Discrete quantum systems from topological field theory

Constantin Teleman; Daniel S. Freed; Michael J. Hopkins

REVIEW 3 major objections 5 minor 1 cited by

Discrete quantum systems from topological field theory

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that any Reshetikhin–Turaev topological field theory, in particular any 3d Chern–Simons theory, is the low-energy limit of a gapped lattice model built from point defects, with a Hamiltonian that is a sum of commuting…

desk verdict New defect-to-Hamiltonian recipe that plausibly embeds any RT theory into a commuting-projector lattice model; exposition is uneven where it matters most. read the letter →

arxiv 2506.05131 v1 pith:SPEPLRSY submitted 2025-06-05 hep-th cond-mat.str-elmath-phmath.ATmath.MP

Daniel S. Freed , Michael J. Hopkins , Constantin Teleman This is my paper

classification hep-thcond-mat.str-elmath-phmath.ATmath.MP MSC 81T4581T2557R56

keywords topologicalfieldtheorygappedlatticemodelscommutingprojectorHamiltoniansChern-SimonstoriccodeLevin-Wenmodeldefectsmodulartensorcategory

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 introduces a general recipe for turning a 3-dimensional topological field theory into a discrete gapped quantum system. Starting from a Reshetikhin–Turaev theory with modular tensor category $\mathcal{B}=F(S^1)$, one decorates a surface with finitely many point defects labeled by objects $1+b$ of $\mathcal{B}$; the state space is the value of the theory on the decorated surface. The Hamiltonian is a sum of operators $H_y$, each obtained by evaluating the theory on a cylinder with an extra defect insertion, and the central result is that the $H_y$ are commuting self-adjoint projections, that the spectrum of $H$ is contained in $\mathbb{Z}_{\ge0}$, and that the kernel of $H$ is exactly the original theory's state space on the undecorated surface. The models are therefore gapped, and their low-energy limit is the original topological field theory, so arbitrary Chern–Simons theories acquire lattice realizations. The same defect technology reproduces the toric code and gives a topological-field-theory construction of Levin–Wen models.

What carries the argument

The central machinery is the evaluation of a topological field theory on bordisms with codimension-two point defects. The key identity is the eigenspace decomposition (2.6), which makes each $H_y$ a projector: inserting the endomorphism $h_y=\begin{pmatrix}0&0\\0&\mathrm{id}_b\end{pmatrix}$ splits the state space into a kernel summand (defect replaced by $1$) and an image summand (defect replaced by $b$). For the Levin–Wen part, the machinery is the dimensional reduction of the Turaev–Viro theory on an interval with Dirichlet endpoints, together with the Frobenius trace $\tau_x(\mathrm{id}_x)=\dim(x)$ and the defect $p=\bigoplus_i 1/\dim(c_i)$, which together force the central component of the projector calculation to evaluate to the identity operator.

What would settle it

For a case beyond the toric code, such as SU(2)$_k$ Chern–Simons on a torus with two point defects, compute the Hamiltonian from the TFT state sum; the claim predicts integer spectrum and a kernel equal to the Chern–Simons Hilbert space of the torus. A negative eigenvalue, a non-integer spectrum, or a kernel dimension different from that Hilbert space would refute the central claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a construction. For any Reshetikhin–Turaev theory $F$ on the bordism category of $1$-, $2$- and $3$-manifolds with $(w_1,p_1)$-structure, with modular tensor category $\mathcal{B}=F(S^1)$, fix a finite set $D$ of objects of the form $1+b$. For a closed surface $Y$ with a finite set of points $\Delta$ and normal framings, a labeling $\delta:\Delta\to D$ defines the state space $H(Y,\Delta,\delta)$. Each point $y$ contributes an operator $H_y$: evaluate $F$ on the cylinder $[0,1]\times Y$ with the defect lines and one extra point defect at $\{1/2\}\times\{y\}$ labeled by the endomorphism $\begin{pmatrix}0&0\\0&\mathrm{id}_b\end{pmatrix}$ of $1+b$. The paper invokes standard TFT gluing arguments to show the $H_y$ are self-adjoint projections and pairwise commute, with an eigenspace decomposition in which the defect at $y$ is replaced either by the transparent object $1$ or by $b$. Hence the spectrum of $H=\sum_y H_y$ is contained in $\mathbb{Z}_{\ge0}$ and its kernel is $F(Y)$. The paper then verifies that the toric code is the special case of finite $\mathbb{Z}/2$ gauge theory with defect set $\{1+e,1+m\}$, and constructs Levin–Wen models from Turaev–Viro theories with a Dirichlet boundary theory, again with spectrum in $\mathbb{Z}_{\ge0}$ and kernel $T_{\mathcal{C}}(Y)$.

Load-bearing premise

The load-bearing premise is that a fully extended Reshetikhin–Turaev topological field theory with point defects exists and satisfies the standard gluing rules used to compute the Hamiltonian operators; the paper defers this framework to cited work rather than proving it here, so a failure of those gluing identities would undermine the projection, commutation, and kernel claims.

Editorial extensions

If this is right

Every 3d Chern–Simons theory (with a $p_1$-structure) admits gapped lattice models with commuting-projector Hamiltonians and ground-state space equal to the Chern–Simons state space on the spatial surface.
The models are local in a generalized sense: cutting the surface along a circle decomposes the state space as $\bigoplus_a (H_1)_a\otimes (H_2)_{a^\vee}$ over simple objects; for Hopf-algebra Turaev–Viro models this improves to a plain tensor product indexed by edges.
The toric code is exactly the special case of finite $\mathbb{Z}/2$ gauge theory with defects $1+e$ and $1+m$, so the toric code's gapped behavior follows from the same TFT mechanism.
For any spherical unitary fusion category $\mathcal{C}$, the Turaev–Viro theory $T_{\mathcal{C}}$ gives a Levin–Wen Hamiltonian whose spectrum is contained in $\mathbb{Z}_{\ge0}$ and whose kernel is $T_{\mathcal{C}}(Y)$; applied to $\mathrm{Vect}[G]$ this is another toric-code realization.
Because the Hamiltonian is a sum of commuting projections with integer spectrum, the models are gapped with a spectral gap bounded below by $1$ independently of the lattice.

Reading between the lines

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

An implicit consequence is discretization independence: since the Hamiltonian is defined by evaluating a TFT, the ground-state sector should be invariant under refining or changing the defect layout, though the paper does not formulate this as a theorem.
The same defect-insertion recipe is explicitly said to generalize to other dimensions, so the natural next step is a 4d version producing gapped lattice models for 4d TFTs; the paper does not carry this out.
The Ising reprise suggests a broader principle: any TFT boundary theory or domain wall can be converted into a discrete statistical-mechanical model, with dualities of the TFT descending to dualities of the lattice models; the paper only gestures at this via its earlier Ising work.
A testable extension is to run the construction on a nonabelian modular tensor category such as SU(2)$_k$ on a torus and compare excited-state degeneracies with the anyonic content; the paper explicitly checks only the toric-code case.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my read.

The one thing to know: this note has a genuinely new idea in Section 2, a defect-based recipe for turning any 3d TFT (e.g., Chern-Simons) into a lattice model with commuting-projector Hamiltonian whose ground states are exactly the TFT's Hilbert spaces. If correct, that's a substantial step beyond toric code and Levin-Wen, which appear as special cases. It deserves a serious referee, but with a clear instruction to fill in the proofs.

What's good: the construction itself is elegant and likely right. The Hamiltonian terms are TFT evaluations, so projectivity and commutation are not accidents; they come from the TFT gluing and semisimplicity. The paper is honest that Section 4 is a recap of [FT1], and the Levin-Wen section leans on [KK] and the authors' own prior work, which is fine because those are published, citable results.

The soft spots are real but mostly expository. The central assertion (2.5)-(2.6) that H_y is a projection with the stated eigenspace decomposition is dispatched as 'elementary standard arguments.' That's the load-bearing step for everything that follows: it gives spec H subset Z_{\ge 0} and ker H = F(Y). The stress-test note is right that this is asserted, not shown. I don't think it's wrong—it's the kind of statement that follows from standard gluing for codim-2 defects in a fully extended RT theory, and these authors know how to prove it—but the note doesn't give the argument or a precise reference. A referee should ask for it.

The toric code identification is also sketchier than the rest: the semiclassical path-integral remark is plausible but the match to (1.7) is not demonstrated term by term. The Levin-Wen projector proof is a real computation, and the central step (Figures 14-17) is only sketched: it depends on the dimensional reduction and trace formula from [FT1]. Again, probably correct, but too terse for a reader who isn't already inside the framework.

Bottom line: this is a research note from people who know the machinery. The novelty is concentrated in Section 2 and is worth taking seriously. I'd send it to peer review with the condition that the missing arguments be provided or precisely cited. If I worked on topological phases or TFT, I'd cite it. I'd bring it to the reading group, though maybe after the proofs are cleaned up.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method for constructing gapped lattice models from 3-dimensional Reshetikhin–Turaev topological field theories by placing codimension-two point defects on a surface. For a defect label of the form 1+b, the Hamiltonian term H_y is defined as the value of the TFT on a cylinder with an endomorphism h_y=diag(0,id_b) inserted on the defect line; the authors assert that these are commuting projections with the eigenspace decomposition (2.6), so that the total Hamiltonian H has integer spectrum and kernel equal to the no-defect TFT state space F(Y). They then argue that the toric code arises from finite Z/2 gauge theory with defects 1+e and 1+m, and give a TFT construction of Levin–Wen models using a Dirichlet boundary theory, with projection operators P_{v,e} and face terms H_f. Section 4 recaps the authors' earlier TFT treatment of the 1+1 dimensional Ising model.

Significance. If the key projection/decomposition lemma in Section 2 holds, the paper gives a clean and general construction of commuting-projector Hamiltonians whose ground-state sectors are the state spaces of arbitrary Chern–Simons theories, unifying the toric code and Levin–Wen models under one TFT framework. This is conceptually valuable and likely correct. The paper is honest about deferring technical foundations to [F], [FT2], [FT1], and [FMT], and the identification ker H = F(Y) is an implementation rather than a circular argument. However, the most load-bearing step—the proof of (2.6)—is not supplied, and the toric code and Levin–Wen sections contain further sketched identifications; in its current form the note is too terse to be a complete research paper.

major comments (3)

[Section 2, Eqs. (2.5)–(2.7)] The claim that H_y is a projection with eigenspace decomposition (2.6) is the foundation of the spectral and ground-state statements, but the proof is replaced by the phrase 'elementary standard arguments in topological field theory.' What is needed is a precise state-splitting/gluing statement: the bordism of Figure 3 with h_y inserted acts on the local object 1+b at y and induces an isomorphism H(Y,∆,δ) ≅ H(Y,∆,δ') ⊕ H(Y,∆,δ''), compatible with cutting and gluing along circles. This is not a formal consequence of the Atiyah–Segal axioms alone for codimension-two point defects with normal framings. Please either prove this lemma or give a theorem-and-verse reference in [F], [FT2], or [FMT]; without it, the assertions 'spec H ⊂ Z_{≥0}' and 'ker H = F(Y)' are unsupported. The text should also state whether H_y is self-adjoint and whether the summands in (2.6) are orthogonal.
[Section 2, toric code identification] The verification that the D={1+e,1+m} model is the toric code is a series of assertions: the evaluations of the bordisms in Figure 8 to 1+e and 1+m are not computed, and the final step says the semiclassical description 'reproduces precisely' the toric code of Section 1. Remark 2.12 then concedes that the analogous quantum gluing argument is 'too naive' and requires a blow-up construction as in [D]. Since the toric code is the paper's only explicit check that the general Section 2 construction yields a known lattice model, this identification should be proved in detail or replaced by a precise citation to a computation in [FMT] or elsewhere.
[Section 3, P_{v,e}, H_v, H_f] The proof that P_{v,e} is a projection is a sketch: the statement that the central component of Figure 14 evaluates to the identity is justified by a chain of assertions about dimensional reduction, Hochschild (co)homology, and the compensating effect of p in (3.7) that are not fully demonstrated. In addition, the operator H_v = id − ∏_e P_{v,e} is claimed to have spectrum {0,1}, which requires that the P_{v,e} for edges e at a fixed vertex v commute; this is not proved. The face operators H_f are introduced by reference to [FT1, §7A], and the assertion that the Hamiltonian (3.12) has integer spectrum and kernel T_C(Y) is supported only by 'The construction makes clear...' Please supply the missing projection and commutativity proofs, or state and prove the precise lemma from [FT1] that applies to this Levin–Wen setting.

minor comments (5)

[Eq. (2.6)] Define δ' and δ'' explicitly: δ' replaces the defect at y by the transparent object 1, and δ'' replaces it by b.
[Section 2, footnote 2] The foundational setup is deferred to [F], which is listed as lecture notes; if the key gluing lemma is not in a published source, the paper should include a short proof or state precisely which axiom of [FT2] covers point defects with normal framings.
[Figure 15(b)] The phrase 'the vertical sides of the square are dotted in the sense of [FT2,§2.1.2]' is too cryptic for a reader who does not have [FT2] at hand; please explain what the dotted sides mean.
[Figure 19] The term 'antisphere' is used without definition.
[Abstract and Introduction] The paper advertises 'gapped systems whose low energy limit is Chern–Simons theory,' but the precise statement proven (modulo (2.6)) is only that the kernel of H is F(Y); please clarify whether 'low energy limit' is meant to include excited states and anyonic data or only the ground-state sector.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Section 2 is an explicit construction whose kernel is F(Y) by design; cited prior work is independent published mathematics.

full rationale

The paper's central construction in Section 2 is definitional rather than predictive: H_y is the TFT evaluation of a bordism carrying the defect endomorphism h_y = diag(0,id_b), and the asserted direct-sum decomposition (2.6) is the eigenspace decomposition of that idempotent. Consequently `spec H subset Z_{>=0}` and `ker H = F(Y)` are structural consequences of the projector setup, not fitted outputs; no parameter is tuned to data and then renamed a prediction. The toric-code recovery in Section 2 is verified by an explicit semiclassical path-integral computation, and the Levin-Wen construction in Section 3 is an independent reformulation following [KK], with the computations of the Frobenius trace and the value d(C) taken from the published [FT1] and [ENO]; these citations are external theorems with stated assumptions, not self-referential validations of the present claim. The only flagged weakness is a proof gap: after (2.5) the paper asserts that `elementary standard arguments` prove idempotence, commutativity, and the splitting (2.6) without supplying the full gluing/state-splitting identity for codimension-two defects with normal framings. If that identity failed, the spectral and ground-state conclusions would not be established. This is a correctness risk, not circularity, because the asserted identity is not assumed as an input equivalent to the conclusion. The self-citations to [F], [FT2], and [FT1] are load-bearing for technical background but do not reduce the central derivation to a prior statement of the same result.

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

The central construction introduces no fitted constants; it depends on the full machinery of extended TFT, including background p1-structures, defect evaluations, and the Turaev-Viro fully local theory. Several key computations are quoted from the authors' earlier publications ([FT1], [FT2], [FMT]) rather than proved. No new entities such as particles or forces are introduced; the point defects and boundary theories are standard TFT notions.

assumptions (5)

standard math There exists a well-defined extended Reshetikhin-Turaev TFT F: Bord<1,2,3>(F) -> Cat_C for each modular tensor category B with central charge lift (Eq. (2.1)-(2.3)).
The entire lattice model is defined as an evaluation of F on bordisms with defects; the paper cites [RT1, RT2], [F], and [FT2] rather than proving existence.
domain assumption F satisfies Atiyah-Segal gluing and locality on stratified manifolds with codimension-two defects and normal framings (used in Eq. (2.5), (2.6), (2.9), and Figures 3, 8, 12, 15).
Gappedness, commutation of H_y, and the vacuum identification all depend on F's evaluations of the specific bordisms; the note invokes 'elementary standard arguments in topological field theory' and cites [FMT].
standard math The Turaev-Viro theory T_C associated to a spherical unitary fusion category C extends to a fully local TFT with Dirichlet boundary theory rho (Eq. (3.1); [TV], [FMT, Definition 3.2]).
The Levin-Wen state space and projectors are defined as T_C evaluations on disks and 3-manifolds with corners.
domain assumption The dimensional reduction of T_C on an interval with rho-endpoints has value C, and its Frobenius trace is tau_x(id_x)=dim(x) ([FT1, Proposition 6.18], Eq. (3.8)).
The proof that P_{v,e} is a projection uses this formula and the Hochschild cohomology isomorphisms (3.9)-(3.10); details are in the authors' prior paper.
standard math The cobordism hypothesis and classification of fully extended TFTs are valid (used in Section 4 for Kramers-Wannier duality and Ising duals).
The consequences listed at the end of Section 4 are drawn from the cobordism hypothesis; the paper does not prove them.

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Cite this review

Pith. "Pith review of Discrete quantum systems from topological field theory." pith.science (2026). https://pith.science/paper/SPEPLRSY

@misc{pith2026250605131,
  author       = {Pith},
  title        = {Pith review of: Discrete quantum systems from topological field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPEPLRSY}},
  note         = {Machine review of arXiv:2506.05131}
}

read the original abstract

We introduce a technique to construct gapped lattice models using defects in topological field theory. We illustrate with 2+1 dimensional models, for example Chern-Simons theories. These models are local, though the state space is not necessarily a tensor product of vector spaces over the complex numbers. The Hamiltonian is a sum of commuting projections. We also give a topological field theory construction of Levin-Wen models.

Figures

Figures reproduced from arXiv: 2506.05131 by the authors.

**Figure 1.** A closed surface Y with an embedded graph Λ Let Y be a closed smooth 2-dimensional manifold, and suppose Λ ⊂ Y is a finite graph (or “lattice”) that is smoothly embedded in Y , as depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** A closed (w1, p1)-surface with codimension 2 defects (normal framing not drawn) Let Y be a closed 2-manifold equipped with a (w1, p1)-structure, and fix a finite subset ∆ ⊂ Y as well as a framing3 of the normal bundle to ∆ ⊂ Y ; see [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The 3-dimensional bordism ( ↳ Y, ∆, δ) → (Y, ∆, δ) that computes Hy EndB(1 + b), where 1 + b ∈ B is the defect δ(y) at y ∈ Y , and we take it to be the endomorphism given by the matrix (2.4) hy = 0 0 0 idb . Let (2.5) Hy : H(Y, ∆, δ) −→ H(Y, ∆, δ) be the value of F on the resulting bordism. Elementary standard arguments in topological field theory prove that Hy is a projection operator and that the operators {Hy… view at source ↗

Figures from the paper (20 more)

**Figure 4.** Figure 4: Locality of the state space; see (2.9) (1) If we split Y = Y1∪Z Y2 along a circle, and correspondingly split ∆ = ∆1∪∆2 and δ = δ1∪δ2, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: The 3-dimensional bordism that computes e −τH/ℏ (3) The Hamiltonian operator is a sum of evaluations of F on different objects of the bordism category (allowing defects). On the other hand, there is a single evaluation of F that computes the evolution for imaginary tim…

**Figure 6.** Figure 6: A closed surface with two species 1 + e and 1 + m of point defects We conclude by explaining how to recover the toric code as a special case of our construction. Let G be the finite cyclic group of order two, and let F = FG be 3-dimensional finite G-gauge theory. The m…

**Figure 7.** Figure 7: Construction of (Y, ∆, δ) from (Y,Λ) To verify this claim, use the extension (2.10) FeG : Bord⟨0,1,2,3⟩ −→ Cat⊗ C of G-gauge theory to a fully local theory of unoriented manifolds with codomain the 3-category of tensor categories. This theory assigns the tensor categor…

**Figure 8.** Figure 8: A cylinder with Dirichlet/Neumann (green/red) boundary theory at one end Quite generally, boundary theories produce defects supported on arbitrary stratified manifolds. Apply this to points to redraw [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: we have opted instead to draw the equivalent picture with small disks excised around each defect, and we have colored the resulting boundary circles with Dirichlet and Neumann boundary theories. The semiclassical description of G-gauge theory with Dirichlet and Neumann…

**Figure 10.** Figure 10: Simpler locality along edges in Hopf algebra models [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: A disjoint union of disks over vertices and edges of each disk are colored with the Dirichlet boundary theory ρ (green), and there is are embedded point defects (yellow) at the intersection points of disk boundaries and edges. The boundary circles inherit an orientati…

**Figure 12.** Figure 12: The operator Pv,e This matches the state space of the Levin–Wen model (up to isomorphism). As in the toric code, the Hamiltonian of the Levin–Wen model is a sum of terms attached to vertices and faces. For each vertex v and abutting edge e consider the bordism depicte…

**Figure 13.** Figure 13: The link of the embedded point defect in [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]

**Figure 14.** Figure 14: The square Pv,e ◦ Pv,e We claim4 that Pv,e is a projection operator, i.e., Pv,e ◦ Pv,e = Pv,e. The squared operator is shown in [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 15.** Figure 15: (a) The central component of [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 16.** Figure 16: The Hochschild (co)homology HH0(C) ∼= HH0 (C) Return now to the computation of the bordism in [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

**Figure 17.** Figure 17: The element of HH0 (C) produced in [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]

**Figure 18.** Figure 18: The surface that evaluates to the kernel of P v Hvu The Hamiltonian of the Levin–Wen model is the sum of (3.11) over the vertices and the sum over faces f of a projection operator Hf . The construction of Hf follows [FT1, §7A]. Namely, the vector space TC (S 2 ) is on…

**Figure 19.** Figure 19: The operator id −Hf , including the purple antisphere 4. Reprise: a stat mech model in 1 + 1 dimensions The models in §2 have continuous time. One can also use topological field theory with defects to construct models with discrete time, that is, stat mech models. Thi…

**Figure 20.** Figure 20: A 2-dimensional bordism with embedded graphs The Ising model [C, ID] can be regarded as a 2-dimensional field theory on manifolds equipped with an appropriate embedded graph. A bordism Y : S0 → S1 of this type is depicted in [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]

**Figure 21.** Figure 21: Ising as a boundary of 3-dimensional finite gauge theory The group G acts as a symmetry of the Ising model: left translate all spins simultaneously. Thus Ising can be realized as a boundary theory of the 3-dimensional finite gauge theory FeG that appeared in (2.10); s…

**Figure 22.** Figure 22: A piece of surface colored with ρ, ϵ, δ · ( ( - ⑤ [PITH_FULL_IMAGE:figures/full_fig_p018_22.png]

**Figure 23.** Figure 23: The incoming disk De boundary consequence of 3-dimensional finite electromagnetic duality. Second, one finds duals to Ising models for nonabelian G. These duals are constructed using the cobordism hypothesis. Third, one can make predictions for the gapped phases of th…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Cobordism of nested manifolds
math.AT 2025-12 conditional novelty 6.0 of 10

Nested cobordism of manifolds is, whenever the outer submanifold has a framed normal direction, identical to link cobordism, transferring Wang's link invariants to nested manifolds.

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