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arxiv: 2405.04649 · v2 · submitted 2024-05-06 · 🧮 math.AT · math-ph· math.MP

The Smith Fiber Sequence and Invertible Field Theories

Arun Debray , Sanath K. Devalapurkar , Cameron Krulewski , Yu Leon Liu , Natalia Pacheco-Tallaj , Ryan Thorngren This is my paper

Pith reviewed 2026-05-24 01:36 UTC · model grok-4.3

classification 🧮 math.AT math-phmath.MP
keywords Smith homomorphismsThom spectrabordism groupslong exact sequenceAnderson dualityinvertible field theoriestangential structurescofiber
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The pith

Smith homomorphisms unify examples via equivalent definitions and yield long exact sequences in bordism groups and invertible field theories.

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

The paper aims to unify the many examples of Smith homomorphisms in the literature by giving three definitions and proving they are equivalent, including one as maps of Thom spectra. This equivalence lets the authors identify the cofiber of the spectrum-level Smith map and produce a long exact sequence of bordism groups. The sequence generalizes known constructions such as Wood's and Wall's sequences. Taking Anderson duals of the sequence then produces a corresponding long exact sequence of invertible field theories that carries a physical interpretation.

Core claim

We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups. Taking Anderson duals yields a long exact sequence of invertible field theories.

What carries the argument

The Smith homomorphism, realized as a map of Thom spectra that changes both dimension and tangential structure, which induces the cofiber identification and the long exact sequence.

Load-bearing premise

The three definitions of Smith homomorphisms remain equivalent when applied to arbitrary tangential structures on manifolds.

What would settle it

A specific manifold with tangential structure where two of the three definitions of the Smith homomorphism produce different maps, or where the claimed long exact sequence in bordism groups fails to hold.

Figures

Figures reproduced from arXiv: 2405.04649 by Arun Debray, Cameron Krulewski, Natalia Pacheco-Tallaj, Ryan Thorngren, Sanath K. Devalapurkar, Yu Leon Liu.

Figure 1
Figure 1. Figure 1: Long exact sequence of field theories associated to Equation (7.9d). Observe that all maps in low degrees are determined by exactness. This long exact sequence also appears in [DDK+25, §IV.C] Example 8.22. In degree k = 4, there is a map of invertible field theories (8.23) IZΩ 3 Spin×Z/2 ∼= Z ⊕ Z/8 Defσ −−−→ Z/16 ∼= IZΩ 4 Pin+ . The Z/16 classifies anomalies of Majorana fermions in 2 + 1 dimensions. An ass… view at source ↗
Figure 2
Figure 2. Figure 2: Long exact sequence in spin bordism partially determining Ω Spin ∗ (RP1 , σ) Remark A.16. To address the question as to whether A is isomorphic to Z/4 or Z/2 ⊕ Z/2, one could appeal to geometric arguments or an Adams spectral sequence calculation, but it turns out that the Smith long exact sequence that we will study in [PITH_FULL_IMAGE:figures/full_fig_p055_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bordism Long Exact Sequence for Pin− ⇝ Pin+ (a) ∗ = 0: The group Ω Spin 0 (RP1 , σ) ∼= Z/2 is generated by the class of the point equipped with the inclusion i into RP1 . The condition of Tpt ⊕ i ∗σ being spin is satisfied since i ∗σ is trivial. The map f forgets i, so sends this generator to the point with its pin− structure, which is a generator of Ω Pin− 0 ∼= Z/2. (b) ∗ = 1: Consider the circle with spi… view at source ↗
read the original abstract

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper.

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 / 2 minor

Summary. The paper gives a general account of Smith homomorphisms between bordism groups that change both dimension and tangential structure. It supplies three definitions (geometric on manifolds, maps of Thom spectra, and a third), proves their equivalence, identifies the cofiber of the spectrum-level Smith map to obtain a long exact sequence of bordism groups, relates the sequence to Wood's and Wall's constructions, and applies Anderson duality to produce a long exact sequence of invertible field theories.

Significance. If the equivalences are established for arbitrary tangential structures, the work unifies scattered examples from the literature and supplies a new computational device for bordism groups and invertible field theories. The spectrum-level cofiber identification and the resulting fiber sequence constitute a concrete strength when rigorously verified.

major comments (2)
  1. [§3] §3 (equivalence of definitions): the proof that the geometric Smith homomorphism coincides with the map of Thom spectra is given for reductions of structure group, but the abstract and §1 claim the result for arbitrary tangential structures (maps of spaces to BO). The argument does not explicitly address fibrancy or homotopy-colimit issues that arise for general structure maps; this equivalence is load-bearing for the cofiber computation in §4.
  2. [§4] §4 (cofiber identification): the long exact sequence of bordism groups is derived from the cofiber of the spectrum-level Smith map once the three definitions are shown equivalent. If the equivalence fails to hold in full generality, the sequence and its Anderson dual do not follow in the breadth stated in the abstract and §5.
minor comments (2)
  1. [Abstract] The third definition of the Smith homomorphism is mentioned in the abstract but not named until §2; an explicit label in the introduction would aid readability.
  2. [§1] Notation for the tangential structure map (e.g., the space X → BO) is introduced in §1 but used without reminder in later sections; a short notation table would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the generality of the equivalence and cofiber results. We agree that the proof in §3 requires explicit extension to arbitrary tangential structures and will revise accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (equivalence of definitions): the proof that the geometric Smith homomorphism coincides with the map of Thom spectra is given for reductions of structure group, but the abstract and §1 claim the result for arbitrary tangential structures (maps of spaces to BO). The argument does not explicitly address fibrancy or homotopy-colimit issues that arise for general structure maps; this equivalence is load-bearing for the cofiber computation in §4.

    Authors: We agree that the detailed proof of equivalence between the geometric definition and the Thom spectrum map in §3 is presented in the context of reductions of structure group. To address the general case of arbitrary maps of spaces to BO, we will add an explicit discussion in a new subsection of §3. This will cover the necessary fibrancy conditions by replacing the structure map with a fibration and handling the relevant homotopy colimits using standard techniques in stable homotopy theory. This revision will confirm that the equivalence holds in full generality, supporting the claims in the abstract and §1. revision: yes

  2. Referee: [§4] §4 (cofiber identification): the long exact sequence of bordism groups is derived from the cofiber of the spectrum-level Smith map once the three definitions are shown equivalent. If the equivalence fails to hold in full generality, the sequence and its Anderson dual do not follow in the breadth stated in the abstract and §5.

    Authors: We acknowledge that the cofiber identification and the resulting long exact sequence in §4 depend on the equivalence of definitions established in §3. With the planned extension of the equivalence proof to arbitrary tangential structures, the cofiber computation will apply in the general setting. We will update the text in §4 and §5 to explicitly reference the general equivalence and note that the long exact sequence of bordism groups and its Anderson dual hold for arbitrary tangential structures as claimed. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalences and cofiber identification rest on independent spectrum-level constructions.

full rationale

The paper constructs three distinct definitions of Smith homomorphisms (geometric on manifolds, as maps of Thom spectra, and a third) and proves their equivalence using standard tools of algebraic topology and bordism. The cofiber identification and resulting long exact sequence of bordism groups (and Anderson duals) are then derived as consequences of this equivalence and the properties of Thom spectra, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The argument is self-contained and does not rename known results or smuggle ansatzes via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard constructions of Thom spectra, bordism groups, and Anderson duality in stable homotopy theory; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of Thom spectra and bordism groups in the stable homotopy category
    Invoked implicitly when defining Smith maps as spectrum maps and identifying cofibers.
  • standard math Existence and properties of Anderson duality for spectra
    Used to pass from the bordism sequence to the invertible field theory sequence.

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