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Exotic families of embeddings

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Pith's one-line read The paper proves that 3-manifolds admit topologically trivial but smoothly distinct families of embeddings in 4-manifolds, with infinitely many examples in $S^4$ sharing diffeomorphic complements.

desk verdict Theorem 1.1 is solid and self-contained; Theorem 1.3 is plausible but uncheckable without the unpublished companion [7]. read the letter →

arxiv 2501.12673 v2 pith:KLB7GLWS submitted 2025-01-22 math.GT

Dave Auckly , Daniel Ruberman This is my paper · ORCID
classification math.GT MSC 57M2557Q60
keywords exoticembeddings3-manifoldsin4-manifoldsfamilyDonaldsoninvariantscorkshomologyspherestopologicalisotopystabilizationembeddingspaces
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

The paper constructs topologically trivial yet smoothly nontrivial families of embeddings of 3-manifolds into 4-manifolds. Its first theorem produces, for infinitely many integer homology 3-spheres $Y_n$, infinitely many smooth embeddings $j_{p,n}:Y_n\to S^4$ that are topologically ambient isotopic and have diffeomorphic complements, yet no diffeomorphism of $S^4$ carries one to another unless $p=q$. Its main theorem builds, for any 3-manifold $Y$ embedded in $E(2)\#\mathbb{CP}^2$ away from a fiber, 4-manifolds $Z^r_k$ supporting spherical families of embeddings of $Y$ whose homology classes generate an arbitrarily large rank summand of the kernel of the map from smooth to topological embedding spaces. If correct, this shows that smooth 4-dimensional embedding spaces carry nontrivial homotopy in arbitrarily high degrees that is invisible to topological isotopy, and gives a general stabilization-plus-gauge-theory recipe for producing such exotica.

What carries the argument

The central mechanism is a recursive commutator construction. One starts with a diffeomorphism $\varphi:X\#U\to X'\#U$ that induces the same homology isomorphism as a homeomorphism $\psi$, and a reflection $f$ of the stabilising summand $U$; the commutator $\alpha_0=[\varphi,1\#f]$ is an exotic diffeomorphism, and higher spherical families $\alpha_k:S^k\to\mathrm{Diff}(Z_k)$ are built by $\alpha_{k+1}=[F_k,1_{Z_k}\#_T \alpha]$, where $F_k$ is a smooth contraction of the stabilised family. The family of embeddings is then obtained by composing a fixed codimension-one embedding $\iota:Y\to V$ with these diffeomorphisms and passing through the family submanifold sum $Z\rr J\ss$, whose preferred component carries the family Donaldson invariant $D^{H^{k+1}\mathrm{emb}}_{c_q,\beta}$. Computability rests on the anti-holomorphic blow-up formula $|D^{\pi^{k+1}}_{c+\nu}(F(1_Z\#f)F^{-1})|=2|D^{\pi^k}_c(\beta)|$ and on the identity $D^{H^{k+1}\mathrm{emb}}_{c_q,\beta}(J^{k+1}_p)=\gamma'_p\delta_{pq}$, both taken from the authors' manuscript [7].

What would settle it

Falsifying Theorem 1.1 would require exhibiting a diffeomorphism $\psi:S^4\to S^4$ with $\psi\circ j_{p,n}=j_{q,n}$ for some $p\neq q$; falsifying Theorem 1.3 would require computing the degree-zero family Donaldson invariant $D_{c_q,\beta}(Z\rr J^{1}_p\ss)$ for $p\neq q$ and finding a nonzero value, since Lemma 4.7 predicts zero. Both checks are concrete and do not depend on choices left unspecified in the paper.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that stabilisation turns a finite collection of exotic smooth structures on homeomorphic 4-manifolds into spherical families of embeddings of any 3-manifold that embeds in an elliptic K3 surface blown up once, disjointly from a fiber. Theorem 1.1 gives, for infinitely many integer homology 3-spheres $Y_n$, embeddings $j_{p,n}:Y_n\to S^4$ whose complements are diffeomorphic and that are topologically ambient isotopic, yet no diffeomorphism $\psi$ of $S^4$ with $\psi\circ j_{p,n}=j_{q,n}$ exists unless $p=q$. Theorem 1.3 states that for any such $Y$ and any $r>0$, $k\ge 0$ there are simply connected 4-manifolds $Z^r_k$ with spherical classes $J^{j+1}_p\in H_{j+1}(\mathrm{emb}(Y,Z^r_k))$, $j\le k$, $j\equiv k\pmod 2$, $p=1,\dots,r$, generating a rank $r$ summand of the kernel of the map induced by the inclusion of smooth into locally flat topological embeddings; the classes are also trivial in the topological embedding space and after summing with $S^2\times S^2$ when $Y$ is a homology sphere.

Load-bearing premise

The load-bearing premise is that the unpublished manuscript [7] actually defines the family Donaldson invariants and proves the two identities used here, namely the Kronecker-delta computation and the anti-holomorphic blow-up formula, and the paper says the stabilization step that produces the fixed target manifolds is written out only there; if that manuscript is wrong or unavailable, the proof of Theorem 1.3 cannot be checked.

Editorial extensions

If this is right

  • For any $r$, the kernel $\ker[H_{j+1}(\mathrm{emb}(Y,Z^r_k))\to H_{j+1}(\mathrm{emb}_{\mathrm{top}}(Y,Z^r_k))]$ contains a free abelian summand of rank $r$, so the smooth embedding space is arbitrarily more complicated than the topological one in infinitely many homotopy degrees of one parity.
  • The embeddings $j_{p,n}:Y_n\to S^4$ have diffeomorphic complements for all $p$, so the complement of a 3-manifold in $S^4$ does not determine its smooth isotopy class, even up to ambient diffeomorphism.
  • The construction applies verbatim to any 3-manifold embedding disjointly from a fiber in $E(2)\#\mathbb{CP}^2$, and the authors note it adapts to boundaries of nuclei, Brieskorn spheres, the Akbulut-Mazur cork boundary, and blowups of $E(n)$, so the phenomenon is not confined to one example.
  • When the marking ambiguity of a family of submanifolds is controlled, as for $\Sigma(2,3,11)$ or for asymmetric hyperbolic 3-manifolds, the same invariants distinguish families of submanifolds rather than only families of embeddings.
  • The rank is finite only because the Donaldson parameter set $\widehat{C^k_Z}$ is finite; the paper identifies the Seiberg-Witten analogue, with its infinite spin$^c$ structures, as the route to infinite-rank summands, which it treats in a forthcoming paper.

Reading between the lines

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

  • If the quoted family Donaldson invariants behave as claimed, the same stabilisation-plus-family-invariant recipe should work for any separating codimension-one embedding into a 4-manifold with a non-trivial gauge-theoretic invariant; the paper's Question 6.2 makes this dependence explicit, and a positive answer would vastly broaden the class $Y$.
  • The marking-multiset device of Section 7 suggests a general transfer principle: whenever $\mathrm{Diff}(Y)$ has finitely many path components, any homomorphism from $\pi_*(\mathrm{emb}(Y,Z))$ to an abelian group descends to $\pi_*(\mathrm{Sub}(Y,Z))$ by summing over homotopy markings, so embedding-space exotica can be converted into submanifold exotica whenever the marking set is finite.
  • Because the paper locates all of its exotica in a single Akbulut cork twist, a future parameterized cork theorem would have to reconcile the intuition that exotic behaviour localizes with evidence that some families survive arbitrary stabilizations; testing this tension is a natural next step.
  • An immediate testable reduction would be to verify the unpublished computations for the $k=0$, $r=1$ case in explicit handle diagrams of $Z^1_0$: if the family Donaldson invariant of $J^1_1$ computed directly equals $\gamma'_1$ and vanishes for the other classes, the bridge from [7] to Theorem 1.3 would be independently checked.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. Theorem 1.1 gives infinitely many integer homology 3-spheres Y_n, each admitting infinitely many embeddings j_{p,n}: Y_n -> S^4 whose complements are diffeomorphic, which are topologically ambient isotopic, and are such that no diffeomorphism of S^4 carries j_{p,n} to j_{q,n} for p ≠ q. Theorem 1.3 asserts that for any 3-manifold Y embedding in Ep(2)#CP^2 disjointly from a fiber, and any r>0, k≥0, there are fixed simply connected 4-manifolds Z^r_k and spherical classes J_p^{j+1} in H_{j+1}(emb(Y,Z^r_k)) generating a rank-r summand of the kernel of the map to H_{j+1}(emb_top(Y,Z^r_k)), with a homotopy-group analogue when Y is a homology sphere. Section 7 offers a method for converting these families of embeddings into families of submanifolds. The proof of Theorem 1.1 is essentially self-contained modulo standard quoted results on Gompf corks, Tange twisted doubles, Casson surgery formulas, and JSJ decomposition. The proof of Theorem 1.3, however, depends at several explicit points on the authors' unpublished companion manuscript [7] for the definition and computation of family Donaldson invariants.

Significance. If Theorem 1.3 and the Section 7 submanifold conclusions hold, the paper establishes a significant new phenomenon: high-dimensional, topologically trivial families of embeddings of 3-manifolds into fixed 4-manifolds, with linear-algebraic rank control of the kernel comparing smooth and topological embedding spaces. Theorem 1.1 is a clean and checkable result that already answers the motivating question about non-isotopic embeddings with diffeomorphic complements. The paper also gives useful conceptual recipes—submanifold sums, stabilization, and conversion from embedding families to submanifold families—that are likely to be reusable. The main weakness is verifiability: the detection machinery for Theorem 1.3 is not present in this manuscript, being deferred to the authors' own unpublished work [7]. This is an incompleteness rather than a demonstrated error, but it is load-bearing for the paper's central theorem.

major comments (3)
  1. [§4, Lemma 4.7] The proof of Theorem 1.3 cannot be checked from this manuscript alone because the family Donaldson invariant D^{H^{k+1} emb}_{c,β} is only described informally in §4, with the statement that rigorous transversality details are given in [7]. The key identity D^{H^{k+1} emb}_{c_q,β}(J_p^{k+1}) = γ'_p δ_{pq} in Lemma 4.7 is the sole mechanism for proving that the classes J_p^{k+1} generate a rank-r summand, and it is quoted from the unpublished [7]. Since the definition, the transversality setup, and the computation of this identity are all deferred, the linear independence claim in Theorem 1.3 is not verifiable as written. If this identity carries an unstated hypothesis, or if the invariant is not defined for the manifolds Z^r_k constructed here, the theorem collapses.
  2. [§5, Theorem 5.4] Two load-bearing computations are quoted from [7] without derivation. First, the assertion that the degree-zero Donaldson invariants of the manifolds X_p^r satisfy (q_{X_p^r,c_q}) = 4(p−1)^r times the identity matrix appears in §5 with citation [7] but no computation. Second, Theorem 5.4, the anti-holomorphic blow-up formula |D^{π^{k+1}}_{c+ν}(F(1_Z # f)F^{-1})| = 2|D^{π^k}_c(β)|, is imported from [7] and is required to propagate non-zero invariants through the recursive construction. No proof or precise hypotheses on b_2^+, parity, the contraction F, or the admissible data c,ν,β are given. These computations are central: without them the constants γ'_p and the non-vanishing of the family invariants for the constructed embeddings are unsupported.
  3. [§6] The stabilization step needed to make the target manifold independent of k is asserted but not proved. The text states that for an upper bound n one can choose manifolds V_k so that Z^1_k #_{Σ_1} V_k ≅ Z^2_n independent of k, and then says 'This argument is written out in more detail in [7].' This identification is essential: Theorem 1.3 is a statement about fixed manifolds Z^r_k, while the construction up to that point produces growing manifolds Z^1_k. The additional claim that the stabilization changes the invariants only by a non-zero factor is also not justified here. Without a complete proof of this stabilization, the stated fixed-target form of Theorem 1.3 is not established.
minor comments (4)
  1. [§4, Lemma 4.4] The statement of Lemma 4.4 appears garbled: it reads 'The classes [J_p^{k+1}] and [J_0^{k+1}] π_{k+1}(emb_top(Y,Z^1_k))' and seems to be missing an equality and the phrase 'are equal in'. Please correct the statement, since this lemma is used for topological triviality of the constructed families.
  2. [§5] The notation in §5 is confusing: the symbols X and Y are reused for both manifolds and for the cork-twist partner, and then one reads 'We use X_1 = X, X = Y, and Z_0 = Y # M'. This makes the construction harder to follow; a consistent renaming would improve readability.
  3. [References] Because Theorem 1.3 depends essentially on [7], the reference 'manuscript in preparation' is not adequate for verification. The authors should either include the needed definitions and computations in this paper or provide a publicly available version of [7] with precise statements of the quoted results.
  4. [§7, Example 7.3] The computation of Diff(Y) for the Akbulut-Mazur cork boundary uses SnapPy and is explicitly described as not completely rigorous; the text says a rigorous method appears in forthcoming work. Since this example is used to illustrate the submanifold discussion rather than to prove Theorem 1.3 or 1.1, this is acceptable, but the provisional nature of the computation should be stated more prominently in the example.

Circularity Check

4 steps flagged · score 7.0 of 10

Theorem 1.3 rests on an unverified self-citation chain: the family Donaldson invariant, its δ_{pq} evaluation, the anti-holomorphic blow-up formula, and the final stabilization are all deferred to the authors' unpublished companion [7].

  1. self citation load bearing [Section 5, 'Specific choices', paragraph after defining c_q]
    "Furthermore, the Donaldson invariants satisfy the assumption that pqXrp,cqq is 4p´1qr times the identity matrix [7]."

    This matrix identity is the entire invariant input for Lemma 4.7: the conclusion that the classes J_p^{k+1} generate a rank-r summand uses exactly the diagonal form γ'_p δ_pq. The computation is not performed in this paper; it is cited to [7], an unpublished manuscript by the same two authors. Thus the central non-vanishing result in Theorem 1.3 rests on an unverified self-citation rather than on a derivation contained in this manuscript.

  2. self citation load bearing [Section 5, 'Specific choices', paragraph introducing family Donaldson invariants]
    "In [7] the degree zero Donaldson invariant is extended to an invariant for families of diffeomorphisms."

    The invariant that supplies the non-zero evaluations is not constructed in this paper. The text gives an informal chain definition and then says that a rigorous transversality version is described in [7]. Since [7] is the authors' own in-preparation manuscript, the detection mechanism for all of the family phenomena in Theorem 1.3 is imported from an inaccessible self-citation, not independently verifiable here.

2 more flagged steps
  1. self citation load bearing [Section 5, Theorem 5.4 (Anti-holomorphic blow-up)]
    "Theorem 5.4 (Anti-holomorphic blow-up). Let β : Sk Ñ DiffpZ, Dq be a family so that β # 1M smoothly contracts via a contraction F . We then have |Dπk`1 c`ν pFp1Z # f qF´1q| “ 2|Dπk c pβq|."

    This formula is the engine that shows the recursively constructed families have non-zero invariants. It is introduced as 'the following result from [7]' and no proof or independent derivation is given. Without Theorem 5.4, the computation of the invariant on the examples does not go through, so the core assertion of Theorem 1.3 reduces to a self-citation whose content cannot be checked in this paper.

  2. self citation load bearing [Section 6, Proof of Theorem 1.3]
    "As we have defined them, the manifolds Z1 k grow as k grows. Picking an upper bound n we can pick a new family of manifolds Vk so that Z1 k #Σ1 Vk– Z2 n independent of k for k ď n. Provided this only changes the invariants by a non-zero factor, the same argument will work. This argument is written out in more detail in [7]."

    The fixed target manifold Z_n^r required by Theorem 1.3 is only obtained through this stabilization claim. The paper neither proves the diffeomorphism Z^1_k #_{\Sigma_1} V_k ≅ Z^2_n nor verifies the non-zero factor; it explicitly defers the argument to [7]. This is the final load-bearing step of the proof, so the theorem statement depends on the authors' unpublished companion.

full rationale

The paper splits into two very different parts. Theorem 1.1 is self-contained in the manuscript: it is proved from Proposition 2.1, which in turn cites external and available works by Gompf, Tange, and others, and the argument is written out in detail here. No circularity is present there. The high score is entirely due to Theorem 1.3, whose proof is a chain of self-citations to the in-preparation companion [7]. Specifically, the definition of the family Donaldson invariant is only informal and its rigorous version is in [7]; the key diagonal evaluation (q_{X^r_p,c_q}) = 4(p−1)^r I is cited to [7]; the anti-holomorphic blow-up formula of Theorem 5.4 is stated as a result from [7]; and the stabilization needed to obtain a fixed target manifold Z is deferred to [7]. Each of these is load-bearing: without them, Lemma 4.7 does not produce the rank-r summand, and Theorem 1.3 is not established. Under the stated rules, a citation to one's own unpublished manuscript is not independent support, because it is neither machine-checked, code-reproduced, nor externally falsifiable. This is not a tautology or a fitted-parameter prediction, but it is a clear case of load-bearing self-citation, so the central claim of the paper cannot be verified from the manuscript alone. The score reflects the fact that a substantial independent theorem (Theorem 1.1) is included, while the main families theorem depends on an inaccessible companion.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper fits no parameters to data. Its constructions use integer choices (n, p, q, r, log transform multiplicities) as design variables, not free parameters. The central claims rest on published results (Gompf's corks, Tange's theorem, Konno's invariants, Freedman's homeomorphism classification, Wall-Kreck-Perron-Quinn stabilization) and on the unpublished companion [7], which provides the family Donaldson invariant machinery. No new physical entities are introduced.

assumptions (7)
  • domain assumption Gompf's infinite order corks X+(n)=C(2,1;-2n) with boundary Y_n and torus twist f(n) satisfy the properties stated in Proposition 2.1(1)(2).
    Quoted from [28], [29], and [66] in Section 2; the paper does not reprove these results.
  • domain assumption The twisted double X+(n) union_{f(n)} (-X+(n)) is diffeomorphic to S^4 (Tange [66]).
    Needed to define the embeddings into S^4 in Theorem 1.1; quoted from the published result [66].
  • ad hoc to paper Existence and computation of family Donaldson invariants D^{H^k Diff}_{c,beta} on H_k(Diff(Z)) and pi_k(Diff(Z)) from the authors' unpublished manuscript [7].
    The paper cites [7] (manuscript in preparation) for the invariant and for the key identity D^{H^{k+1} emb}_{c_q,beta}(J^{k+1}_p)=gamma'_p delta_{pq}; no independent verification is available.
  • ad hoc to paper Anti-holomorphic blow-up formula (Theorem 5.4): |D^{pi^{k+1}}_{c+nu}(F(1_Z # f)F^{-1})| = 2|D^{pi^k}_c(beta)|.
    Quoted from [7] and used to compute family invariants for the commutator construction in Section 5.
  • domain assumption Konno's family Donaldson invariants D_{c,beta}(Z) for families of manifolds are well-defined and behave as characteristic classes.
    Published in [40]; used as the invariant for family submanifold sums in Section 4.
  • standard math Freedman's homeomorphism psi: X -> Y rel the nucleus N(1), and the stabilizability X#M congruent Y#M with phi homotopic to psi # 1.
    Standard consequences of Freedman's classification and Wall's theorem, cited in Section 5; used to set up the stable isotopy construction.
  • domain assumption The 'key stable isotopy' from [5] can be chosen to be the identity on N(Sigma_1).
    Quoted from [5, Section 2] and used in Section 4 to produce topological isotopies G_p that extend over the family.

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Cite this review

Pith. "Pith review of Exotic families of embeddings." pith.science (2026). https://pith.science/paper/KLB7GLWS

@misc{pith2026250112673,
  author       = {Pith},
  title        = {Pith review of: Exotic families of embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLB7GLWS}},
  note         = {Machine review of arXiv:2501.12673}
}
abstract

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in $S^4$ that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3-manifold that embeds in a blown-up K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.

Figures

Figures reproduced from arXiv: 2501.12673 by the authors.

Figure 1
Figure 1. Infinite order corks X`pnq . 1 2n [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The homology spheres Yn . the left is the (mirror) of the exterior of the stevedore knot. Put coordinates on this torus as px, yq P R 2 {Z 2 where R{Zˆpoint is a longitude and pointˆR{Z is a meridian. Coordinates for a tubular neighborhood of the torus will then be px, y, tq and the diffeomorphism fpnq is given by fpnqpx, y, tq “ px ` t, y, tq in the neighborhood and the identity elsewhere. This map is called a toru… view at source ↗
Figure 3
Figure 3. The homology sphere Yn as a splice . the Alexander-Conway polynomial of K is ∇Kpzq “ p2z 2 ´ 1q 2 . Clearly, S 3 1{0 pKq “ S 3 , so the Casson invariant satisfies λpS 3 1{0 pKqq “ 0. Casson’s surgery formula [1] reads: λpS 3 1{pm`1q pKqq ´ λpS 3 1{mpKqq “ 1 2 ∇2 Kp0q, so an induction argument gives λpYnq “ λpS 3 1{2n pKqq “ ´8n. Because Casson’s invariant changes sign under orientation reversal [1], the manifolds Yn… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Construction of the embedding jp It remains to show that the embeddings jp are smoothly distinct. Assuming that there is a diffeomorphism f : X0 #Σ Vp Ñ X0 #Σ Vp so that f ˝ jp “ jq, one would conclude that ´ pX0 #Σ VpqzjppY q ¯ 0 ď Y V1 – ´ pX0 #Σ VpqzjqpY q ¯ 0 ď Y V…
Figure 5
Figure 5. Figure 5: Construction cartoon diffeomorphic to Z and structure group DiffpZq, and hence is equivalent to a map from B to the classifying space B DiffpZq. We discuss some of the relations between these three types of families here and then turn to the definition of various invar…
Figure 6
Figure 6. Figure 6: Codimension one submanifold sum Let us assume that that the basic Icq extends to an invariant of families of manifolds, denoted Icq . Given this assumption, the invariant for a family of embeddings is given by I Hk`1 emb cq pJq “ Icq pZrrJssq. Lemma 4.7. The classes rJ…
Figure 7
Figure 7. Figure 7: The boundary of the Akbulut-Mazur manifold . where σ and τ are exhibited in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: The Gompf nucleus after log transform N p1,3q 2p`1 . By the work of Freedman, there is a homeomorphism rel boundary ζ : N p1,3q 2p`1 Ñ Np1,3q that matches the multiple fiber class f p3q in the transformed nucleus with the fiber class in the nucleus on the other side. F…
Figure 9
Figure 9. Figure 9: The Pp´3, 3, ´3q knot with DS tangle T and augmentation A . It remains to construct the invertible cobordism; the construction is similar to the use of the ‘KT grabber’ in [12]. Note first that doing ´1{n surgery on A turns K into a knot Kn that has n full twists in th…

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