arxiv: 2605.00316 · v1 · submitted 2026-05-01 · 🧮 math-ph · cond-mat.str-el· hep-th· math.AT· math.MP

Recognition: unknown

Unraveling the Bott spiral

Arun Debray , Cameron Krulewski , Luuk Stehouwer

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:13 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.str-elhep-thmath.ATmath.MP

keywords Bott spiralsymmetry-protected topological phasesinvertible field theoriesAtiyah-Bott-Shapiro orientationK-theorydimensional reductionAltland-Zirnbauer classeshomotopy theory

0 comments

The pith

A homotopy model maps free fermionic SPTs to interacting ones via a twisted Atiyah-Bott-Shapiro orientation and computes the Bott spiral.

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

The paper builds a homotopy-theoretic model of the Bott spiral for symmetry-protected topological phases of fermions. Free phases are modeled by K-theory while interacting phases use reflection-positive invertible field theories. A twisted generalization of the Atiyah-Bott-Shapiro orientation supplies the map from free to interacting data, and spiral maps of the field theories track dimensional reduction. This matters because it clarifies how approximations that ignore interactions relate to the full interacting classification and answers questions about reduction in interacting systems. The analysis also shows that invertible field theories need more symmetry information than an Altland-Zirnbauer class and that the remaining periodicity on the interacting side depends on a specific group isomorphism.

Core claim

We construct and compute a homotopy-theoretic model for the Bott spiral of symmetry-protected topological phases studied by Queiroz-Khalaf-Stern. We model free and interacting fermionic SPTs using K-theory and reflection-positive invertible field theories, respectively, and define a twisted generalization of the Atiyah-Bott-Shapiro orientation to produce a free-to-interacting map. We also define and compute spiral maps of IFTs to model dimensional reduction, answering a question of Hason-Komargodski-Thorngren. Our analysis uses a novel 4-periodic description of a sector of the twisted ko-homology of elementary abelian 2-groups.

What carries the argument

The twisted generalization of the Atiyah-Bott-Shapiro orientation, which produces the free-to-interacting map between K-theory and the homotopy theory of reflection-positive invertible field theories, together with the spiral maps that implement dimensional reduction.

If this is right

The free-to-interacting map relates K-theory classes of fermionic phases to classes in the homotopy theory of reflection-positive invertible field theories.
Spiral maps of invertible field theories give explicit rules for how phases transform under dimensional reduction in the interacting regime.
Invertible field theories are sensitive to symmetry data beyond the Altland-Zirnbauer class, so some phases that look equivalent in K-theory may be distinguished when interactions are included.
The periodicity structure that survives on the interacting side is controlled by the isomorphism between two extraspecial groups of order 32.
Computations rely on a 4-periodic description of a sector of twisted ko-homology for elementary abelian 2-groups.

Where Pith is reading between the lines

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

The extra symmetry data required by invertible field theories suggests that some experimentally realized phases may split into distinct interacting classes even when their free approximations coincide.
The same spiral maps could be used to predict how phases behave when the physical system is reduced to lower dimensions in a laboratory setting.
The underlying group isomorphism may point to algebraic structures that control periodicity in other topological invariants outside this specific spiral.

Load-bearing premise

The specification of an Altland-Zirnbauer class alone does not give enough data to define the symmetry type for an invertible field theory, and the remnant of Bott periodicity on the interacting side depends on an isomorphism of two extraspecial groups of order 32.

What would settle it

An explicit low-dimensional calculation of the classification of interacting SPTs for a fixed symmetry type that produces a different group from the one obtained by applying the twisted orientation map to the corresponding K-theory class.

Figures

Figures reproduced from arXiv: 2605.00316 by Arun Debray, Cameron Krulewski, Luuk Stehouwer.

**Figure 1.** Figure 1: The Bott spiral starting with 1d class BDI′ : our mathematical model for [QKS16, view at source ↗

**Figure 2.** Figure 2: The Bott spiral starting with 1d class CII: our mathematical model for [QKS16, view at source ↗

**Figure 3.** Figure 3: This figure shows the sequence of interacting SPT phases in the complex version of the Bott spiral, starting with the d = 1 class AIII insulator, which is predicted to generate the group ℧ 2 Pinc ∼= Z/4 in the top right. The leftmost column indicates the Altland–Zirnbauer class of the SPT at each step. The middle column encodes spectra of the form MTSpinc ∧ MEℓ,k and the maps between them. The last column … view at source ↗

**Figure 4.** Figure 4: In this figure, we abbreviate ℧ i ℓ,k := ℧ i Spin(MEℓ,k). This grid shows the groups of (deformation classes of) invertible field theories on spin-(ℓ, k) manifolds that can be reached by repeatedly applying the spiral maps of Definition 4.51 along with identifications resulting from Corollary 2.31. Physically, these represent the groups of interacting SPTs able to be reached by the creation of a Z/2-symmet… view at source ↗

**Figure 5.** Figure 5: As in view at source ↗

**Figure 6.** Figure 6: The third Bott spiral: as in [QKS16, view at source ↗

**Figure 7.** Figure 7: If one tries to model the Bott spiral starting with continuous class CII in dimension 1, the first several maps in the Bott spiral work out the same as in the discrete case, but once we get to dimension 5, there is no map spψ that would allow us to continue the spiral. Here ϕ and ψ mean the corresponding spϕ and spψ maps smashed with MTPinh±. We discuss this further in Remark 4.73, where we interpret this … view at source ↗

**Figure 8.** Figure 8: The Z 2 -graded Z/2-algebra E := Ext∗,∗ A(1)(Z/2, Z/2) described in Proposition 5.23. The following theorem is surely well-known, but we were unable to find it in the literature, so we have included a proof. Theorem 5.25 (Margolis’ theorem for ko-modules). Let M be a bounded below ko-module of finite type. Any isomorphism of A(1)-modules (5.26a) φ: H∗ ko(M) ∼=−→ N ⊕ F, view at source ↗

**Figure 9.** Figure 9: Left: the A(1)-module N1 := H∗ ((BZ/2)σ−1 ;Z/2). This is also He ∗ (RP∞;Z/2) shifted down in degree by 1. Right: Ext(N1), the E2-page of the Adams spectral sequence for ko ∧ (BZ/2)σ−1 . All differentials and extension questions are trivial. We discuss N1 and its Adams spectral sequence in Example 5.31 view at source ↗

**Figure 10.** Figure 10: Left: the A(1)-module N2, the unique non-free summand of Σ −2He ∗ (BA4; Z/2). Right: Ext(N2), the E2-page of the Adams spectral sequence for a summand of ko ∧ Σ −2BA4. All differentials and extension questions are trivial. We discuss N2 and its Adams spectral sequence in Example 5.34. Yu [Yu95, Theorem 3.1] exhibits an E-module isomorphism (5.38a) Ext(N3) ∼= E{γ, ci | i ≥ 0}/R3, with γ ∈ Ext2,8 , ci ∈ Ext… view at source ↗

**Figure 11.** Figure 11: Left: the A(1)-module N3, the unique non-free summand of Σ −3He ∗ (BA4 ∧ BZ/2;Z/2). Right: Ext(N3), the E2-page of the Baker–Lazarev Adams spectral sequence for a ko-module summand M3 of Σ −3 ko ∧ BA4 ∧ BZ/2. All differentials and extension questions are trivial. See Example 5.37 for more information on N3, M, and this Adams spectral sequence. We draw this in view at source ↗

**Figure 12.** Figure 12: Left: the A(1)-module N0 := Σ−1H∗ ((BZ/2)1−σ ;Z/2). Right: Ext(N0), the E2-page of the Adams spectral sequence for Σ −1 ko ∧ (BZ/2)1−σ . All differentials and extension questions are trivial. See Example 5.40 for more information. looks nothing like the Ni , and its Adams spectral sequence has both differentials and hidden extensions (ibid., Theorem 4). ⋄ Lemma 5.45 (Yu [Yu95, Lemma 2.5]). There are stabl… view at source ↗

**Figure 13.** Figure 13: The result of superimposing the right-hand sides of Figures 9, 10, 11, and 12 on the same axes, illustrating the close relationship between the Ext charts of the corresponding four A(1)-modules. See Remark 5.47. applying this iteratively to the BZ/2 pieces of BV , one deduces a splitting48 (5.50) Σ ∞BV ≃ _n k=1 ( n _k) i=1 Σ ∞((BZ/2)∧k ). We showed ko ∧ BZ/2 is of EA-type in Example 5.31; then Lemmas 5.43… view at source ↗

**Figure 14.** Figure 14: The effect of the spiral map ϕℓ,k on mod 2 cohomology. This map factors as the zero section map eσ composed with the crush map c, all smashed with the identity on MEℓ,k−1. We use this map on cohomology in the proof of Lemma 5.66. Lemma 5.68. Theorem 5.58 is true for all values of k and ℓ. We will prove this with another inductive argument, this time inducting on ℓ. To do this, we need a map relating ko ∧ … view at source ↗

**Figure 15.** Figure 15: A picture of the fiber sequence including the maps ς and υ from [Bru99, §2], introduced in the text in Lemma 5.71. We also label the A(1)-modules appearing as the mod 2 cohomology of the spectra in this fiber sequence, including the infinite seagull Υ∞ from Definition 5.72. Definition 5.74. Recall the map λσ : (BZ/2)σ−1 → Σ −1KO from Definition 3.34 (specifically (3.37)). Because (BZ/2)σ−1 is (−2)-connect… view at source ↗

**Figure 5.** Figure 5: ], and view at source ↗

**Figure 16.** Figure 16: The A(1)-modules Fi from Definition 5.92. Figure adapted from Pearson [Pea14, (8-14)]. Definition 5.93. Let R2 denote the kernel of the unique nonzero A(1)-module map Σ −1A(1) → Σ −1Z/2, let Ω: ModA(1) → ModA(1) be ΣR2 ⊗ –, and let Ω n denote the result of applying Ω n times view at source ↗

**Figure 17.** Figure 17: ExtA(1)(Fi), as calculated in Proposition 5.99. The entire chart is Ext(F0); for Ext(Fi), move everything i degrees down and to the left, then delete all classes not in the first quadrant. Proof. Use the Adams spectral sequence as usual. Corollary 5.97 gives us the A(1)-module structure on H∗ (Σ−1RP2 ∧ MEℓ,k; Z/2) modulo free summands, and Proposition 5.99 calculates the Ext of this module (again, modulo … view at source ↗

**Figure 18.** Figure 18: Left: the E(1)-module structure on N1; straight lines are Q0-actions and dashed curved lines are Q1-actions. Right: ExtE(1)(N1), the E2-page of the Adams spectral sequence for ku∧(BZ/2)σ−1 . Gray diagonal lines denote v1-actions. All differentials and extension questions are trivial. We discuss this Adams spectral sequence in Example 6.71. This figure adapted from [Deb21, view at source ↗

read the original abstract

We construct and compute a homotopy-theoretic model for the Bott spiral of symmetry-protected topological phases (SPTs) studied by Queiroz--Khalaf--Stern. We model free and interacting fermionic SPTs using K-theory and reflection-positive invertible field theories (IFTs), resp., and define a twisted generalization of the Atiyah--Bott--Shapiro orientation to produce a free-to-interacting map. We also define and compute spiral maps of IFTs to model dimensional reduction in this context, answering a question of Hason--Komargodski--Thorngren. Our analysis highlights two general aspects of homotopical free-to-interacting maps. First, IFTs are more sensitive than K-theory is to the input symmetry data; in particular, the specification of an Altland--Zirnbauer class is insufficient information to define symmetry type for an IFT. Second, the remnant of Bott periodicity on the interacting side relies on an isomorphism of two extraspecial groups of order 32. Our computations use a novel 4-periodic description of a sector of the twisted ko-homology of elementary abelian 2-groups.

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.

Desk Editor's Note private letter to a colleague

This paper gives a homotopy model for the Bott spiral in fermionic SPTs via a twisted ABS map and new spiral maps, with the main technical load on a specific 2-group isomorphism.

read the letter

This paper sets up a homotopy model for the Bott spiral of fermionic SPTs. It connects free fermion classifications in K-theory to interacting ones in reflection-positive invertible field theories by defining a twisted ABS orientation as the free-to-interacting map. It also introduces spiral maps to capture dimensional reduction, which directly addresses a question raised by Hason, Komargodski, and Thorngren. The new pieces are the spiral maps themselves and the 4-periodic description of twisted ko-homology for elementary abelian 2-groups. These constructions and the resulting computations for the Bott spiral do not appear in the cited prior literature. The observation that an Altland-Zirnbauer class alone does not determine the symmetry type for an IFT is a useful distinction that the paper draws out clearly. The work does a good job of using standard homotopy tools to organize the material and produce explicit maps. The self-contained nature of the mathematical construction, relying on external facts like the group isomorphism rather than fitting parameters, keeps the circularity low. The potential weak point is the load-bearing role of the isomorphism between two extraspecial groups of order 32. This is said to supply the remnant of Bott periodicity on the interacting side. The paper needs to show rigorously that this isomorphism induces the claimed periodicity in the IFT setting. If that step has any gaps, the spiral map computations would not fully model the physical Bott spiral. The novel ko-homology description must also correctly encode the symmetry data beyond what AZ classes provide. These are not minor details but central to the claims. The paper is for condensed-matter theorists and mathematical physicists who care about SPT classifications and homotopical methods. A reader with background in algebraic topology will get the most from the constructions and computations. It deserves serious refereeing because the ideas are substantive and the computations could be checked by experts in the area. I recommend sending it out for peer review. The technical concerns are addressable with more detail on the derivations.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs a homotopy-theoretic model for the Bott spiral of symmetry-protected topological phases (SPTs) using K-theory for free fermionic SPTs and reflection-positive invertible field theories (IFTs) for interacting ones. It defines a twisted generalization of the Atiyah-Bott-Shapiro orientation to produce a free-to-interacting map and introduces spiral maps of IFTs to model dimensional reduction, answering a question of Hason-Komargodski-Thorngren. The analysis highlights that Altland-Zirnbauer classes are insufficient to define symmetry type for IFTs and that the remnant of Bott periodicity on the interacting side relies on an isomorphism of two extraspecial groups of order 32, with computations based on a novel 4-periodic description of twisted ko-homology of elementary abelian 2-groups.

Significance. If the constructions hold, this provides a rigorous homotopy-theoretic bridge between free and interacting SPT classifications, with explicit maps that could facilitate further computations in the field. The novel 4-periodic ko-homology description and the sensitivity of IFTs to symmetry data beyond AZ classes are technical contributions that strengthen the mathematical foundations for studying the Bott spiral. The work ships explicit constructions using standard homotopy tools and addresses open questions on dimensional reduction.

major comments (1)

[Abstract] Abstract: The central claim that the remnant of Bott periodicity relies on an isomorphism of two extraspecial groups of order 32 is presented as an output of the analysis, but this appears to be a load-bearing step for the free-to-interacting map and the computed spiral maps. The manuscript must explicitly show (in the section deriving the IFT periodicity) how the isomorphism induces the claimed periodicity in the IFT category without additional assumptions, as failure here would mean the computed maps do not model the Queiroz-Khalaf-Stern Bott spiral.

minor comments (1)

[Introduction] The abstract is highly technical and dense; consider adding a short non-technical overview paragraph at the start of the introduction to clarify the physical motivation for readers outside homotopy theory.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to strengthen the explicit connection between the extraspecial group isomorphism and the IFT periodicity. We have revised the paper accordingly to address this point directly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the remnant of Bott periodicity relies on an isomorphism of two extraspecial groups of order 32 is presented as an output of the analysis, but this appears to be a load-bearing step for the free-to-interacting map and the computed spiral maps. The manuscript must explicitly show (in the section deriving the IFT periodicity) how the isomorphism induces the claimed periodicity in the IFT category without additional assumptions, as failure here would mean the computed maps do not model the Queiroz-Khalaf-Stern Bott spiral.

Authors: We agree that the link between the extraspecial group isomorphism and the resulting 4-periodicity in the IFT category requires a more explicit derivation to serve as a load-bearing step. In the revised manuscript we have expanded the relevant section (now Section 4.3, 'Derivation of IFT Periodicity') with a dedicated subsection that walks through the argument in detail: we first recall the two extraspecial groups of order 32 and their explicit isomorphism φ, then show that φ induces an equivalence on the relevant twisted ko-homology groups that commutes with the spiral maps and preserves the reflection-positive structure. This equivalence is constructed directly from the symmetry-type data and the novel 4-periodic description of twisted ko-homology of elementary abelian 2-groups already developed in the paper; no further assumptions are introduced. The free-to-interacting map and the computed spiral maps are thereby shown to reproduce the Queiroz-Khalaf-Stern Bott spiral on the nose. We have also added a brief clarifying sentence to the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained mathematical construction

full rationale

The paper defines a homotopy-theoretic model for the Bott spiral using K-theory for free SPTs and reflection-positive IFTs for interacting ones, along with a twisted generalization of the Atiyah-Bott-Shapiro orientation and spiral maps for dimensional reduction. These constructions rely on external inputs from homotopy theory and group theory, including the isomorphism of two extraspecial groups of order 32 as an independent fact (not derived or defined within the paper's equations). The novel 4-periodic description of twisted ko-homology is computed from the model rather than presupposing the target results. No steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the derivation is self-contained against external benchmarks in algebraic topology and group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard axioms of K-theory, reflection-positive field theories, and homotopy theory, plus one external group-theoretic fact; no free parameters or new postulated entities are introduced.

axioms (3)

domain assumption K-theory classifies free fermionic SPTs and reflection-positive IFTs classify interacting ones
Invoked to model the two regimes and define the free-to-interacting map
standard math The Atiyah-Bott-Shapiro orientation admits a twisted generalization
Used as the basis for the free-to-interacting map
standard math Two extraspecial groups of order 32 are isomorphic
Required for the remnant of Bott periodicity on the interacting side

pith-pipeline@v0.9.0 · 5512 in / 1651 out tokens · 27513 ms · 2026-05-09T19:13:02.718190+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages · 3 internal anchors

[1]

31 [ABG18] Matthew Ando, Andrew J

https://arxiv.org/abs/1403.4320. 31 [ABG18] Matthew Ando, Andrew J. Blumberg, and David Gepner. Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map.Geom. Topol., 22(7):3761–3825, 2018.https://arxiv.org/abs/ 1112.2203. 31 [ABK21] David Aasen, Parsa Bonderson, and Christina Knapp. Characterization and classification of fermionic sym...

work page arXiv 2018
[2]

95 [AHS17] Adhip Agarwala, Arijit Haldar, and Vijay B

https://arxiv.org/abs/2107.02837. 95 [AHS17] Adhip Agarwala, Arijit Haldar, and Vijay B. Shenoy. The tenfold way redux: fermionic systems with N-body interactions.Ann. Physics, 385:469–511, 2017.https://arxiv.org/abs/1606.05483. 3 [AK12] Gilles Abramovici and Pavel Kalugin. Clifford modules and symmetries of topological insulators. International Journal o...

work page arXiv 2017
[3]

34, 42 [And64a] D

https://arxiv.org/abs/1903.06782. 34, 42 [And64a] D. W. Anderson. The real K-theory of classifying spaces.Proceedings of the National Academy of Sciences of the United States of America, 51(4):634–636, 1964. 96 [And64b] Donald Werner Anderson.A new cohomology theory. PhD thesis, University of California, Berkeley,

work page arXiv 1903
[4]

Anderson

110 [And69] D.W. Anderson. Universal coefficient theorems for K-theory. 1969. https://faculty.tcu.edu/ gfriedman/notes/Anderson-UCT.pdf. 9, 36, 53 [AP76] J. F. Adams and S. B. Priddy. Uniqueness ofBSO. Math. Proc. Cambridge Philos. Soc., 80(3):475–509,

1969
[5]

Products in spinc-cobordism

100, 108, 110, 123 [AS25] Hassan Abdallah and Andrew Salch. Products in spinc-cobordism. Int. Math. Res. Not. IMRN, (18):Paper No. rnaf291, 19, 2025.https://arxiv.org/abs/2407.10045. 119 [Ati61a] M. F. Atiyah. Bordism and cobordism. Proc. Cambridge Philos. Soc., 57:200–208, 1961. 46 [Ati61b] M. F. Atiyah. Thom complexes. Proc. London Math. Soc. (3), 11:29...

work page arXiv 2025
[6]

82, 99, 107, 110, 117, 123 [Bal22] William Balderrama.K-theory equivariant with respect to an elementary abelian 2-group.New York J

https://arxiv.org/abs/2003.12003. 82, 99, 107, 110, 117, 123 [Bal22] William Balderrama.K-theory equivariant with respect to an elementary abelian 2-group.New York J. Math., 28:1531–1553, 2022.https://arxiv.org/abs/2201.03095. 27 [Bay94] Dilip Kumar Bayen. The connective real K-homology of finite groups. PhD thesis, Wayne State University, 1994. 90, 121 [...

work page arXiv 2003
[7]

Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors.Phys

3 [CC21] Eyal Cornfeld and Shachar Carmeli. Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors.Phys. Rev. Res., 3:013052, Jan 2021.https://arxiv.org/ abs/2009.04486. 3 [CF64] P.E. Conner and E.E. Floyd.Differentiable periodic maps. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, 1964....

work page arXiv 2021
[8]

’t Hooft Anomalies and Defect Conformal Manifolds: Topological Signatures from Modulated Effective Actions,

43 [Cop25] Christian Copetti. ’t Hooft anomalies and defect conformal manifolds: Topological signatures from modulated effective actions. 2025.https://arxiv.org/abs/2507.15466. 7, 45 [COSY20] Clay Córdova, Kantaro Ohmori, Shu-Heng Shao, and Fei Yan. DecoratedZ2 symmetry defects and their time-reversal anomalies.Physical Review D, 102(4), aug 2020.https://...

work page arXiv 2025
[9]

https://arxiv.org/abs/1803.11183

©2018. https://arxiv.org/abs/1803.11183. 37, 38, 42, 55, 74, 81 [DK70] P. Donovan and M. Karoubi. Graded Brauer groups andK-theory with local coefficients.Inst. Hautes Études Sci. Publ. Math., (38):5–25, 1970. 17, 26, 33, 48 [DK25] Arun Debray and Cameron Krulewski. Smith homomorphisms andspinh structures. Proc. Amer. Math. Soc., 153(2):897–912, 2025.http...

work page arXiv 2018
[10]

32 [DY23] Arun Debray and Matthew Yu

https://arxiv.org/abs/math/0402082. 32 [DY23] Arun Debray and Matthew Yu. Adams spectral sequences for non-vector-bundle Thom spectra. 2023. https://arxiv.org/abs/2305.01678. 19, 29, 31, 32 [DY24a] Arun Debray and Matthew Yu. Type IIA string theory and tmf with level structure. 2024.https: //arxiv.org/abs/2411.07299. 37, 45 [DY24b] Arun Debray and Matthew...

work page arXiv 2023
[11]

Onsiteability of Higher-Form Symmetries

https://arxiv.org/abs/2510.23701. 42 [FF26] Gregory Faurot and Giovanni Ferrer. The homotopy 3-type of abelianC∗-algebras. 2026. https: //arxiv.org/abs/2603.29985. 43 [FH20] Daniel S. Freed and Michael J. Hopkins. Invertible phases of matter with spatial symmetry.Adv. Theor. Math. Phys., 24(7):1773–1788, 2020.https://arxiv.org/abs/1901.06419. 35, 61 [FH21...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[12]

51 132 ARUN DEBRAY, CAMERON KRULEWSKI, AND LUUK STEHOUWER [Her17] Thorsten Hertl

https://arxiv.org/abs/1306.0389. 51 132 ARUN DEBRAY, CAMERON KRULEWSKI, AND LUUK STEHOUWER [Her17] Thorsten Hertl. Pin bordism in low degrees of classifying spaces. Master’s thesis, University of Göttingen,

work page arXiv
[13]

52, 58 [HHY22] Sheng-Jie Huang, Chang-Tse Hsieh, and Jiabin Yu

https://thorsten-hertl.github.io/Research/Masterarbeit.pdf. 52, 58 [HHY22] Sheng-Jie Huang, Chang-Tse Hsieh, and Jiabin Yu. Effective field theories of topological crystalline insulators and topological crystals.Phys. Rev. B, 105:045112, Jan 2022.https://arxiv.org/abs/2107. 03409. 61 [HHZ05] P. Heinzner, A. Huckleberry, and M. R. Zirnbauer. Symmetry class...

work page arXiv 2022
[14]

Topological Elliptic Genera I -- The mathematical foundation

https://arxiv.org/abs/2412.02298. 94 [Mac66] E. A. M. MacAlpine. Duality in a twisted homology theory.Bull. Amer. Math. Soc., 72:1051–1054,

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Ring spectra which are Thom complexes

29 [Mah79] Mark Mahowald. Ring spectra which are Thom complexes. Duke Math. J., 46(3):549–559, 1979. 105 UNRAVELING THE BOTT SPIRAL 135 [Mah82] Mark Mahowald. The image ofJ in the EHP sequence. Ann. of Math. (2), 116(1):65–112, 1982. Correction inAnnals of Mathematics, 120:399–400, 1984. 85, 96 [Mal11] Arjun Malhotra.The Gromov-Lawson-Rosenberg conjecture...

work page arXiv 1979
[16]

15, 62, 63 [Mil63] John W

https://arxiv.org/abs/1505.06341. 15, 62, 63 [Mil63] John W. Milnor. Spin structures on manifolds. Enseign. Math. (2), 9:198–203, 1963. 42 [Mil24] Keith Mills. The structure of the spinh bordism spectrum.Proc. Amer. Math. Soc., 152(8):3605–3616,

work page arXiv 1963
[17]

15, 104, 109, 110, 122, 123 [Miy25] Jin Miyazawa

https://arxiv.org/abs/2306.17709. 15, 104, 109, 110, 122, 123 [Miy25] Jin Miyazawa. Localization of a KO∗(pt)-valued index and the orientability of thePin−(2) monopole moduli space.Algebr. Geom. Topol., 25(2):887–918, 2025.https://arxiv.org/abs/2109.10579. 51 [MK12] Jack Morava and Nitu Kitchloo. The Baker-Richter spectrum as cobordism of quasitoric manifolds

work page arXiv 2025
[18]

45 [MM76] M

With an appendix by Nitu Kitchloo.https://arxiv.org/abs/1201.3127. 45 [MM76] M. Mahowald and R. James Milgram. Operations which detect Sq4 in connectiveK-theory and their applications. Quart. J. Math. Oxford Ser. (2), 27(108):415–432, 1976. 58, 110 [MNN15] Akhil Mathew, Niko Naumann, and Justin Noel. On a nilpotence conjecture of J. P. May.J. Topol., 8(4)...

work page arXiv 1976
[19]

42 [Oga22a] Yoshiko Ogata

https://arxiv.org/abs/2101.00426. 42 [Oga22a] Yoshiko Ogata. An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional fermion systems.Comm. Math. Phys., 395(1):405–457, 2022.https: //arxiv.org/abs/2110.04672. 42 [Oga22b] Yoshiko Ogata. Classification of symmetry protected topological phases in quantum sp...

work page arXiv 2022
[20]

53, 54 [RNQ+24] Xing-Yu Ren, Shang-Qiang Ning, Yang Qi, Qing-Rui Wang, and Zheng-Cheng Gu

https://arxiv.org/abs/1408.1581. 53, 54 [RNQ+24] Xing-Yu Ren, Shang-Qiang Ning, Yang Qi, Qing-Rui Wang, and Zheng-Cheng Gu. Stacking group structure of fermionic symmetry-protected topological phases.Phys. Rev. B, 110:235117, Dec 2024. https://arxiv.org/abs/2310.19058. 61 [Ros12] Achim Rosch. Unwinding of a one-dimensional topological superconductor.Phys....

work page arXiv 2024
[21]

2, 3, 4 UNRAVELING THE BOTT SPIRAL 137 [RW14] Oscar Randal-Williams

https://arxiv.org/abs/0912.2157. 2, 3, 4 UNRAVELING THE BOTT SPIRAL 137 [RW14] Oscar Randal-Williams. Homology of the moduli spaces and mapping class groups of framed,r-Spin and Pin surfaces.J. Topol., 7(1):155–186, 2014.https://arxiv.org/abs/1001.5366. 81 [RWG14] Oscar Randal-Williams and Mark Grant. Vector bundle for prescribed Stiefel-Whitney classes, ...

work page arXiv 2014
[22]

Interacting topological phases and modular invariance.Phys

15, 57, 62 [RZ12] Shinsei Ryu and Shou-Cheng Zhang. Interacting topological phases and modular invariance.Phys. Rev. B, 85:245132, Jun 2012.https://arxiv.org/abs/1202.4484. 10, 15, 60, 61, 62 [SAC21] Daniel Sheinbaum and Omar Antolín Camarena. Crystallographic interacting topological phases and equivariant cohomology: to assume or not to assume.J. High En...

work page arXiv 2012
[23]

61 [SAC25] Daniel Sheinbaum and Omar Antolín Camarena

https://arxiv.org/abs/2007.06595. 61 [SAC25] Daniel Sheinbaum and Omar Antolín Camarena. Failure of the crystalline equivalence principle for weak free fermions.Phys. Rev. B, 111:L081118, Feb 2025.https://arxiv.org/abs/2408.07203. 61, 64 [Sal23] A. Salch. The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective. 2023. https:/...

work page arXiv 2007
[24]

38, 55, 74 [TW79] Laurence Taylor and Bruce Williams

https://arxiv.org/abs/1811.12654. 38, 55, 74 [TW79] Laurence Taylor and Bruce Williams. Surgery spaces: formulae and structure. InAlgebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978), volume 741 ofLecture Notes in Math., pages 170–195. Springer, Berlin, 1979. 96 [TW21] Ryan Thorngren and Yifan Wang. Anomalous symmetries e...

work page arXiv 1978
[25]

Fermion Path Integrals And Topological Phases

117 [Wil73b] W. Stephen Wilson. A new relation on the Stiefel-Whitney classes of spin manifolds.Illinois J. Math., 17:115–127, 1973. 100 [Wit16] Edward Witten. Fermion path integrals and topological phases.Rev. Mod. Phys., 88:035001, Jul 2016. https://arxiv.org/abs/1508.04715. 55, 74 [WS14] Chong Wang and T. Senthil. Interacting fermionic topological insu...

work page Pith review arXiv 1973
[26]

Correspondence of boundary theories between internal and crystalline symmetry protected topological phases

https://arxiv.org/abs/2310.19266. 61 [ZNQG25] Jian-Hao Zhang, Shang-Qiang Ning, Yang Qi, and Zheng-Cheng Gu. Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems. Phys. Rev. X, 15:031029, Jul 2025.https://arxiv.org/abs/2204.13558. 61 [ZWY+20] Jian-Hao Zhang, Qing-Rui Wang...

work page internal anchor Pith review Pith/arXiv arXiv 2025