{"id":"8036e123-5a44-46e9-b4ea-31e18c4e3da8","arxiv_id":"2412.15359","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Certain homology classes in Sp2 and in sphere bundles over spheres are representable by immersions but not by embeddings, with unavoidable nontrivial double points.","lead":"Mathematicians have long asked whether every shape that can be drawn with self-intersections can also be drawn without them. This paper finds the first classes where self-intersections are unavoidable: some homology classes are immersed but not embedded.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weight-bearing algebraic error: in Theorem C the class p_1^2-2p_2 reduces mod 2 to w_2^4, not w_4^2, so the obstruction identity for Theorem B is unproved.","rationale":"The reader's weakest assumption concerned the identification of the 7-skeleton K as a Thom space via the J-homomorphism in Proposition 2.1. That step is likely sound, as the Blakers-Massey generator is known to lie in the image of J: \\pi_3(SO(3)) \\to \\pi_6(S^3). The more pressing issue is an internal inconsistency in the algebraic proof of Theorem C: the same integral class p_1 is reduced mod 2 in two incompatible ways within a single calculation. Since Theorem C provides the obstruction that rules out embeddings in Theorem B, and Theorem B is one of the two families of examples highlighted in the abstract and the reader's strongest claim, this computational error is load-bearing. The central existence result of the paper (Theorem A) may still be correct, but the published proof of Theorem B is incomplete as written. A targeted recomputation will settle whether the error is merely typographical or actually invalidates the obstruction. Therefore the appropriate verdict is CONDITIONAL: accept subject to verification of the mod 2 identity in Theorem C, or withdrawal of the H_{13}(N) claims.","tokens_in":15425,"tokens_out":46771,"duration_ms":357140,"concrete_test":"Recompute the left-hand side of the equation in Theorem C's proof with \\tilde\\beta := w_{11} w_2^4 (and \\tilde\\delta := w_{11} w_2^2), using the same Wu/Cartan expansions, and compare with the right-hand side w_{11}(w_{10}w_3 + w_9w_2^2) in H^{24}(MSO_{11};Z_2). If equality fails, Theorem C is false; if equality holds, the paper contains only a typographical error and Theorem B survives.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem C (Section 3), the universal classes are \\tilde\\beta = t(p_1^2 - 2p_2) and \\tilde\\delta = t p_1. Since for an oriented bundle p_i \\equiv w_{2i}^2 (mod 2), one has p_1 \\equiv w_2^2, hence p_1^2 - 2p_2 \\equiv p_1^2 \\equiv w_2^4 (mod 2). The paper computes Sq^5 \\tilde\\beta = Sq^5(w_{11} w_4^2), while for \\tilde\\delta it correctly uses Sq^9 \\tilde\\delta = Sq^9(w_{11} w_2^2). These two usages require p_1 \\equiv w_4 and p_1 \\equiv w_2^2 simultaneously, which is false in H^*(BSO_{11};Z_2) (where w_4 is independent of w_2^2). Because the displayed identity in H^{24}(MSO_{11};Z_2) is the only justification for the obstruction in Theorem C, the non-embeddability part of Theorem B for H_{13}(N) is not established as written. This is a concrete computational error, not merely an omitted proof; the identity must be rechecked with \\tilde\\beta \\equiv w_{11} w_2^4.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15735,"tokens_out":24934,"duration_ms":208138,"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":[{"comment":"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.","section":null}],"minor_comments":[{"comment":"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].","section":"Throughout"},{"comment":"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":"References"},{"comment":"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":"Section 3"},{"comment":"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.","section":"Section 5.2"},{"comment":"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.","section":"Theorem 5.3"}],"recommendation":"major_revision","confidential_remarks":"The reader's report missed a concrete algebraic error in the proof of Theorem C: the mod 2 reduction of p_1^2 − 2p_2 is w_2^4, not w_4^2. This leaves Theorem C unproven and the Section 3 proof of Theorem B's non-embeddability invalid, although Section 5 gives an independent route for the specific manifolds. The authors should be asked to correct the computation or revise the statement of Theorem C. The rest of the paper, in particular Theorems A and D and the double-point results, appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper does give the first documented immersed-but-not-embedded homology classes: the generators of H_7(Sp2) are immersed (new) and not embedded (Bohr-Hanke-Kotschick), and the double-point class is nonzero. Second, the proof of Theorem B has a concrete computational error in Theorem C that invalidates the non-embeddability statement for H_13(N) as written.\n\nWhat's good: Theorem A is solid. Proposition 2.1 is a clean argument using the 7-skeleton as a Thom space and J-homomorphism surjectivity. The double-point analysis in Section 5 is a nice geometric complement, and Theorem D (Steenrod representable but not immersed) is well-supported from Grant-Szűcs. The exposition is clear and the question is well contextualized.\n\nWhere it breaks: In the proof of Theorem C, ~β = t(p_1^2 - 2p_2) is treated mod 2 as t w_4^2, but the correct reduction is t w_2^4 (since ρ(p_i)=w_{2i}^2). The same proof correctly uses ρ(p_1)=w_2^2 for ~δ. You can't have both. The displayed identity in H^24(MSO_11;Z_2) is the only justification for the obstruction, and the 'reader can easily check' hides the issue. Unless the identity can be re-established with the correct term, Theorem B's non-embeddability part is unproved.\n\nThat's a serious defect in a load-bearing result, but it's a local algebraic slip, not a conceptual failure. The paper's approach is sound, and I'd bet the computation can be repaired—the right-hand side is simple, and the left-hand side with the correct term might still sum to it. But as is, the paper overstates what it proves.\n\nBottom line: this deserves a serious referee, but the referee must verify Theorem C. If the identity fails, Theorem B should be split off or the paper published with only Theorems A and D. I'd tell the editor to treat the computation as the review's main task.","headline":"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.","tokens_in":16218,"tokens_out":6040,"would_cite":false,"duration_ms":49652,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R42","57R40","55S10","57R20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["immersions","embeddings","homology classes","Steenrod representability","Thom spaces","double points","symplectic group Sp(2)","Postnikov towers"],"falsifier":"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.","tokens_in":15260,"feed_emoji":"","tokens_out":13353,"duration_ms":74629,"temperature":0.7,"pith_summary":"The paper answers a foundational question about how finely the regularity of a representing map distinguishes homology classes: it gives the first documented examples of closed oriented manifolds with homology classes that are representable by immersions but not by embeddings. The main examples are the generators of $H_7(\\mathrm{Sp}_2)$ and the generators of $H_{13}(N)$, where $N$ is a linear $S^{11}$-bundle over $S^{13}$ with nonzero $\\mathrm{Sq}^2$ in degree 11. For these classes, every self-transverse immersion representing the class has double points that represent a nonzero homology class in the source manifold, so the failure to embed is not an artifact of a particular immersion. The paper further shows that the hierarchy is strict in the other direction as well: there are Steenrod representable 2-torsion classes in high-dimensional manifolds that are not immersed.","feed_headline":"First immersed-but-not-embedded homology classes found","feed_subtitle":"Generators of H7(Sp2) and H13(N) are representable by immersions but never by embeddings, settling a question of Zhenhua Liu.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Shows the generator of $H_7(\\mathrm{Sp}_2)$ is not embedded; the non-embeddability half of Theorem A that this paper complements with immersability.","marker":"[1]"},{"why":"Supplies the stable splitting $\\Sigma^2\\mathrm{Sp}_2 \\simeq (S^5 \\cup \\Sigma^2\\omega' e^9) \\vee S^{12}$, which lets the immersion problem be reduced to the 7-skeleton.","marker":"[17]"},{"why":"Gives the lemma that the Thom space of a rank-$q$ bundle over $S^{p+1}$ with clutching function $\\alpha$ is the mapping cone of $J(\\alpha)$, used to identify the skeleton as a Thom space.","marker":"[15]"},{"why":"Provides the minimal cell structure $\\mathrm{Sp}_2 \\simeq (S^3 \\cup_{\\omega'} e^7) \\cup e^{10}$ and the Blakers–Massey generator $\\omega'$, the starting point for the Thom-space identification.","marker":"[11]"},{"why":"Classifies linear $S^{11}$-bundles over $S^{13}$ and identifies the unique (up to connected sum with homotopy spheres) total space with non-zero $\\mathrm{Sq}^2$ on $H^{11}$, the manifold $N$ in Theorem B.","marker":"[10]"},{"why":"Supplies the Wu and Cartan formulae and the $P^i_p$ identities for Thom classes used to derive the obstruction equation in Theorem C.","marker":"[16]"},{"why":"Original source of the identities $\\rho_5(\\beta) = P^1_5 t$, $\\rho_3(\\beta') = P^2_3 t$, $\\rho_3(\\delta) = P^1_3 t$ for the universal Thom class, used in Theorem C.","marker":"[29]"},{"why":"Provides the mod-2 obstruction $\\beta_2 \\mathrm{Sq}^I$ for immersions used in Theorem D to build Steenrod-representable classes that are not immersed.","marker":"[6]"},{"why":"Provides the Whitney self-intersection formula $f^*(x) = e(\\nu_f) + m_2(f)$ used in Section 5 to prove double-point classes are non-zero.","marker":"[8]"},{"why":"Massey–Peterson theorem that $w_{11}(\\nu_M) = 0$ for any closed 13-manifold, used in the proof of Theorem 5.3 for the double points in $N$.","marker":"[MP]"}],"fun_headline_variants":["Immersed not embedded: first examples found","Homology that immerses but never embeds","Double points prove immersion-only homology","Steenrod representable? Still not immersed","Answering Liu: immersion without embedding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Immersed not embedded: first examples found","Homology that immerses but never embeds","Double points prove immersion-only homology","Steenrod representable? Still not immersed","Answering Liu: immersion without embedding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000223,"raw_usage":{"total_tokens":1434,"prompt_tokens":898,"completion_tokens":536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":470}},"tokens_in":514,"tokens_out":536,"duration_ms":3516,"temperature":1.0,"reasoning_tokens":470,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:29:59.135752+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the generator of $H_7(\\mathrm{Sp}_2)$ is not embedded; the non-embeddability half of Theorem A that this paper complements with immersability."},{"cited_title":"Mimura, On the number of multipications on SU (3) and Sp2, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the stable splitting $\\Sigma^2\\mathrm{Sp}_2 \\simeq (S^5 \\cup \\Sigma^2\\omega' e^9) \\vee S^{12}$, which lets the immersion problem be reduced to the 7-skeleton."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the lemma that the Thom space of a rank-$q$ bundle over $S^{p+1}$ with clutching function $\\alpha$ is the mapping cone of $J(\\alpha)$, used to identify the skeleton as a Thom space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the minimal cell structure $\\mathrm{Sp}_2 \\simeq (S^3 \\cup_{\\omega'} e^7) \\cup e^{10}$ and the Blakers–Massey generator $\\omega'$, the starting point for the Thom-space identification."},{"cited_title":"Ishimoto, On the classification of (n − 2)-connected 2n-manifolds with torsion free homol- ogy groups, Publ","cited_arxiv_id":null,"evidence_quote":"Classifies linear $S^{11}$-bundles over $S^{13}$ and identifies the unique (up to connected sum with homotopy spheres) total space with non-zero $\\mathrm{Sq}^2$ on $H^{11}$, the manifold $N$ in Theorem B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Wu and Cartan formulae and the $P^i_p$ identities for Thom classes used to derive the obstruction equation in Theorem C."},{"cited_title":"W u, On Pontrjagin classes","cited_arxiv_id":null,"evidence_quote":"Original source of the identities $\\rho_5(\\beta) = P^1_5 t$, $\\rho_3(\\beta') = P^2_3 t$, $\\rho_3(\\delta) = P^1_3 t$ for the universal Thom class, used in Theorem C."},{"cited_title":"Grant, A","cited_arxiv_id":null,"evidence_quote":"Provides the mod-2 obstruction $\\beta_2 \\mathrm{Sq}^I$ for immersions used in Theorem D to build Steenrod-representable classes that are not immersed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Whitney self-intersection formula $f^*(x) = e(\\nu_f) + m_2(f)$ used in Section 5 to prove double-point classes are non-zero."}],"review_version":1}