be 'Hurewicz'.
  2. [Section 3.2, Proposition 3.2] The statement that Omega_*^{Spin-Spin(16)} has no odd torsion because H^*(BSpin(16);Z) has none does not immediately follow; H^*(BSs(16);Z) is a quotient of a free ring and could in principle have torsion. The argument needs a justification for the quotient.
  3. [Section 4.2] The text refers to a 'blue Z2' and a 'purple Z2' in Table 5, but Table 5 is blank, so these color-coded references are unverifiable.
  4. [Figures 1-3] The notation U in the figures is never defined; it appears to denote the free A1-module on one generator, but this should be stated explicitly.
  5. [Reference [10]] Reference [10] is a URL to nLab/MathOverflow; this should be replaced by a proper citation to the literature.
  6. [Remark 3.6] The statement that the next relation lies in dimension 13 and is not addressed is too vague; either provide the computation or remove the remark.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central computations are new applications of external theorems and no step reduces to its own input.

full rationale

The paper's derivation chain is not circular. The core theorem 3.7 is asserted via an Adams spectral sequence computation that uses Debray-Yu [6, Theorem 2.28(3)] and [7], both by external authors; these sources do not already contain the Spin-Sp(4) or Spin-Spin(16) bordism groups, so invoking them is independent support rather than a self-citation chain. The A1-modules in Figures 1-3 are presented as replacements that agree with the true module only in low degrees; even if this truncation is insufficient to determine Ext in the desired range (the E2 tables 5-7 are blank and differential collapse is asserted without a chart), the modules are not defined from the claimed answer, so the argument is incomplete but not circular. Lemma 3.5 uses the 9-dimensional relation pulled back from the independently known BSs(16) cohomology ring, not the formula Sq^2(y4)=x2y4 it proves; the later substitution that simplifies the relation to the form stated in Theorem 3.3 is the reverse, non-circular direction. The SU(8) case is explicitly attributed to prior work [7]; repeating a known result with citation is a novelty concern, not a circularity concern. No fitted parameters are renamed as predictions and no uniqueness theorem is imported from the present author's prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no numerically fitted parameters and no new physical entities. It relies on external spectral sequence technology, cited cohomology rings, and several unshown algebraic computations, which are the main burden on the central claim.

assumptions (5)
  • domain assumption The Debray-Yu spectral sequence method [6, Theorem 2.28(3)] applies to the three Spin-G Thom spectra even though no vector bundle V over BH with w2(V)=zeta exists.
    Section 2.3: the author notes these are examples without such V and says the computation remains valid by [6, Theorem (2)]. This external theorem is load-bearing.
  • standard math The cohomology rings of BSp(4), BSU(8), BSpin(16), and B2Z2 in degrees up to 11 are as stated in Proposition 3.4, taken from Mimura-Toda [15].
    The proof of Theorem 3.3 uses these rings as input; they are cited from standard references.
  • ad hoc to paper The Steenrod module structure and the 9-dimensional relation in the cohomology rings of B(Sp(4)/Z2), B(SU(8)/Z2), and BSs(16) are correctly computed in Theorem 3.3.
    This is the paper's own computation; an error here propagates to the A1-modules and the final bordism groups.
  • ad hoc to paper The A1-modules displayed in Figures 1-3 coincide with M ko f0,x2 up to degree 8.
    Section 3.4: the author states they replace M ko f0,x2 with an A1-module satisfying certain conditions, but the correctness of this replacement is asserted rather than proved.
  • ad hoc to paper The E2-term of the Adams spectral sequence is as claimed and all differentials vanish because they commute with the Ext_{A1}(Z2,Z2)-action and h0.
    Section 3.4: the E2-term is claimed but not displayed, and differential vanishing is justified in a single sentence.

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Cite this review

Pith. "Pith review of Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7." pith.science (2026). https://pith.science/paper/NRJKGIQZ

@misc{pith2026250415014,
  author       = {Pith},
  title        = {Pith review of: Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRJKGIQZ}},
  note         = {Machine review of arXiv:2504.15014}
}
abstract

We investigate the structure of Spin-$G$ bordism groups, focusing on the interplay between Spin and additional twisting symmetries such as $Sp(4)$, $SU(8)$ and $Spin(16)$. Using techniques from spectral sequences, obstruction theory, and cohomology operations, we compute explicit generators for the Spin-$G$ bordism groups in dimensions up to 7.

Figures

Figures reproduced from arXiv: 2504.15014 by the authors.

Figure 1
Figure 1. The case of Spin-Sp(4) 0 1 2 3 4 5 6 7 8 9 10 11 12 U Ux2 Ux3 U(x5 + x2x3) Ux2 2 U(x 2 3 + x 3 2 ) Ux2 2x3 Ux2x3 Ux2 3 Ux2x5 U(x3x5 + x2x 2 3 ) Ux2x 2 3 Uy4 U(y6 + x2y4) Ux2y4 Ux3y4 Ux2y6 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The case of Spin-SU(8) 0 1 2 3 4 5 6 7 8 9 10 11 12 U Ux2 Ux3 U(x5 + x2x3) Ux2 2 U(x 2 3 + x 3 2 ) Ux2 2x3 Ux2x3 Ux2 3 Ux2x5 U(x3x5 + x2x 2 3 ) Ux2x 2 3 Uy4 U(y6 + x2y4) U(y7 + x3y4) Ux2y4 Ux3y4 Ux2y6 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The case of Spin-Spin(16) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Reference graph

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