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On $h$-cobordisms of complexity $2$

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Pith's one-line read This paper proves that h-cobordisms between exotic closed simply connected 4-manifolds can have arbitrarily large Morgan–Szabó complexity, with factorially many examples obtained from reflections of the intersection form.

desk verdict Genuine new result with detailed Floer computations, but load-bearing reliance on unpublished [Lad22] and a one-sentence b^+=1 extension mean the referee should check those two spots carefully. read the letter →

arxiv 2501.08750 v2 pith:SHTEVR4E submitted 2025-01-15 math.GT

Roberto Ladu This is my paper · ORCID
classification math.GT MSC 57K4057R5857R57
keywords h-cobordismMorgan–SzabócomplexityprotocorkmonopoleFloerhomologySeiberg–Witteninvariantscorksexotic4-manifoldsDolgachevsurfaces
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 aims to show that the Morgan–Szabó complexity of an h-cobordism between two exotic closed simply connected 4-manifolds can be arbitrarily large, not merely for a manifold to itself but between genuinely distinct manifolds. The key step is a complete computation of the monopole Floer homology of the boundary of the protocork $P_0$ and of the action of the twisting involution $\tau$, which yields an obstruction: a complexity-2 h-cobordism can only change Seiberg–Witten invariants in formal dimensions up to 2. Using this, the author constructs, via isometries of the intersection form, many h-cobordisms whose Seiberg–Witten variation has formal dimension $m^2 + m - 2$ diverging as $m \to \infty$, and shows that at least $2^{m_N} m_N!$ distinct examples exist for each $N$. In particular, blowing up the elliptic surface $E(1)$ and a Dolgachev or knot-surgery partner by 17 copies of $\overline{\mathrm{CP}}^2$ already produces h-cobordisms of complexity strictly greater than 2 (hence at least 4).

What carries the argument

The load-bearing object is the protocork $P_0$: a compact 4-manifold with boundary $Y = \partial P_0$ whose 'twist' operation — remove $P_0$ from a 4-manifold and reglue it via the boundary involution $\tau$ — changes the smooth structure while preserving the homeomorphism type. $P_0$ is the unique protocork realizable at Morgan–Szabó complexity 2, and its boundary $Y$ is a graph manifold with plumbing graph given by two trivial normal bundle spheres intersecting three times with signs $+,-,+$. The computation of the monopole Floer homology $\widehat{HM}_\bullet(Y)$ and the action of $\tau_*$ on $\widehat{HM}^{-1}(Y)$ is the engine of the paper; it shows that the 'difference element' $\Delta$ is $U$-torsion ($U\cdot \Delta = 0$), which bounds the variation of Seiberg–Witten invariants under the twist by formal dimension 2. The construction of the h-cobordisms then uses Kreck's theorem (isometry classes of intersection forms parametrize h-cobordisms), composing a homeomorphism-induced isometry $\Phi$ with a reflection $\rho_m$ along a class $\alpha_m = (2m+1)H - \sum E_i$ of square $-1$; Lemma 6.2 guarantees the isometry preserves chamber-dependent Seiberg–Witten invariants, so the variation along the resulting h-cobordism is exactly controlled by the reflection formula.

What would settle it

Find a normal handle decomposition of the h-cobordism $C_2$ (17 blow-ups) with only two excess intersections, or independently compute $\widehat{HM}^{-1}(\partial P_0)$ and show that the action of $\tau$ is not the unipotent matrix of Theorem 1.1 (for example, that $\Delta = 0$ or that $\tau_*$ is diagonalizable); either observation would break the obstruction and contradict Theorem 1.2(1).

Watch

Extended reading notes

Core claim

The paper's central discovery is that the minimal non-trivial Morgan–Szabó complexity, 2, is obstructed by monopole Floer homology: for the unique complexity-2 protocork $P_0$, with boundary $Y = \partial P_0$ and twist involution $\tau$, the reduced Floer homology is $\widehat{HM}^{-1}(Y) \cong \mathbb{F} x_0 \oplus \mathbb{F} \Delta \oplus \mathbb{F} \alpha_1$ with $\tau_*(x_0) = x_0 + \Delta$ and $U\cdot \Delta = 0$. Consequently a protocork twist using $P_0$ can alter Seiberg–Witten invariants only in formal dimension at most 2. The theorem that follows, Theorem 1.2, asserts that for $X_0 = E(1)$ and $X_1$ a Dolgachev surface or the Fintushel–Stern knot surgery on $E(1)$, blowing up to $X_i \# 17\overline{\mathrm{CP}}^2$ yields at least $2^{17}17!$ h-cobordisms of complexity greater than 2, and for every $N>0$ there is some number $m_N$ of blow-ups giving at least $2^{m_N} m_N!$ h-cobordisms of complexity larger than $N$. The same Floer-theoretic input, via Corollary 1.3, yields strongly non-extendable cork involutions for the Akbulut cork and the $(Y_{1,8,1}, \tau_{1,8,1})$ cork.

Load-bearing premise

The argument relies on the claim that at the lowest non-zero complexity there is only one protocork shape, $P_0$, a classification imported from the author's previous work; if a second shape with the same complexity existed, the obstruction in Theorem 1.1 would not rule out complexity-2 h-cobordisms between the constructed pairs.

Editorial extensions

If this is right

  • For every $N > 0$, some number $m_N$ of blow-ups of a fixed exotic pair yields at least $2^{m_N} m_N!$ distinct h-cobordisms of complexity larger than $N$, so complexity is unbounded among non-inertial h-cobordisms.
  • With 17 blow-ups, the construction already rules out the minimal non-trivial complexity: the h-cobordisms have complexity at least 4.
  • The same Floer-theoretic calculation gives a general criterion for strong non-extendability of cork involutions, and the Akbulut cork and the $(Y_{1,8,1}, \tau_{1,8,1})$ cork satisfy it.
  • The computation of $\widehat{HM}_\bullet(Y)$ and of $\tau_*$ covers a cyclic graph manifold case not treated by existing general algorithms for graph manifolds.

Reading between the lines

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

  • The factorial growth in the count of h-cobordisms likely reflects many genuinely distinct 5-dimensional h-cobordisms, not merely different isometries; distinguishing them would require 5-dimensional invariants, which the paper does not compute.
  • The mechanism is not special to $E(1)$: any exotic pair homeomorphic via an isometry that preserves chamber-dependent Seiberg–Witten invariants should admit arbitrarily complex h-cobordisms after enough blow-ups along similar reflection classes.
  • A testable consequence: an explicit normal handle decomposition of one of the constructed h-cobordisms, if it could be found, would either confirm the complexity lower bound or expose a gap in the Floer-theoretic obstruction.
  • The obstruction at complexity 2 is sharp in the sense that the moment $\Delta$ becomes $U$-torsion of higher order, the allowed formal dimension grows; exploiting protocorks with larger torsion order might produce even stronger lower bounds with fewer blow-ups.
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Formalized claims in Lean

  1. Claim #1: The paper's central discovery is that the minimal non-trivial Morgan–Szabó complexity, 2, is obstructed by monopole Floer homology: for the unique complexity-2 protocork $P_0$, with boundary $Y = \partial P_0$ and twist involution $\tau$, the reduced Floer homology is $\widehat{HM}^{-1}(Y) \cong \mathbb{F} x_0 \oplus \mathbb{F} \Delta \oplus \mathbb{F} \alpha_1$ with $\tau_*(x_0) = x_0 + \Delta$ a

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. This paper studies Morgan-Szabó complexity 2 h-cobordisms in dimension 5. The author computes the monopole Floer homology \widehat{HM}_*(Y) of the boundary Y of the protocork P0 and the action of the twisting involution τ on \widehat{HM}_{-1}(Y) (Theorem 1.1). This computation is used to obstruct complexity-2 h-cobordisms and, via a Morgan–Szabó-style construction, to produce h-cobordisms from E(1)#17\overline{CP}^2 to a Dolgachev or knot-surgered E(1)#17\overline{CP}^2 of complexity greater than 2, and more generally of arbitrarily large complexity after sufficiently many blow-ups (Theorem 1.2). The paper also proves a strong-cork non-extendability criterion (Corollary 1.3). The proof of Theorem 1.1 combines surgery exact triangles, a computation of HF^+(Y_{-1,1,8}) via (1,1)-knot mapping cones, and published computations of the Akbulut cork boundary; the proof of Theorem 1.2 imports a protocork classification and a Seiberg-Witten variation bound from the author's unpublished preprint [Lad22].

Significance. If correct, Theorem 1.2 is a significant advance: it provides the first non-inertial h-cobordisms with non-minimal, in fact arbitrarily large, complexity between closed simply connected exotic 4-manifolds, and the factorial growth in the number of blow-ups is new. The Floer-theoretic computation of Theorem 1.1 is largely self-contained, and the reductions to published computations ([Gut24], [AD05], [LRS18]) are credible; Section 5's mapping-cone computation is detailed and checkable. The main caveat is that two load-bearing inputs come from the unpublished manuscript [Lad22], and the b^+=1 extension needed for the arbitrarily-large-complexity claim is asserted rather than proved. These issues are fixable in a revision but must be addressed before acceptance.

major comments (3)
  1. [Remark 2 / Theorem 6.3] The divergent-complexity part of Theorem 1.2 depends on applying [Lad22, Cor. 1.2] to the manifolds X_{0,m}, which have b^+=1. Remark 2 states that the proof goes through using the metric-dependent HM map and the factorization formula [KM07, (3.14)], but it does not address the wall-crossing and chamber issues specific to b^+=1. Since equality (16) concerns chamber-dependent invariants m^\pm and the contradiction step in Theorem 6.3 needs a bound on all Seiberg-Witten variations of formal dimension larger than 2ℓ, the b^+=1 case is load-bearing. A complete proof, or a precise published reference that covers b^+=1, must be supplied.
  2. [§6.1] The assertion that P0 is the only complexity-2 protocork is imported from the unpublished preprint [Lad22]. This uniqueness is explicitly used to conclude that C_2 has complexity >2, and it also underlies the finiteness and the list S_r in (17). Please include the classification statement as a proved lemma in this paper or give a published reference; as it stands, the N=2 claim rests on an external input that the reader cannot verify.
  3. [Lemma 6.2] The proof of Lemma 6.2 uses the gluing formula [KM07, Prop. 27.5.1] for the invariants m_k and then asserts that m_k, together with k and the intersection pairing, determines the chamber-dependent invariants m^\pm. For b^+=1 the invariants m^\pm are metric- and chamber-dependent, and the manuscript does not justify this determination or the applicability of the gluing formula in that case. Since Lemma 6.2 is what produces the key equality (16), this step needs to be expanded rather than cited.
minor comments (3)
  1. [Proof of Proposition 4.3] The sentence "Since (τ_{0,1,8})_* acts as id on HM_red(Y_{0,1,8}) by Proposition 4.2" is misstated: Proposition 4.2 concerns Y_{8,1,8}. The intended argument is that the image of HM_red(Y_{0,1,8}) in HM_red(Y_{8,1,8}) is contained in the fixed subspace FΔ ⊕ Fα of (τ_{8,1,8})_*, so the statement should be reworded accordingly.
  2. [§6.1 and §6.3] There are several typos, including "occour" in §6.1 and "contraddiction" in §6.3; the paper would benefit from a careful proofreading pass.
  3. [Proposition 4.2] The deduction of the displayed matrix for the action of (τ_{8,1,8})_* from the Lefschetz number 2 and the mod-2 Seiberg-Witten variation is compressed into the phrase "as in [LRS18]"; since this input is used for Theorem 1.1, the author should spell out the argument or cite the exact statement in [LRS18] that covers it.

Circularity Check

2 steps flagged · score 4.0 of 10

Core Floer computation is self-contained, but Theorem 1.2 leans on the author's own unpublished [Lad22] for the uniqueness of the complexity-2 protocork and for the b+=1 extension of the SW-variation obstruction.

  1. uniqueness imported from authors [Section 6.1; used in Theorem 6.3 (N=2 case)]
    "It is key to our argument that there is only one such graph with complexity 2. In particular if C(Z)=2 then X0 and X1 are related by a protocork twist using P0, the main subject of this paper."

    The uniqueness of the complexity-2 plumbing graph is not proved in this paper; it is imported from the author's own unpublished preprint [Lad22]. This is load-bearing for the N=2 case of Theorem 1.2: the contradiction for C_2 in Theorem 6.3 requires that every complexity-2 h-cobordism is a P0 protocork twist, so that Theorem 1.1's computation and [Lad22, Cor. 1.2] apply. If a second complexity-2 protocork existed, the no-variation computation for P0 would not obstruct that h-cobordism. The paper frames the choice as forced, but the uniqueness theorem is a same-author citation rather than an independently established fact in this paper.

  2. self citation load bearing [Theorem 6.3 proof; Remark 2]
    "Then by [Lad22, Corollary 1.2], there cannot be any variation of Seiberg-Witten invariants corresponding to moduli spaces of formal dimension larger than 2ℓ along the h-cobordism C_m. ... In [Lad22, Corollary 1.2] we assume b+(X)>1, however the same proof goes through to cover the case b+(X)=1 using the metric dependent HM-map [KM07, pg. 562-566]."

    The arbitrarily-large-complexity statement (Theorem 1.2(2)) is proved by contradiction whose decisive obstruction is [Lad22, Cor. 1.2], a result from the author's own prior preprint. The constructed manifolds have b+=1, so the proof additionally relies on a one-sentence extension asserted in Remark 2; the extension is not proved here, and [KM07] is cited for a metric-dependent map rather than for the b+=1 gluing/chamber statement. Thus the central divergence conclusion reduces, at its decisive step, to an unverified same-author input. This is not a definitional equivalence, but it is load-bearing self-citation; the construction of C_m and Lemma 6.2 supply independent content.

full rationale

Sections 3-5 compute yHM(Y) mostly from external inputs ([Gut24], [AD05], [LRS18], [KM07]), and the nonvanishing of Δ is re-established in Proposition 4.3 via the K3#CP^2 versus 3CP^2#20CP^2 twist, so Theorem 1.1 is not circular in the equation-identity sense. The main self-citation load appears in Theorem 1.2: the N=2 case uses [Lad22]'s uniqueness classification of complexity-2 protocorks, and the arbitrary-N case uses [Lad22, Cor. 1.2] together with a one-line b+=1 extension in Remark 2. These are same-author, unpublished, and load-bearing; if they fail, the corresponding conclusions collapse. Because the h-cobordism construction and the Seiberg-Witten variation computation are independent, and no fitted parameter is renamed as a prediction, the appropriate score is 4 rather than 6+.

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

The central claims rest on the author's own protocork framework in [Lad22] (self-cited, unpublished), while the genuinely new computation here is the Floer homology of Y = ∂P_0 and the action of τ, grounded in published external computations ([Gut24], [AD05], [LRS18], [KM07], [OS08]). No new topological or physical entities are postulated; the auxiliary manifolds Y_{a,b,c} are standard surgical constructions. There are no fitted or empirical free parameters.

assumptions (5)
  • standard math Isomorphism classes of h-cobordisms between closed 1-connected 4-manifolds are in bijection with isometries of their intersection forms [Kre01].
    Used in Section 6.2 to convert the isometry R_m into the h-cobordism C_m and to conclude that distinct isometries give distinct h-cobordisms, which drives the counting.
  • domain assumption Protocork framework from [Lad22]: h-cobordisms correspond to protocork twists with plumbing graphs, and up to isomorphism there is exactly one protocork of complexity 2, namely P_0, with graphs admitting geometric handle cancellation excluded.
    Section 6.1: 'It is key to our argument that there is only one such graph with complexity 2'; imported from the author's unpublished preprint, no proof or pinpointed reference given here.
  • domain assumption [Lad22, Cor. 1.2]: a protocork of complexity smaller than r cannot cause Seiberg-Witten invariant variation for formal dimension larger than 2ℓ, where ℓ is the maximal U-torsion order of the difference elements, with the extension to b^+ = 1 given in Remark 2.
    Load-bearing for both parts of Theorem 1.2; self-cited, not machine-checked, and the b^+ = 1 extension is justified in one sentence via metric-dependent ỸHM maps.
  • domain assumption Grading assumptions AG of Appendix A hold for Y_{-1,8,-1} and Y_{-1,8,0}.
    Proposition 3.2 says this is 'straightforward' for Y_{-1,8,-1} since it bounds a contractible 4-manifold, and for Y_{-1,8,0} 'we can run an argument as in [Lad22, Thm. 1.4]'; the degree shift [−2] in Proposition 3.2 depends on these lemmas.
  • standard math Published external computations: [Gut24] for the Floer homology of the positron cork boundary, [AD05] and [LRS18, Thm.
    Used throughout Sections 3 and 5 as benchmarks and tools; these are published results by other authors.

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Pith. "Pith review of On $h$-cobordisms of complexity $2$." pith.science (2026). https://pith.science/paper/SHTEVR4E

@misc{pith2026250108750,
  author       = {Pith},
  title        = {Pith review of: On $h$-cobordisms of complexity $2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHTEVR4E}},
  note         = {Machine review of arXiv:2501.08750}
}
abstract

We study $5$-dimensional $h$-cobordisms of Morgan-Szab\'o complexity $2$. We compute the monopole Floer homology and the action of the twisting involution of the protocork boundary associated with such $h$-cobordisms, obtaining an obstruction for $h$-cobordisms between exotic pairs to have minimal complexity. We construct the first examples of $h$-cobordisms of non-minimal, in fact, arbitrarily large, complexity between an exotic pair of closed, $1$-connected $4$-manifolds. Further applications include strong corks.

Figures

Figures reproduced from arXiv: 2501.08750 by the authors.

Figure 1
Figure 1. Left: Kirby diagram for the protocork P0, Y “ BP0, the involution τ : Y Ñ Y is induced by a rotation by π in the vertical axis. Right: surgery presentation of the 3-manifold Ya,b,c. Notice that Y » Y0,0,8 » Y0,8,0 » Y8,0,0. If W is a 4-manifold with boundary, we denote by WzB 4 the manifold obtained by removing a ball from W. We can interpret it as a cobordism S 3 Ñ BW, then the relative Seiberg-Witten invariant of … view at source ↗
Figure 2
Figure 2. The torus Σα is obtained by capping the surface of genus one in blue with the disk coming from the Dehn filling of the 0-framed unknot. indeed they are both have vanishing triple cup product. The vertical sequence of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The horizontal sequence is the sequence of manifolds appearing in the long exact triangle sequence induced by surgery along the a-framed curve: Y0,8,0 Ñ Y8,8,0 » S 1 ˆ S 2 Ñ Y´1,8,0 Ñ ¨ ¨ ¨ . The vertical sequence is induced by surgery along the c-framed curve: Y´1,8,´1 Ñ Y´1,8,0 Ñ Y´1,8,8 » S 3 Ñ ¨ ¨ ¨ Proof. It is enough to compute HFredp´Y´1,8,´1; Fq in virtue of [KM07, Prop. 28.3.4] and [cKLT10]. ´pY´1,8,´1q is … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: The horizontal sequence is induced by surgery on the meridian of the b-framed curve Y0,8,8 » S 1 ˆ S 2 Ñ Y0,0,8 Ñ Y0,1,8 Ñ ¨ ¨ ¨ . The vertical sequence is induced by surgery along the a-framed curve: Y1,8,´1 Ñ Y0,1,8 Ñ Y8,1,8 Ñ ¨ ¨ ¨ . Note that Y8,1,8 is the boundary…
Figure 5
Figure 5. Figure 5: (a) The knot Kpϵ, σq is described by the green curve, here ϵ P t˘1u and σ P B4 is a product of pτ 2q ˘2 and pτ 3q ˘1 . (b) Generators of B4. (c) Example to show our composition convention, here σ “ pτ 2q ´2 τ 3pτ 2q 2 and Kp1, σq is the left handed trefoil [PITH_FULL_…
Figure 6
Figure 6. Figure 6: (a) Initial doubly pointed Heegaard diagram, the surface Σ is the torus obtained by identifying the opposite sides of the rectangle and has the specified orientation. (b) The action of pτ 2q 2 on Σ is a full Dehn twist around the curve c. (c) The action of τ 3 on Σ is …
Figure 7
Figure 7. Figure 7: Concatenating a multiple of the meridian µ. to xτ 2 2 , τ 3y by composition. We can therefore produce a new diagram pΣ, σpαq, β, z, wq. Now, if the algebraic intersection of the two curves with respect to our chosen orientation of Σ satisfies σpαq ¨ β “ ϵ, then pΣ, σpα…
Figure 8
Figure 8. Figure 8: Double pointed Heegaard diagram (left) and lifted Heegaard diagram (right) for example [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Floer exact triangle Y´1,1,8 Ñ S 3 ´1 pKq Ñ S 1 ˆ S 2 . This surgery presentation for Y´1,1,8 is obtained from that of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: Left: the framed curve γ. Right: a Seifert surface for γ in S 3 ´1 pKq is obtained by capping the blu surface with the disk bounding the black curve. is that k “ ˘1 because otherwise HF´pY´1,1,8q would be non-trivial in some non-integral degree (Y´1,1,8 bounds a contr…
Figure 11
Figure 11. Figure 11: Derivation of the Heegaard diagram for the knot Kp1, σq on the torus. Top-left: application of ¯τ 2 2 τ¯ ´2 3 τ¯ ´2 2 . Top-right: application of ¯τ 2 2 τ¯ 2 3 . Bottom-left: we perform an isotopy of the α-curve. Bottom-right: another isotopy to get the final diagram.…
Figure 12
Figure 12. Figure 12: Lifted Heegaard diagram for the knot Kp1, σq. The plane is oriented counterclockwise. and we can form A ` :“ à sPZ A ` s B ` :“ à sPZ B ` (13) ˆ tsu. Since B` is a quotient complex of A` s for any s, we have a projection map v ` s : A` s Ñ B`. In [OS04] it is shown th…
Figure 13
Figure 13. Figure 13: The filtered complex CFK8pKp1, σq; Zq is generated over ZrUs by the complex shown in the picture. Each arc represents a ˘1 component of the differential, for example Bx1 “ ´x4 and Bx2 “ x1 ` x3. . Now for n P Zzt0u, we define D ` n : A ` Ñ B ` D ` n ppasqsPZq “ pbsqsP…
Figure 14
Figure 14. Figure 14: Sequence in a Floer exact triangle S 3 Ñ Y0,1,8 Ñ Y1,8,1. The isomorphisms with S 3 and Y1,8,1 become apparent once the 1-framed knot is slid over the 0-framed one thus unlinking it from the blue one. 7. Proof of Corollary 1.3 Proof. With the notation of Corollary 1.3…

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