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Cobordism Utopia: U-Dualities, Bordisms, and the Swampland

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper computes every U-duality bordism group for 8d maximal supergravity and matches each generator to a known string, M-, or F-theory defect.

desk verdict The D=8 bordism computations are the real deal; the spin-lift caveat, though clearly flagged, makes the physical prime-2 predictions provisional. read the letter →

arxiv 2505.15885 v2 pith:FD72IXCF submitted 2025-05-21 hep-th math.AT

Noah Braeger , Arun Debray , Markus Dierigl , Jonathan J. Heckman , Miguel Montero This is my paper · ORCID
classification hep-thmath.AT MSC 55N2255T15
keywords U-dualityspinbordismSwamplandCobordismConjecturemaximalsupergravitystringtheorydefectsnon-geometricbackgroundsAdamsspectralsequenceexceptionalfield
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

The paper tests a sharp prediction of the Swampland Cobordism Conjecture: any global symmetry in a quantum theory of gravity must be broken by some dynamical defect, and in a UV-complete theory the bordism group classifying the corresponding charges must be trivialized. Working with the U-duality groups of maximally supersymmetric supergravity, it computes the spin bordism groups $\Omega_k^{\mathrm{Spin}}(BG_U)$ whose nontrivial classes are exactly the charges that would otherwise be conserved. For every maximal supergravity theory in dimensions 3 through 10 it determines the first bordism group, and for the 8d theory with $G_U = \mathrm{SL}(2,\mathbb{Z}) \times \mathrm{SL}(3,\mathbb{Z})$ it gives the full list for $k = 1, \dots, 7$; for example $\Omega_1 = \mathbb{Z}_2 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_4$ and $\Omega_3 = \mathbb{Z}_3^3 \oplus \mathbb{Z}_2^3 \oplus \mathbb{Z}_8^3$. The central result is that every generator in the computed range can be matched to a known string, M-theory, or F-theory background, including non-geometric twists, strings, and instantons, so that no fundamentally new defect is needed. If correct, this is evidence that the low-energy supergravity spectrum is completed by known ingredients of string theory and that the cobordism conjecture holds for U-duality charges in these dimensions.

What carries the argument

The central object is the spin bordism group $\Omega_k^{\mathrm{Spin}}(BG_U)$, whose elements are closed $k$-manifolds equipped with a principal $G_U$-bundle, modulo bordisms; a nonzero class predicts a symmetry-breaking defect of codimension $k+1$. The computation is organized prime by prime: at $p=3$, stable splittings reduce $B\mathrm{SL}(3,\mathbb{Z})$ to two copies of $BD_6$ and $B\mathrm{SL}(2,\mathbb{Z})$ to $B\mathbb{Z}_3$; at $p=2$, Minami's stable splitting reduces $B\mathrm{SL}(3,\mathbb{Z})$ to $B\mathrm{SL}(3,\mathbb{F}_2) \vee B\mathrm{SL}(3,\mathbb{F}_2) \vee L(2)$, and the Adams spectral sequence over the subalgebra $A(1)$ computes connective real $K$-homology, which agrees with spin bordism in degrees $\leq 7$ by the Anderson-Brown-Peterson isomorphism. A K\"unneth and module structure, induced by the Pontrjagin product on cyclic subgroups, organizes the generators and determines the multiplicative relations. The perfection of $\mathrm{SL}(n,\mathbb{Z})$ for $n \geq 3$ is what makes the reduced first bordism group vanish for $D \leq 7$, so no additional codimension-two duality defects are needed there.

What would settle it

Recompute the bordism groups with the metaplectic lift of $\mathrm{SL}(2,\mathbb{Z})$, replacing the $\mathbb{Z}_4$ class by the Spin-$\mathbb{Z}_8$ structure: if $\Omega_3^{\mathrm{Spin}}(B\mathbb{Z}_4) = \mathbb{Z}_2 \oplus \mathbb{Z}_8$ becomes $\Omega_3^{\mathrm{Spin}-\mathbb{Z}_8}(pt) = \mathbb{Z}_2$, then the proposed $L^3_{4,\gamma_4}$ background is not the physical symmetry-breaking defect. Alternatively, check directly in exceptional field theory whether the section constraint admits the singular fiber claimed for the non-geometric generator $(\gamma_3, \Gamma_3^{(2)})$; if no such solution exists, that generator lacks a UV completion.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that U-duality respects the Swampland Cobordism Conjecture in a strong sense. For the 8d maximally supersymmetric theory the bordism groups $\Omega_k^{\mathrm{Spin}}(B(\mathrm{SL}(2,\mathbb{Z}) \times \mathrm{SL}(3,\mathbb{Z})))$ are completely determined for $1 \leq k \leq 7$, including explicit generators: lens spaces with specified monodromies, products and fibrations of them, K3, and the spaces $W_4$, $A/A'$, $Q_4^5$, and $W_6$. Each generator is exhibited as the boundary of a string, M-, or F-theory configuration whose singular core is the symmetry-breaking defect; where no duality frame geometrizes the full bundle, the defect is declared non-geometric. For dimensions $D \leq 7$ the paper shows the reduced first bordism group vanishes, with $\Omega_1^{\mathrm{Spin}}(BG_U) \cong \mathbb{Z}_2$ coming only from the spin circle, because the duality groups $\mathrm{SL}(5,\mathbb{Z})$, $\mathrm{SO}(5,5,\mathbb{Z})$, $E_{6(6)}(\mathbb{Z})$, $E_{7(7)}(\mathbb{Z})$, and $E_{8(8)}(\mathbb{Z})$ are perfect, so the corresponding codimension-two charges are already trivialized by smooth gravitational solitons. The paper's summary lesson is a completeness statement: string, M-, and F-theory already contain the objects needed to break every U-duality global symmetry in the computed range, and unlike the type IIB case, no new object such as the R7-brane is required.

Load-bearing premise

The load-bearing premise is that the bordism group $\Omega_k^{\mathrm{Spin}}(BG_U)$ with the bosonic U-duality group $G_U$ is the right physical invariant; the true duality group acting on fermions, its Spin or Pin$^+$ lift, is not known, and choosing a different lift changes some generators and the corresponding defects.

Editorial extensions

If this is right

  • Every nontrivial U-duality bordism class in the 8d theory for $1 \leq k \leq 7$ is realized by a known string, M-, or F-theory background, so no fundamentally new object is required to satisfy the Swampland Cobordism Conjecture for these U-duality charges.
  • For maximal supergravities in dimensions $D \leq 7$, the reduced first bordism group vanishes because the U-duality groups are perfect, meaning the corresponding codimension-two duality-bundle charges are trivialized by smooth gravitational solitons rather than by new localized defects.
  • Some of the required symmetry-breaking objects have no purely geometric description in any single duality frame; non-geometric twists, non-geometric strings, and non-geometric instantons are part of the spectrum of defects predicted by the computation.
  • The nontrivial computed groups imply a large set of conserved topological charges in the low-energy supergravity, each of which must be broken by a defect in any complete quantum-gravity theory, exactly as the cobordism conjecture demands.

Reading between the lines

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

  • If the correct fermionic lift of the U-duality group is a nontrivial extension, some of the listed 2-torsion generators and their supersymmetric interpretations would need to be revised; the paper's own example is that $\Omega_3^{\mathrm{Spin}}(B\mathbb{Z}_4) = \mathbb{Z}_2 \oplus \mathbb{Z}_8$ changes to $\Omega_3^{\mathrm{Spin}-\mathbb{Z}_8}(pt) = \mathbb{Z}_2$ when the $\mathbb{Z}_4$ action is
  • A natural next computation is the same bordism census for $E_{6(6)}(\mathbb{Z})$, $E_{7(7)}(\mathbb{Z})$, and $E_{8(8)}(\mathbb{Z})$, or for $k \geq 8$ in eight dimensions; the paper's stable-splitting strategy would need new mathematical input, but the result would likely expose more non-geometric defects and could test whether the completeness statement persists.
  • The observation that $\mathrm{SL}(3,\mathbb{Q})$-conjugate but $\mathrm{SL}(3,\mathbb{Z})$-inequivalent bundles give identical low-energy physics but different high-energy spectra suggests a general construction of discrete $\theta$-angles: any such pair should differ by a topological coupling at low energies, which could be checked by computing the partition-function ratio $T(X)$ for the $\Gamma_
  • If any generator in the computed list fails to survive a calculation with the true Spin or Pin$^+$ lift, that class would become a genuine prediction of a new object, analogous to the R7-brane, rather than a match to a known background.
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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

2 major / 4 minor

Summary. This paper computes spin bordism groups associated with the bosonic U-duality groups of maximally supersymmetric supergravity theories. For all dimensions 3≤D≤10 it computes Ω_1^Spin(BG_U), and for D=8, where G_U=SL(2,Z)×SL(3,Z), it computes Ω_k^Spin(BG_U) for 1≤k≤7. The mathematical work uses stable splittings to reduce BSL(2,Z)×BSL(3,Z) to classifying spaces of finite groups at primes 2 and 3, combined with Adams and Atiyah-Hirzebruch spectral sequences and Künneth/module structures. The authors give explicit manifold representatives for all bordism generators (lens spaces, products, W4, Q5_4, etc.) and propose string/M/F-theory backgrounds that would break the corresponding global symmetries. A key qualitative conclusion is that, unlike the ten-dimensional IIB case, no fundamentally new object such as the R7-brane is needed for D=8.

Significance. If the physical bordism group is indeed Ω_k^Spin(BG_U) with the bosonic group G_U, this is a substantial step forward for the Swampland Cobordism Conjecture: it provides a complete catalogue of U-duality bordism classes in eight dimensions and a physically interpretable set of defect backgrounds. The computations appear original, and the use of stable splittings, spectral sequences, and ring/module structures is technically impressive. The paper is also unusually honest about its main limitation: the spin/Pin lift of the U-duality group is not known, and the authors demonstrate explicitly that prime-2 bordism groups can change under such a lift. This makes the physical interpretation of the prime-2 summands conditional rather than definitive, but the paper still lays the necessary groundwork and the computational methods are likely to be reusable once the lift is understood.

major comments (2)
  1. [§1.2.6, §1.2.1, abstract] Section 1.2.6 (Eq. 1.10) explicitly shows that the prime-2 bordism group of a cyclic duality subgroup changes when its spin lift is included: Ω_3^Spin(BZ4)=Z2⊕Z8 whereas Ω_3^{Spin-Z8}(pt)=Z2. The same section states that the spin/Pin lift of the U-duality group G_U is unknown. Since Table 1 and the generator tables contain many Z2, Z4, Z8, Z16, and Z32 summands, the physical predictions for the corresponding defects depend on an uncomputed fermionic action of the duality group. The abstract and §1.2.1 nevertheless claim a 'full list' of bordism groups and corresponding string/M/F-theory backgrounds, which overstates what is established. I recommend either computing the lifted bordism groups for the relevant cyclic subgroups at prime 2, or substantially qualifying the abstract and the completeness claims to state that the computations are for the bosonic duality group and that the physical defect list is provisional until the spin/Pin lift is determined.
  2. [§6.2, after Eq. (6.14)] The argument that the second T^2 generator with monodromies M^(2)_1 and M^(2)_2 cannot be bounded by a smooth gravitational soliton relies on the assertion 'One can check numerically that this is indeed the case.' This is a load-bearing step for the physical interpretation, because it justifies introducing a new codimension-two defect for the second S4 embedding. The explicit matrix equations, or the numerical code used, should be included so that the check is reproducible and the reader can verify the nonexistence of the commutator decomposition.
minor comments (4)
  1. [Footnote 32] The 'brodism' footnote in Section 6.7 is humorous in tone and out of place in a formal research paper; I recommend deleting it or replacing it with a plain definition of the term 'bordism' as used in that paragraph.
  2. [§8.3, Proposition 8.35] The statement and proof of Proposition 8.35 are compressed for readers who do not work with stable splittings daily; adding a short concrete example or an explicit sentence explaining how the two copies of the BS4 second-component generators (through ρ3 and ρ4) enter the final generating set would improve readability.
  3. [Table 3 and §6.7] The generators L^7_{4,Γ^(i)_4} and their defect interpretations are listed in Table 3 without a caveat, while §6.7 explicitly notes that these configurations are affected by the spin-lift problem; this caveat should be repeated in the table caption or in a footnote so that the table is not read as an unconditional physical prediction.
  4. [§1.2.1, §6.7] The statement that 'we do not need to introduce any fundamentally new object' is stronger than what the paper actually establishes, since supersymmetry, stability, and worldvolume dynamics are explicitly deferred; I suggest replacing it with wording such as 'no new object beyond known string/M/F-theory ingredients has been identified'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bordism computations are self-contained or rest on external mathematical results, and the defect identifications are explicit constructions rather than inputs.

full rationale

The paper's derivation chain is not circular. The central computations are (a) the reduction of first spin bordism to Abelianization (Eq. 7.1), (b) the stable splittings of BSL(2,Z) and BSL(3,Z) (Propositions 8.9, 8.14, 8.16, 8.21) attributed to Soule, Minami, Brown, and Mitchell-Priddy, (c) the Atiyah-Hirzebruch and Adams spectral sequence computations for BZ3, BD6, BZ4, BS4, and their smash products, and (d) the construction of explicit lens-space and torus-orbifold representatives whose cones give string, M-, or F-theory defects. No parameter is fitted to data and then renamed a prediction; the proposed symmetry-breaking defect for each class is a concrete quotient, e.g., (T^2 x C^2)/Z_k, whose boundary is the stated bordism generator by direct construction. The self-citations to the IIBordia paper [28] supply building-block bordism groups such as Omega^Spin_*(BZ4) and Omega^Spin_*(BD6); these are parameter-free published mathematical computations and are not the target result of this paper, so they do not make the derivation circular. The genuinely new SL(3,Z) and product computations are independent of those inputs, and the stable splittings come from external mathematics. The paper also explicitly acknowledges in Section 1.2.6 and Section 6.7 that the Spin/Pin lift of the U-duality group is unknown and that prime-2 bordism classes may change under the correct fermionic lift; this is a stated correctness caveat, not a circular reduction, because the bosonic bordism groups are computed from their stated definitions. For these reasons no circular step is identified and the appropriate score is 0.

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

The central computations are not fitted to data. They rest on external mathematics (Soule, Minami, Brown), prior spin-bordism computations for finite groups, and the physical assumption that the Swampland Cobordism Conjecture converts non-trivial bordism classes into required defects. The paper's own caveat about the unknown spin-lift of G_U is the most serious unproven input.

assumptions (5)
  • domain assumption Swampland Cobordism Conjecture: quantum gravity bordism group Omega_QG_k = 0 for all k, so non-trivial Omega_k^Spin(BG_U) classes must be broken by adding defects or singular objects.
    Used at the start (Section 1, Eq. 1.1) to convert bordism classes into predictions of symmetry-breaking defects; not proven in this paper.
  • domain assumption The discrete U-duality groups are G_D^U = SL(ell,Z) knit-product SO(ell-1,ell-1,Z), from Obers and Pioline.
    This structure underlies the abelianization computations in Section 7 (Lemma 7.8); if wrong, the k=1 table for D=3..6 changes.
  • standard math Stable splittings of BSL(2,Z), BSL(3,Z), and BSL(2,Z) x BSL(3,Z) due to Brown, Soule, and Minami.
    These external theorems reduce bordism of the infinite discrete groups to finite groups such as BZ3, BD6, BZ4, and BS4; they are the backbone of Part II.
  • standard math The spin bordism groups of the finite building blocks BZ3, BD6, BZ4, and BS4 are taken from prior literature, including the authors' own IIBordia paper.
    These are external or prior published computations, not the target result; the new contribution is the combination with SL(3,Z) and the module structure.
  • domain assumption The physical bordism group is Omega_k^Spin(BG_U) with the bosonic U-duality group, and no nontrivial Spin-lift of G_U is used.
    The authors state the Spin and Pin lifts are unknown (Section 1.2.6) and show for SL(2,Z) that the analogous Spin-Z8 computation gives different groups. If a lift is required, some prime-2 generators and their defects may not be physical.

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Cite this review

Pith. "Pith review of Cobordism Utopia: U-Dualities, Bordisms, and the Swampland." pith.science (2026). https://pith.science/paper/FD72IXCF

@misc{pith2026250515885,
  author       = {Pith},
  title        = {Pith review of: Cobordism Utopia: U-Dualities, Bordisms, and the Swampland},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FD72IXCF}},
  note         = {Machine review of arXiv:2505.15885}
}
abstract

The U-dualities of maximally supersymmetric supergravity theories lead to celebrated non-perturbative constraints on the structure of quantum gravity. They can also lead to the presence of global symmetries since manifolds equipped with non-trivial duality bundles can carry topological charges captured by non-trivial elements of bordism groups. The recently proposed Swampland Cobordism Conjecture thus predicts the existence of new singular objects absent in the low-energy supergravity theory, which break these global symmetries. We investigate this expectation in two directions, involving the different choices of U-duality groups $G_U$, as well as $k$, the dimension of the closed manifold carrying the topological charge. First, we compute for all supergravity theories in dimension $3 \leq D \leq 11$ the bordism groups $\Omega_1^{\text{Spin}}(BG_U)$. Second, we treat in detail the case of $D = 8$, computing all relevant bordism groups $\Omega_k^{\text{Spin}}(BG_U)$ for $1 \leq k \leq 7$. In all cases, we identify corresponding string, M-, or F-theory backgrounds which implement the required U-duality defects. In particular, we find that in some cases there is no purely geometric background available which implements the required symmetry-breaking defect. This includes non-geometric twists as well as non-geometric strings and instantons. This computation involves several novel computations of the bordism groups for $G_U = \mathrm{SL}(2,\mathbb{Z}) \times \mathrm{SL}(3,\mathbb{Z})$, which localizes at primes $p=2,3$. Whereas an amalgamated product structure greatly simplifies the calculation of purely $\mathrm{SL}(2,\mathbb{Z})$ bundles, this does not extend to $\mathrm{SL}(3,\mathbb{Z})$. Rather, we leverage the appearance of product / ring structures induced from cyclic subgroups of $G_U$ which naturally act on the relevant bordism groups.

Figures

Figures reproduced from arXiv: 2505.15885 by the authors.

Figure 1
Figure 1. Bordism generator in dimension k as fibration of fiber Fp over the base Bk−p (left). Associated bordism defect interpreted as the twisted compactification of the defect associated to Fp compactified on Bk−p with non-trivial action on normal directions (right). 1.2.2 Twisted compactifications In the analysis below we often encounter bordism generators whose geometry decomposes as a direct product or fibration of two … view at source ↗
Figure 2
Figure 2. Non-geometric defect (red dot) as endpoint of a transition function that acts as [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Sketch of symmetry-breaking defect in codimension ( [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: Bounding one factor with lower codimension object (left). Bounding the entire [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Deformation of the bounding manifold in the low-energy theory can change the [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: M-theory capturing SL(3, Z) ⊂ GU of the U-duality group geometrically. fields arise as follows: graviton : Gµν , vectors : Gαµ , Cαβµ , 2-forms : Cαµν , 3-form : Cµνρ , scalars : Gαβ , Cαβγ . (3.4) Up to field redefinitions, we find the exact same spectrum as in (3.2).…
Figure 7
Figure 7. Figure 7: Type IIA capturing SL(2, Z) ⊂ SL(3, Z) ⊂ GU of the U-duality group geometri￾cally. i.e., ωIIA = B + i vol(T 2 ). (3.6) One can lift that to a complexified volume in M-theory by defining ωM = C + i vol(T 3 ), (3.7) which also transforms under Moebius transformation with…
Figure 8
Figure 8. Figure 8: F-theory capturing SL(2, Z)×SL(2, Z)S ⊂ GU of the U-duality group geometrically. with RR axion C0 and dilaton ϕ. The bosonic fields in (3.2) arise from the type IIB fields as follows:19 graviton: gµν , vectors: gaµ , Baµ , Caµ , 2-forms: C + abµν , Bµν , Cµν , 3-form: …
Figure 9
Figure 9. Figure 9: The two relevant orbifolds T 2/Z3 (left), T 2/Z4 (right), with fundamental domain given by the shaded region, and fixed points marked with black circles (the C/Z2 is indicated by a further circle). For the degenerate fiber on the duality defect the monodromy should lea…
Figure 10
Figure 10. Figure 10: The singular fiber T 3/(Z3) Γ (2) 3 as fibration of T 2/Z3 over S 1 (right), obtained by modding out a Z3 rotation indicated in two different perspectives on the left and in the middle. Things become more interesting for Γ(2) 3 , which does not leave any individual ci…
Figure 11
Figure 11. Figure 11: The singular fiber T 3/(Z4) Γ (2) 4 as fibration of T 2/Z4 over S 1 . For Γ(2) 4 things are more complicated, especially since it is not an element of SO(3) and we need to find a good basis for the torus lattice Λ3. This is done in Appendix A. Similar to the situation…
Figure 12
Figure 12. Figure 12: Sketch of the solution of the section constraint cutting out a singular subspace of [PITH_FULL_IMAGE:figures/full_fig_p043_12.png]
Figure 13
Figure 13. Figure 13: Bounding manifold of S 1 with transition function given by g = g1g2g −1 1 g −1 2 . This implies that there are no defects of real codimension-two necessary to break global symmetries induced by non-trivial duality bundles. The reason for this is that the U-duality gro…
Figure 14
Figure 14. Figure 14: Type IIB realization of the U-duality defects in codimension-two ( [PITH_FULL_IMAGE:figures/full_fig_p046_14.png]
Figure 15
Figure 15. Figure 15: Two-dimensional generator given by T 2 with monodromies M (i) 1 and M (i) 2 . with Γ(i) 4 in (4.9) and R (1) =   0 −1 0 −1 0 0 0 0 −1   , R(2) =   −1 0 0 0 −1 −1 0 0 1   . (6.8) Indeed the two monodromy elements correspond to the even permutations of S4, namel…
Figure 16
Figure 16. Figure 16: Bounding manifold of the generators in dimension-two using a gravitational [PITH_FULL_IMAGE:figures/full_fig_p049_16.png]
Figure 17
Figure 17. Figure 17: Schematic depiction of a non-geometric U-duality defect in M-theory as the [PITH_FULL_IMAGE:figures/full_fig_p058_17.png]
Figure 18
Figure 18. Figure 18: The E 2 -page of the Atiyah-Hirzebruch spectral sequence computing ΩeSpin ∗ ((BZ3)+ ∧ BD6). Each named class generates a Z3. Theorem 11.2. In the range p + q ≤ 7, the Atiyah-Hirzebruch spectral sequence computing ΩeSpin ∗ ((BZ3)+ ∧ BD6) collapses on the E 2 -page. As …
Figure 19
Figure 19. Figure 19: The E-module ExtA(1)(Cη, Z2). Vertical lines represent h0-multiplication, so in this picture h0µ = vκ, h0ξ = vλ, etc. This is a picture of Proposition 13.22. We draw this E-module in [PITH_FULL_IMAGE:figures/full_fig_p091_19.png]
Figure 20
Figure 20. Figure 20: Left: the E2-page of the Adams spectral sequence computing ko∗(BZ4). Gener￾ators and some relations labeled. As we discussed in Proposition 13.38, the May-Milgram theorem establishes d2 differentials between the h0-towers, which we display here. Right: the spectral se…
Figure 21
Figure 21. Figure 21: Left: the A(1)-module structure on He∗ (BSL(3, F2); Z2) in low degrees, as we calculated in Corollary 14.9. This picture includes all classes in degrees 9 and below. Right: the same but for S4 in place of SL(3, F2), realizing the effect on cohomology of the Mitchell￾P…
Figure 22
Figure 22. Figure 22: 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 a b c vb d e vd f g h i 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 a d e vd f g h [PITH_FULL_IMAGE:figures/full_fig_p100_22.png]
Figure 23
Figure 23. Figure 23: The permutation (1 2 3 4) generating a Z4 as an element of S4 acting on a tetrahedron (inside a cube). 14.2.1 Generators coming from ko∗(BZ4) and ko∗(B(Z2 × Z2)) Many of the generators of ko∗(BS4) can be determined by considering subgroups of S4. Indeed, consider the …
Figure 24
Figure 24. Figure 24: The map Φ∗ from the E2-page of the Adams spectral sequence for ko∗(BZ4) to the E2-page for ko∗(BS4) can be calculated summand by summand. This is a picture of the proof of Lemma 14.27; see also [28, [PITH_FULL_IMAGE:figures/full_fig_p106_24.png]
Figure 25
Figure 25. Figure 25: The Serre spectral sequence computing H∗ (W4; Z2). The multiplicative structure is clear except for the relation xy + y 2 = 0. This follows from the fact that x and y pull back from BD8, and x1x2 + x 2 2 = 0 in H∗ (BD8; Z2). A straightforward application of Lemma 14.5…
Figure 26
Figure 26. Figure 26: Left: an A(1)-module extension representing the class b ∈ Ext1,3 (J, Z2) defined in Corollary 14.14. Right: the same extension as A(0)-modules is not split. This is an ingredient in the proof of Lemma 15.4. Lemma 15.6. 48Baker [187, §5] calls this A(1)-module the whis…
Figure 27
Figure 27. Figure 27: Left: the E2 page of the Adams spectral sequence computing kof∗(BZ4 ∧ BSL(3, F2)). The differentials shown here are calculated in Theorem 15.16. Right: the E3 = E∞-page. We draw the submodule coming from BZ4 ∧ BS4 in [PITH_FULL_IMAGE:figures/full_fig_p117_27.png]
Figure 28
Figure 28. Figure 28: The singular fiber T 3/(Z3) Γ (2) 3 in terms of its triple cover on the left, where the translational symmetry on the fixed points of T 2/Z3 is indicated by shaded blue arrows, and its fibration structure on the right. Modding out by this Z s 3 translational action, w…
Figure 29
Figure 29. Figure 29: The singular fiber T 3/(Z3) Γ (2) 4 in terms of its double cover on the left, where the translational symmetry on the fixed points of T 2/Z4 is indicated by shaded blue arrows, and its fibration structure on the right. from which we can define the invariant λe1 = λ1 +…
Figure 30
Figure 30. Figure 30: The singular fiber T 3/(Z2)M (2) 1 in terms of its double cover on the left, where the translational symmetry on the fixed points of T 2/Z2 is indicated by shaded blue arrows, and its fibration structure on the right. on a 6-torus (the auxiliary internal space).49 We …
Figure 31
Figure 31. Figure 31: Sketch of the singular fiber of the non-geometric quotient [PITH_FULL_IMAGE:figures/full_fig_p134_31.png]

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