REVIEW 1 major objections 5 minor 3 cited by
Immersed but not embedded homology classes
T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read This paper proves the first known examples of closed oriented manifolds with homology classes representable by immersions but not by embeddings, answering a question of Zhenhua Liu.
desk verdict Smart, readable paper that answers Liu's question for H_7(Sp2), but the proof of Theorem B rests on a mod-2 algebra identity that is wrong as written. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central machinery is the pair of universal Thom classes. A codimension-$k$ class $z$ in an oriented $n$-manifold $N$ with Poincaré dual $x \in H^k(N)$ is embedded exactly when the map $N_+ \to K(\mathbb{Z},k)$ representing $x$ lifts through the Thom class $t_k \colon MSO_k \to K(\mathbb{Z},k)$, and immersed exactly when it lifts through its extension $\tilde{t}_k \colon QMSO_k \to K(\mathbb{Z},k)$ to the free infinite loop space $QMSO_k = \lim_\ell \Omega^\ell \Sigma^\ell MSO_k$. The proof that the generators are immersed reduces the problem to the low-dimensional skeleton: because $\mathrm{Sp}_2 \simeq (S^3 \cup_{\omega'} e^7) \cup e^{10}$ with $\omega'$ in the image of the J-homomorphism, the 7-skeleton is the Thom space of a rank-3 vector bundle over $S^4$, and the stable splitting $\Sigma^2\mathrm{Sp}_2 \simeq \Sigma^2 K \vee S^{12}$ transfers the Thom class to a stable map $\mathrm{Sp}_{2+} \rightsquigarrow MSO(3)$; the same pattern with the 13-skeleton $S^{11} \cup_\eta e^{13} = \mathrm{Th}(\zeta)$ handles $N$. Non-embeddability is detected by cohomology operations: the Steenrod power $P^1_3$ kills the relevant class on $MSO(3)$, and Theorem C packages a mod-2 equation (involving $\mathrm{Sq}^4$, $\mathrm{Sq}^5$, $\mathrm{Sq}^8$, $\mathrm{Sq}^9$ applied to auxiliary classes) that any embedded codimension-11 class must satisfy but the classes in $N$ do not. The double-point conclusions follow from Whitney's self-intersection formula $f^*(x) = e(\nu_f) + m_2(f)$, where $m_2(f)$ is the Poincaré dual of the homology class of the double-point manifold in the source.
What would settle it
Find a closed oriented 7-manifold $M$ and a self-transverse immersion $f \colon M^7 \looparrowright \mathrm{Sp}_2$ with $f_*[M]$ a generator of $H_7(\mathrm{Sp}_2)$ whose double-point class $m_2(f) \in H^3(M)$ is zero; Theorem 5.2 asserts this is impossible, so such an immersion would disprove the double-point rigidity (though not necessarily the mere existence of an immersed representative). Alternatively, compute the k-invariant obstruction to lifting $\mathrm{Sp}_{2+} \to K(\mathbb{Z},3)$ through $QMSO(3)$; a nonzero obstruction would show the generator is not immersed, contradicting Theorem A.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that the classical chain of regularities for representing an integral homology class — embedded, immersed, Steenrod representable — has strict inclusions at both ends, and specifically that 'immersed but not embedded' really occurs. For the compact Lie group $\mathrm{Sp}_2$, whose homology is that of $S^3 \times S^7$, each generator of $H_7(\mathrm{Sp}_2)$ is shown to be the image of the fundamental class under an immersion from a closed oriented 7-manifold, while no embedding can represent it; the non-embeddability was already known, and the new content is the existence of the immersion. The same statement holds for the generators of $H_{13}(N)$ for $N$ a linear $S^{11}$-bundle over $S^{13}$ with $\mathrm{Sq}^2$ nonzero on $H^{11}(N;\mathbb{Z}_2)$. In both cases, the paper proves a stronger rigidity: any representing self-transverse immersion has a non-trivial double-point homology class in the source manifold, detected by the Whitney self-intersection formula. Finally, Theorem D provides closed oriented $n$-manifolds for all $n \geq 27$ with 2-torsion classes in dimension $n-4$ that are Steenrod representable but not immersed, so Steenrod representability does not imply immersability.
Load-bearing premise
The immersability proof hinges on a specific structural fact about $\mathrm{Sp}_2$: its 7-skeleton is a Thom space — the total space of a 3-dimensional vector bundle over the 4-sphere — and this is true only because a certain classical map, the J-homomorphism, is surjective; if this coincidence failed, the stable map that produces the immersion would not exist.
Editorial extensions
If this is right
- Corollary 1.2: for every $n \geq 10$, a closed oriented $n$-manifold carries an $(n-3)$-dimensional homology class that is immersed but not embedded.
- The three regularity notions are pairwise distinct: Steenrod representable classes need not be immersed (Theorem D), and immersed classes need not be embedded (Theorems A and B).
- In the examples, any representing self-transverse immersion has non-zero double-point homology class in the source, so the non-embeddability is witnessed by unavoidable double points, not by a poor choice of immersion.
- Theorem C provides a new general obstruction: any embedded codimension-11 class in a closed oriented manifold must satisfy a specific mod-2 cohomology equation, which can be checked in other 24-manifolds.
- The construction shows that when the attaching map of a skeleton lies in the image of the J-homomorphism, the skeleton becomes a Thom space and immersability of the dual class follows, suggesting a template for further examples.
Reading between the lines
- The same Thom-space template could produce immersed-but-not-embedded classes in other Lie groups or sphere bundles whose attaching maps are J-images and whose cohomology has an operation like $P^1_3$ or $\mathrm{Sq}^2$ obstructing embeddings; the paper's examples may be the first members of a larger family.
- The double-point non-triviality raises a quantitative question the paper leaves open: what is the minimum number of double points among immersions representing these classes, and how does it relate to the Hatcher–Quinn invariant mentioned in Remark 5.4?
- Theorem D's thickening construction could potentially be adapted to lower codimensions using other Eilenberg–MacLane spaces or other excess-$k$ admissible sequences, although the paper notes the method does not reach codimension 3.
- Because immersability is proved stably, the explicit formal immersions of Section 5.2 may be homotoped to actual immersions with controlled geometry, opening the door to studying the regular homotopy classes of these representatives.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses the representability of integral homology classes by embeddings versus immersions. It claims the first documented examples of immersed but not embedded homology classes: the generators of H_7(Sp_2) (Theorem A) and the generators of H_13(N) for a certain linear S^11-bundle N over S^13 (Theorem B). It also proves that for any self-transverse immersion representing these classes, the double-point class in the source is nonzero, and it constructs Steenrod representable classes that are not immersed (Theorem D). The main techniques are Thom's representability theorems, stable homotopy methods, Postnikov towers for MSO_k and QMSO_k, and Whitney's self-intersection formula.
Significance. If the proofs are correct, the paper answers a question of Zhenhua Liu and establishes a genuine distinction between immersion representability and embedding representability of homology classes. The double-point nonvanishing theorems are a nice addition, and Theorem D usefully shows that Steenrod representability does not imply immersability. The constructions are explicit and the paper makes good use of external benchmarks such as Bohr-Hanke-Kotschick and Grant-Szűcs. However, the proof of Theorem C contains a concrete algebraic error in the mod 2 reduction of a Pontryagin class combination; as a result the general obstruction theorem is not established as written.
major comments (1)
- The mod 2 reduction of the class β̃ = t(p_1^2 − 2p_2) is computed incorrectly. For an oriented bundle, p_i ≡ w_{2i}^2 (mod 2), so p_1 ≡ w_2^2 and p_2 ≡ w_4^2; hence p_1^2 − 2p_2 ≡ w_2^4 (mod 2), not w_4^2. Therefore the displayed computation Sq^5 β̃ = Sq^5(w_11 w_4^2) should instead involve w_11 w_2^4. With the correct reduction, the term Sq^5(w_11 w_2^4) contributes a monomial w_11 w_5 w_2^4 that is not present on the right-hand side w_11(w_10 w_3 + w_9 w_2^2), and no other term in the sum can cancel it. Thus the claimed identity in H^24(MSO_11; Z_2) is not established, and the proof of Theorem C is invalid. Since Theorem C is used in the Section 3 proof of the non-embeddability part of Theorem B, that proof path also fails; the alternative argument in Section 5 (Theorem 5.3) may establish the specific non-embeddability claim, but the general obstruction theorem needs a corrected computation or a revised statement.
minor comments (5)
- [Throughout] There are several typos: 'W e' in the abstract, 'anlaysis' in Section 1.1, 'decribed' and 'repsectively' in Section 3, and 'multipications' in reference [17].
- [References] The reference '[MP]' is cited in the text but is not integrated into the numbered bibliography; it appears as an unnumbered entry. This should be regularized.
- [Section 3] The phrase 'as the reader can easily check' is inappropriate given the erroneous identity; after correcting the computation, the authors should either spell out the verification or provide a reproducible calculation.
- [Section 5.2] The notation 'V' is used in the sentence 'Here the double points are represented by a section S^2 → V' but V is not defined. Please clarify.
- [Theorem 5.3] The statement of Theorem 5.3 is immediately followed by '□ Proof.' and the proof is attached without a clear break; the formatting should be corrected.
Circularity Check
No significant circularity: the new immersed-but-not-embedded claims are built on external theorems and independent stable-map constructions, not on the paper's own conclusions.
full rationale
The derivation chain is not circular in any way that reduces a claimed prediction to its inputs. The non-embeddability half of Theorem A is explicitly quoted from Bohr–Hanke–Kotschick [1], an external published theorem; the novel immersability half in Proposition 2.1 is constructed directly from the cell structure Sp2 ≃ (S3 ∪_ω' e7) ∪ e10, the splitting Σ2Sp2 ≃ Σ2K ∨ S12, and the Thom-space identification K ≃ Th(ξ), with the relevant J-homomorphism surjectivity cited to independent sources. Theorem B similarly separates non-embeddability, which is derived from the universal obstruction identity in Theorem C, from immersability, which is proved by showing N is stably parallelizable and that its 13-skeleton is a Thom complex. Theorem C's obstruction identity is checked in the universal example H*(MSO_11;Z2) using Wu–Cartan formulas and the Wu Wen-Tsun theorem cited from Milnor–Stasheff; it is not assumed as an input and then re-exported to the examples. Theorem D uses the prior Grant–Szűcs paper [6], one author of which is the present second author, but that paper is an independent published theorem with its own proof, and the current manifold construction via thickenings of K(Z2,3)^(14) and Poincaré–Lefschetz duality is independent of the target claim. No fitted parameters are renamed as predictions, no universal class is defined in terms of the examples, and no uniqueness theorem is imported from the authors' own prior work. The possible algebraic error in the mod 2 reduction of p_1^2 − 2p_2 raised by a skeptical reading would be a correctness flaw, not circularity, because the identity is still being verified rather than assumed. Overall the paper's central claims have independent content.
Assumptions & free parameters
assumptions (6)
- standard math J-homomorphism J: π3(SO(3)) → π6(S3) is surjective, so the 7-skeleton of Sp2 is the Thom space of a rank 3 bundle over S4.
- domain assumption Sp2 has minimal cell structure Sp2 ≃ (S3 ∪_{ω'} e7) ∪ e10, with ω' the Blakers-Massey generator.
- domain assumption The 24-manifold N has cell structure N ≃ (S11 ∪_η e13) ∪ e24, with η generating π12(S11).
- domain assumption The bundle ξ over S13 used to construct N is stably trivial, so N is stably parallelizable.
- standard math Every 2-torsion integral homology class is Steenrod representable.
- domain assumption Grant-Szűcs obstruction theory: if a mod 2 homology class is immersed, certain Steenrod operations on its Poincaré dual are reductions of integral classes.
Cite this review
Pith. "Pith review of Immersed but not embedded homology classes." pith.science (2026). https://pith.science/paper/3N2GUMJR
@misc{pith2026241215359,
author = {Pith},
title = {Pith review of: Immersed but not embedded homology classes},
year = {2026},
howpublished = {\url{https://pith.science/paper/3N2GUMJR}},
note = {Machine review of arXiv:2412.15359}
}
read the original abstract
We provide the first documented examples of immersions of closed oriented manifolds which are not homologous to embeddings, thus answering a question posed by Zhenhua Liu. In these examples we show that for any representing self-transverse immersion the double points must represent a non-trivial homology class in the source manifold. We also provide examples of Steenrod representable integral homology classes which are not represented by immersions.
Forward citations
Cited by 3 Pith papers
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