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Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

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Pith's one-line read Seiberg-Witten theory builds infinite-rank families of diffeomorphisms invisible to homeomorphisms

desk verdict Ambitious and likely correct paper giving Z^∞ summands in higher homotopy/homology of diffeomorphism groups that die topologically; the analytical gluing is the main thing to check. read the letter →

arxiv 2501.11892 v1 pith:U6B3IMFB submitted 2025-01-21 math.GT math.ATmath.DG

Dave Auckly , Daniel Ruberman This is my paper · ORCID
classification math.GTmath.ATmath.DG MSC 57K4157R5253C21
keywords 4-manifoldsdiffeomorphismgroupsfamilySeiberg-WitteninvariantshomeomorphismTorelligrouppositivescalarcurvaturemetricsembeddingspacesgluingtheorem
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 establishes that the smooth and topological categories in dimension four differ sharply at the level of families: for every p>0 there are 4-manifolds whose diffeomorphism groups contain infinite-rank free abelian summands in homotopy and homology groups in a range of degrees, all lying in the kernel of the natural map to the homeomorphism group. The construction is recursive, starting from non-isotopic but pseudoisotopic diffeomorphisms on elliptic surfaces and producing spherical families of diffeomorphisms of arbitrarily high dimension. Non-triviality is detected by family Seiberg-Witten invariants, computed through a new gluing theorem adapted to the inductive construction. The same families yield infinite-rank summands in the homotopy and homology of spaces of embeddings of spheres and 3-manifolds, and in spaces of positive scalar curvature metrics on standard PSC 4-manifolds.

What carries the argument

The load-bearing object is the parameterized family Seiberg-Witten invariant, in its homotopy form SWπ_k and homology form SWH_k, together with the new Parameterized Irreducible-Reducible Gluing Theorem (Theorem 5.4). That theorem computes the mod-2 family invariant of a connected sum N_1 # N_2 as an $S^{1}$-fibered product of the family moduli spaces on the two summands, one carrying only isolated irreducible solutions and the other only isolated reducible solutions; a Suspension Theorem (Theorem 5.7) then turns a k-dimensional family on Z into a (k+1)-dimensional family on Z # ($S^{2}$×$S^{2}$) while preserving the invariant. The commutator construction with the reflection diffeomorphism on $S^{2}$×$S^{2}$ raises the sphere dimension, and the finite-set-of-basic-classes vanishing result converts mod-2 detection into infinite rank.

What would settle it

Compute the mod-2 family Seiberg-Witten invariant of one claimed generator α^p[q] in the $\mathrm{Spin}^c$ structure $K+2\Sigma\tilde T+2\ell\tilde T_1$ with $|\ell|\leq q$; a value of 0 where the recursive formula predicts 1 would disprove the gluing theorem and with it the construction. Equivalently, exhibit irreducible-reducible good data on a connected sum for which the parameterized moduli space is not the $S^1$-fibered product.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1: for any p>0 there are 4-manifolds Z_p such that for all 0<j≤p with j≡p mod 2, the groups π_j(Diff_0(Z_p)) and H_j(Diff_0(Z_p)) contain Z^∞ summands lying in the kernel of the comparison map to the corresponding homeomorphism groups, and likewise for H_{j+1}(B TDiff(Z_p)). The proof builds spherical families by a commutator construction: a k-dimensional family on one manifold is converted into a (k+1)-dimensional family on a stabilized manifold, using reflection diffeomorphisms on $S^{2}$×$S^{2}$ summands and the fact that the seed diffeomorphisms become isotopic after one stabilization. The non-vanishing is established by family Seiberg-Witten invariants, with a mod-2 gluing computation promoted to integer-valued invariants through the composition law, and infinite rank follows because only finitely many basic classes can contribute to each family.

Load-bearing premise

Everything rests on the analytic gluing theorem: if the family moduli space of a connected sum is not the glued product of the two pieces' moduli spaces for irreducible-reducible good data, the recursive computation of the Seiberg-Witten invariants collapses.

Editorial extensions

If this is right

  • For each parity class of degrees up to p, the diffeomorphism group of Z_p has a free abelian summand of countably infinite rank that becomes trivial in the homeomorphism group, so the smooth and topological classifications of 4-manifold families diverge in every such degree.
  • An infinite-rank subgroup of the constructed families becomes smoothly trivial after a single connected sum with S^2×S^2, and each element is topologically isotopic, indeed pseudoisotopic, to the identity.
  • The sphere bundles built by clutching along these families are smoothly non-trivial but topologically trivial, and their total spaces are diffeomorphic to products.
  • The same invariants give infinite-rank summands in the homotopy and homology of embedding spaces of S^2, S^3, and S^1×S^2 in stabilized manifolds, and the generators that become trivial after one external stabilization are concordant.
  • Spaces of positive scalar curvature metrics on standard spin and non-spin PSC 4-manifolds acquire infinite-rank summands in their homotopy and homology groups, including spin examples not covered by earlier methods.

Reading between the lines

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

  • As an extension, tracking orientations through Theorem 5.4 would likely yield an integer-valued gluing formula directly, letting the Z^∞ summands be seen without the mod-2-to-integer composition step that the paper currently needs.
  • As an extension, because the constructed families are supported away from the distinguished nucleus and are stably trivial, the same recursive construction should transplant to any manifold obtained by fiber-summing along that nucleus, potentially making the phenomenon generic among sufficiently stabilized 4-manifolds.
  • As an extension, the paper leaves open what happens in the quotient R_+(Z)/Diff(Z) of the PSC-metric space; evaluating the same family invariants on the homotopy quotient would test whether the infinite-rank classes survive after forgetting the diffeomorphism parameter.
  • As an extrapolation, the recursive suspension construction suggests a stable-range phenomenon: the same manifold carries summands in all degrees j≡p mod 2 up to p, and one might expect infinite generation to persist or concentrate in a stable limit as p grows.
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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

2 major / 5 minor

Summary. The paper constructs spherical families of diffeomorphisms of certain 4-manifolds and shows that they give Z^∞ summands in π_j(Diff_0(Z_p)) and H_j(Diff_0(Z_p)) which lie in the kernel of the natural maps to the corresponding groups for Homeo_0(Z_p); analogous statements are obtained for H_{j+1}(B TDiff(Z_p)). The construction is inductive, starting from the second author's 1998 non-isotopic diffeomorphisms, and the non-triviality is detected by family Seiberg-Witten invariants. The central analytic input is a new parameterized irreducible-reducible gluing theorem (Theorem 5.4), from which a suspension theorem (Theorem 5.7) and concrete invariant computations (Theorem 5.8) are derived. Applications include infinite generation in homotopy and homology of embedding spaces of S^2, S^3, and S^1×S^2, and of spaces of positive scalar curvature metrics.

Significance. If the analytic core is valid, the paper is a substantial advance: it establishes infinite-rank summands in higher homotopy and homology groups of diffeomorphism groups of 4-manifolds that vanish topologically, in contrast with known finiteness results in higher dimensions, and it gives new applications to embedding spaces and PSC metrics. The paper is commendably explicit about its limitations, e.g., Remark 5.5 notes that the gluing theorem is proved only for zero-dimensional parameterized moduli spaces and without a full orientation analysis. The recursive construction is concrete, and the paper correctly explains the passage from mod 2 to integer invariants via Lemma 2.9 and Lemma 2.21. The main risk is the depth of the analytic gluing argument, which I could not independently verify in full detail.

major comments (2)
  1. [9.2.1, Proposition 9.5] The proof of Proposition 9.5, which deforms good data to locally metric independent data, is the principal unverified step in the paper's analytic core. The text states that a version is proved in [MRS11, Appendix A] for 1-parameter families and that the argument 'adapts readily' to k-dimensional parameter spaces, and the proof given works locally around one exceptional point in a top-dimensional simplex. It is not explained how several exceptional points in different simplices are treated simultaneously, why the modified metric cannot create or destroy solutions on the perturbed region, or why the resulting globally defined data remains irreducible-reducible good. Since Proposition 9.5 is used in the proof of Theorem 5.4 and then in Corollary 5.6 and the Suspension Theorem 5.7, this is load-bearing. Please provide a complete proof or a reference containing exactly this k-parameter, multi-point statement, rather than an assertion of adaptation.
  2. [9.3.2, Lemma 9.8] The parameterized linear gluing result Lemma 9.8 is asserted as a modification of [Nic02] with the proof summarized in a single paragraph. In particular, the treatment of the finite-rank perturbation n(θ) inside the operator rD, the claimed bounds used in the bootstrapping argument, and the exactness of the displayed sequence with the new parameter summand are not written out. Because this lemma is the linearization step underlying Theorem 5.4, a failure or gap here would invalidate the gluing theorem and hence the main computation. The proof should be expanded to the same level of detail as the rest of Section 9, or a precise reference covering this parameterized perturbation should be supplied.
minor comments (5)
  1. [Abstract and throughout] The notation 'Z8' is used for what is clearly meant to be an infinite direct sum of copies of Z; please use \mathbb{Z}^\infty or \bigoplus_{\mathbb{N}}\mathbb{Z} to avoid confusion with the cyclic group of order 8, which appears nowhere else in the paper.
  2. [Abstract] The abstract contains the typo 'we we obtain'; please correct it.
  3. [Section 2.4.1] In the paragraph following Definition 2.8, 'For n = 0' should be 'For k = 0', since the parameter sphere is S^k throughout the section.
  4. [Proposition 9.5] The statement of Proposition 9.5 does not explicitly record the hypothesis b_2^+(X) > dim(Ξ), although the proof and the surrounding discussion use this to ensure that the family data can be chosen good; please add this condition to the statement.
  5. [Section 9.4, proof of Corollary 5.6] The proof of Corollary 5.6 is very terse: it says the parameterized moduli space is the product of the moduli space on Z and the one-point moduli space for S^2×S^2, but it does not explicitly invoke the identification of the S^1-fibered product with the ordinary product when one factor is a point; a sentence making this identification would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the family Seiberg-Witten invariants are defined from first principles and computed via a gluing theorem proved in the paper; self-citations supply established base cases and tools, not the target conclusion.

full rationale

The derivation is self-contained in the relevant sense. The family Seiberg-Witten invariants SWπk and SWHk are defined from the parameterized moduli spaces in Sections 2.4 and 2.5, with independence from choices proved in Lemma 2.9 and Proposition 2.19; there are no fitted parameters and no invariant is defined in terms of the target diffeomorphism or homology groups. The non-triviality computation (Theorem 5.8) is a recursive calculation whose main analytical input, the Parameterized Irreducible-Reducible Gluing Theorem 5.4, is proved in Section 9 via neck-stretching, linear gluing (Lemma 9.8), and a contraction mapping argument. The base case uses the log-transform Seiberg-Witten computations in Equation (11), the diffeomorphisms of [Rub98] and [BK20], and Park's manifolds as reformulated in [Auc23]; these are established external inputs rather than assumptions of the present conclusion. Self-citations such as [Rub98], [AR23], [Auc23], and [MRS11] occur, but none is invoked to assume the Z-infinite-summand conclusion: [Rub98] supplies the starting diffeomorphisms, [Auc23] supplies building-block manifolds, and [MRS11] is cited only for a one-parameter version of the local metric independence lemma, whose k-parameter adaptation is argued directly in Proposition 9.5. Even if that adaptation is terse, it is a technical completeness issue, not a circular one. No fitted input is renamed a prediction, no known result is merely relabeled, and no uniqueness theorem is imported from the authors' prior work to force the choice. Therefore the central claim does not reduce to its own inputs.

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

The paper introduces no fitted parameters or new physical entities. It relies on standard results in 4-manifold topology and gauge theory, which are listed as axioms. The central construction is explicit and does not tune constants to obtain the claimed summands.

assumptions (7)
  • standard math Seiberg-Witten moduli space theory, including regularity, compactness, and wall-crossing for generic data
    Invoked throughout Section 2 to define family invariants; standard in the field.
  • standard math Wall's theorem that diffeomorphisms of S^2×S^2 act as certain reflections and that all automorphisms of the intersection form are realized after stabilization
    Used in Section 3 to construct the model diffeomorphisms R_{A±B}.
  • standard math Freedman's homeomorphism classification of simply connected 4-manifolds
    Used to obtain homeomorphisms ψ_q between elliptic surfaces in Section 3.
  • domain assumption Gompf's stabilization results for elliptic surfaces and log transforms
    Used to assert that X_q and X_0 become diffeomorphic after one stabilization; a key input for the base case.
  • domain assumption Park's geography theorem providing symplectic manifolds with prescribed intersection forms
    Theorem 4.4 provides the building blocks V, U, W; depends on results from [Par02].
  • standard math Taubes' theorem that symplectic manifolds with b_2^+ > 1 have SW invariant ±1 on the canonical class
    Used in Section 4.2 to guarantee non-vanishing SW invariants for V.
  • standard math Adjunction inequality for Seiberg-Witten basic classes
    Used in Section 5 to show vanishing of SW invariants for certain Spin^c structures.

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Pith. "Pith review of Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory." pith.science (2026). https://pith.science/paper/U6B3IMFB

@misc{pith2026250111892,
  author       = {Pith},
  title        = {Pith review of: Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6B3IMFB}},
  note         = {Machine review of arXiv:2501.11892}
}
abstract

We construct infinite rank summands isomorphic to $\mathbb{Z}^\infty$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with $S^2 \times S^2$. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give $\mathbb{Z}^\infty$ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and $3$-manifolds in various $4$-manifolds, and for the space of positive scalar curvature metrics on standard PSC $4$-manifolds.

Figures

Figures reproduced from arXiv: 2501.11892 by the authors.

Figure 1
Figure 1. Independence from metric and perturbation The proof that SWπk is a homomorphism is straightforward, given the independence from all choices. It can also be deduced from Proposition 2.11 below. The last sentence in the lemma is also straightforward. □ Remark 2.10. A very useful consequence of the independence of SWπk X pα, sq from all choices is that it can often be calculated using constant initial (good) data ϖ0 θ … view at source ↗

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Cited by 1 Pith paper

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

  1. Exotic families of embeddings

    math.GT 2025-01 conditional novelty 6.0 of 10

    Smooth embeddings of 3-manifolds in 4-manifolds that are topologically trivial but smoothly exotic, both as individual embeddings and in families parameterized by spheres, are constructed and detected.

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Works this paper leans on

109 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    Antonelli, Dan Burghelea, and Peter J

    Peter L. Antonelli, Dan Burghelea, and Peter J. Kahn. The concordance-homotopy groups of geometric automorphism groups . Lecture Notes in Mathematics, Vol. 215. Springer-Verlag, Berlin-New York, 1971

    1971

  2. [2]

    Adams and John J

    Robert A. Adams and John J. F. Fournier. Sobolev spaces , volume 140 of Pure and Applied Mathematics (Amsterdam) . Elsevier/Academic Press, Amsterdam, second edition, 2003

    2003

  3. [3]

    Isotopy of surfaces in 4-manifolds after a single stabilization

    Dave Auckly, Hee Jung Kim, Paul Melvin, Daniel Ruberman, and Hannah Schwartz. Isotopy of surfaces in 4-manifolds after a single stabilization. Adv. Math. , 341:609--615, 2019. https://doi.org/10.1016/j.aim.2018.10.040 doi:10.1016/j.aim.2018.10.040

    doi:10.1016/j.aim.2018.10.040 2019

  4. [4]

    Stable isotopy in four dimensions

    Dave Auckly, Hee Jung Kim, Paul Melvin, and Daniel Ruberman. Stable isotopy in four dimensions. J. Lond. Math. Soc. (2) , 91(2):439--463, 2015. https://doi.org/10.1112/jlms/jdu075 doi:10.1112/jlms/jdu075

    doi:10.1112/jlms/jdu075 2015

  5. [5]

    Families of diffeomorphisms and embeddings via D onaldson theory

    Dave Auckly and Daniel Ruberman. Families of diffeomorphisms and embeddings via D onaldson theory. manuscript in preparation, 2023

    2023

  6. [6]

    Exotic families of embeddings

    Dave Auckly and Daniel Ruberman. Exotic families of embeddings. In Frontiers in geometry and topology , volume 109 of Proc. Sympos. Pure Math. , pages 71--98. Amer. Math. Soc., Providence, RI, [2024] 2024

    2024

  7. [7]

    Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023

    David Auckly. Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023. http://arxiv.org/abs/2307.16266 arXiv:2307.16266

    arXiv 2023

  8. [8]

    Non-trivial smooth families of K3 surfaces

    David Baraglia. Non-trivial smooth families of K3 surfaces. Math. Ann. , 387(3-4):1719--1744, 2023. https://doi.org/10.1007/s00208-022-02508-3 doi:10.1007/s00208-022-02508-3

    doi:10.1007/s00208-022-02508-3 2023

Show all 109 references
  1. [9]

    On the mapping class groups of simply-connected smooth 4-manifolds, 2023

    David Baraglia. On the mapping class groups of simply-connected smooth 4-manifolds, 2023. http://arxiv.org/abs/2310.18819 arXiv:2310.18819

    2023 arXiv

  2. [10]

    An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds

    David Baraglia. An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds. Math. Res. Lett. , 31(2):329--352, 2024. https://doi.org/10.4310/mrl.241024232537 doi:10.4310/mrl.241024232537

    2024 doi

  3. [11]

    Infinite loop spaces and positive scalar curvature

    Boris Botvinnik, Johannes Ebert, and Oscar Randal-Williams. Infinite loop spaces and positive scalar curvature. Invent. Math. , 209(3):749--835, 2017. https://doi.org/10.1007/s00222-017-0719-3 doi:10.1007/s00222-017-0719-3

    2017 doi

  4. [12]

    A stable cohomotopy refinement of S eiberg- W itten invariants

    Stefan Bauer and Mikio Furuta. A stable cohomotopy refinement of S eiberg- W itten invariants. I . Invent. Math. , 155(1):1--19, 2004. https://doi.org/10.1007/s00222-003-0288-5 doi:10.1007/s00222-003-0288-5

    2004 doi

  5. [13]

    Knotted 3-balls in S^4 , 2019

    Ryan Budney and David Gabai. Knotted 3-balls in S^4 , 2019. http://arxiv.org/abs/1912.09029 arXiv:1912.09029

    2019 arXiv

  6. [14]

    A gluing formula for families S eiberg- W itten invariants

    David Baraglia and Hokuto Konno. A gluing formula for families S eiberg- W itten invariants. Geom. Topol. , 24(3):1381--1456, 2020. https://doi.org/10.2140/gt.2020.24.1381 doi:10.2140/gt.2020.24.1381

    2020 doi

  7. [15]

    On the B auer- F uruta and S eiberg- W itten invariants of families of 4-manifolds

    David Baraglia and Hokuto Konno. On the B auer- F uruta and S eiberg- W itten invariants of families of 4-manifolds. J. Topol. , 15(2):505--586, 2022. https://doi.org/10.1112/topo.12229 doi:10.1112/topo.12229

    2022 doi

  8. [16]

    Finiteness properties of automorphism spaces of manifolds with finite fundamental group

    Mauricio Bustamante, Manuel Krannich, and Alexander Kupers. Finiteness properties of automorphism spaces of manifolds with finite fundamental group. Math. Ann. , 388(4):3321--3371, 2024. https://doi.org/10.1007/s00208-023-02594-x doi:10.1007/s00208-023-02594-x

    2024 doi

  9. [17]

    Torsion in H -spaces

    William Browder. Torsion in H -spaces. Ann. of Math. (2) , 74:24--51, 1961. https://doi.org/10.2307/1970305 doi:10.2307/1970305

    1961 doi

  10. [18]

    Families of diffeomorphisms and concordances detected by trivalent graphs

    Boris Botvinnik and Tadayuki Watanabe. Families of diffeomorphisms and concordances detected by trivalent graphs. J. Topol. , 16(1):207--233, 2023. https://doi.org/10.1112/topo.12283 doi:10.1112/topo.12283

    2023 doi

  11. [19]

    La stratification naturelle des espaces de fonctions diff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie

    Jean Cerf. La stratification naturelle des espaces de fonctions diff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie. Publ.\ I.H.E.S. , 39:5--173, 1970

    1970

  12. [20]

    Representation theory and homological stability

    Thomas Church and Benson Farb. Representation theory and homological stability. Adv. Math. , 245:250--314, 2013. https://doi.org/10.1016/j.aim.2013.06.016 doi:10.1016/j.aim.2013.06.016

    2013 doi

  13. [21]

    Cappell, R

    S.E. Cappell, R. Lee, and E.Y. Miller. Self-adjoint elliptic operators and manifold decompositions. Part I : Low eigenmodes and stretching. Comm. Pure Appl. Math. , 49:825--866, 1996

    1996

  14. [22]

    Moore, Martin Ro c ek, and Saxenam Vivek

    Jay Cushing, Gregory W. Moore, Martin Ro c ek, and Saxenam Vivek. Superconformal gravity and the topology of diffeomorphism groups, 2023. http://arxiv.org/abs/2311.08394 arXiv:2311.08394

    2023 arXiv

  15. [23]

    More concordance homomorphisms from knot F loer homology

    Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. More concordance homomorphisms from knot F loer homology. Geom. Topol. , 25(1):275--338, 2021. https://doi.org/10.2140/gt.2021.25.275 doi:10.2140/gt.2021.25.275

    2021 doi

  16. [24]

    An infinite-rank summand of the homology cobordism group

    Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong. An infinite-rank summand of the homology cobordism group. Duke Math. J. , 172(12):2365--2432, 2023. https://doi.org/10.1215/00127094-2022-0082 doi:10.1215/00127094-2022-0082

    2023 doi

  17. [25]

    Donaldson and Peter B

    Simon K. Donaldson and Peter B. Kronheimer. The Geometry of Four-Manifolds . Clarendon Press, Oxford, 1990

    1990

  18. [26]

    S. K. Donaldson. An application of gauge theory to four-dimensional topology. J. Differential Geom. , 18(2):279--315, 1983

    1983

  19. [27]

    S. K. Donaldson. Yang- M ills invariants of four-manifolds. In Geometry of low-dimensional manifolds, 1 (Durham, 1989) , pages 5--40. Cambridge Univ. Press, Cambridge, 1990

    1989

  20. [28]

    S. K. Donaldson. The S eiberg- W itten equations and 4 -manifold topology. Bull. Amer. Math. Soc. (N.S.) , 33(1):45--70, 1996

    1996

  21. [29]

    Exotic families of embedded spheres, 2023

    Joshua Drouin. Exotic families of embedded spheres, 2023. to appear

    2023

  22. [30]

    Generalised M iller- M orita- M umford classes for block bundles and topological bundles

    Johannes Ebert and Oscar Randal-Williams. Generalised M iller- M orita- M umford classes for block bundles and topological bundles. Algebr. Geom. Topol. , 14(2):1181--1204, 2014. https://doi.org/10.2140/agt.2014.14.1181 doi:10.2140/agt.2014.14.1181

    2014 doi

  23. [31]

    The positive scalar curvature cobordism category

    Johannes Ebert and Oscar Randal-Williams. The positive scalar curvature cobordism category. Duke Math. J. , 171(11):2275--2406, 2022. https://doi.org/10.1215/00127094-2022-0023 doi:10.1215/00127094-2022-0023

    2022 doi

  24. [32]

    Rational homotopy theory , volume 205 of Graduate Texts in Mathematics

    Yves F\' e lix, Stephen Halperin, and Jean-Claude Thomas. Rational homotopy theory , volume 205 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0105-9 doi:10.1007/978-1-4613-0105-9

    2001 doi

  25. [33]

    Freedman

    Michael H. Freedman. The topology of four--dimensional manifolds. J. Diff.\ Geo. , 17:357--432, 1982

    1982

  26. [34]

    Dirac operators in R iemannian geometry , volume 25 of Graduate Studies in Mathematics

    Thomas Friedrich. Dirac operators in R iemannian geometry , volume 25 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. https://doi.org/10.1090/gsm/025 doi:10.1090/gsm/025

    doi

  27. [35]

    Ronald Fintushel and Ronald J. Stern. Instanton homology of Seifert--fibered homology 3 --spheres. Proc.\ Lond.\ Math.\ Soc. , 61:109--137, 1990

    1990

  28. [36]

    Ronald Fintushel and Ronald J. Stern. Rational blowdowns of smooth 4 -manifolds. J. Differential Geom. , 46(2):181--235, 1997. URL: http://projecteuclid.org/euclid.jdg/1214459932

    1997

  29. [37]

    Freed and Karen Uhlenbeck

    Daniel S. Freed and Karen Uhlenbeck. Instantons and 4-Manifolds . MSRI Lecture notes. Springer-Verlag, 1984

    1984

  30. [38]

    Homology cobordism group of homology 3 -spheres

    Mikio Furuta. Homology cobordism group of homology 3 -spheres. Invent. Math. , 100:339--355, 1990

    1990

  31. [39]

    Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell

    David Gabai, David T. Gay, Daniel Hartman, Vyacheslav Krushkal, and Mark Powell. Pseudo-isotopies of simply connected 4-manifolds, 2023. http://arxiv.org/abs/2311.11196 arXiv:2311.11196

    2023

  32. [40]

    Blaine Lawson, Jr

    Mikhael Gromov and H. Blaine Lawson, Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) , 111(3):423--434, 1980

    1980

  33. [41]

    Gompf and T

    R. Gompf and T. Mrowka. Irreducible 4-manifolds need not be complex. Ann. of Math. , 138:61--111, 1993

    1993

  34. [43]

    Robert E. Gompf. A new construction of symplectic manifolds. Ann. of Math. (2) , 142(3):527--595, 1995

    1995

  35. [44]

    Homological stability for moduli spaces of high dimensional manifolds

    S ren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. II . Ann. of Math. (2) , 186(1):127--204, 2017. https://doi.org/10.4007/annals.2017.186.1.4 doi:10.4007/annals.2017.186.1.4

    2017 doi

  36. [45]

    Homological stability for moduli spaces of high dimensional manifolds

    S ren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. I . J. Amer. Math. Soc. , 31(1):215--264, 2018. https://doi.org/10.1090/jams/884 doi:10.1090/jams/884

    2018 doi

  37. [46]

    A. Hatcher. On the diffeomorphism group of S 1 S 2 . Proc. Amer. Math. Soc. , 83(2):427--430, 1981. https://doi.org/10.2307/2043543 doi:10.2307/2043543

    1981 doi

  38. [47]

    Allen E. Hatcher. A proof of the S male conjecture, Diff (S^ 3 ) O (4) . Ann. of Math. (2) , 117(3):553--607, 1983. https://doi.org/10.2307/2007035 doi:10.2307/2007035

    1983 doi

  39. [48]

    Algebraic Topology

    Allen Hatcher. Algebraic Topology . Cambridge University Press, Cambridge, 2002

    2002

  40. [49]

    A 50-Year View of Diffeomorphism Groups , 2012

    Allen Hatcher. A 50-Year View of Diffeomorphism Groups , 2012. Slides from 50 th C ornell T opology F estival. URL: https://pi.math.cornell.edu/ hatcher/Papers/Diff\

    2012

  41. [50]

    A note on exotic families of 4-manifolds

    Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura. A note on exotic families of 4-manifolds. Proc. Amer. Math. Soc. , 151(6):2695--2705, 2023. https://doi.org/10.1090/proc/16356 doi:10.1090/proc/16356

    2023 doi

  42. [51]

    Homological instability for moduli spaces of smooth 4-manifolds, 2022

    Hokuto Konno and Jianfeng Lin. Homological instability for moduli spaces of smooth 4-manifolds, 2022. http://arxiv.org/abs/2211.03043 arXiv:2211.03043

    2022 arXiv

  43. [52]

    P. B. Kronheimer and T. S. Mrowka. Monopoles and Three-Manifolds . Cambridge University Press, Cambridge, UK, 2008

    2008

  44. [53]

    P. B. Kronheimer and T. S. Mrowka. The D ehn twist on a sum of two K3 surfaces. Math. Res. Lett. , 27(6):1767--1783, 2020. https://doi.org/10.4310/MRL.2020.v27.n6.a8 doi:10.4310/MRL.2020.v27.n6.a8

    2020 doi

  45. [54]

    Exotic codimension-1 submanifolds in 4-manifolds and stabilizations, 2022

    Hokuto Konno, Anubhav Mukherjee, and Masaki Taniguchi. Exotic codimension-1 submanifolds in 4-manifolds and stabilizations, 2022. http://arxiv.org/abs/2210.05029 arXiv:2210.05029

    2022 arXiv

  46. [55]

    Exotically knotted closed surfaces from donaldson's diagonalization for families, 2024

    Hokuto Konno, Abhishek Mallick, and Masaki Taniguchi. Exotically knotted closed surfaces from donaldson's diagonalization for families, 2024. http://arxiv.org/abs/2409.07287 arXiv:2409.07287

    2024 arXiv

  47. [56]

    Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations

    Hokuto Konno. Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations. Journal of Topology , 12(4):1246--1265, 2019. https://doi.org/10.1112/topo.12117 doi:10.1112/topo.12117

    2019 doi

  48. [57]

    Characteristic classes via 4-dimensional gauge theory

    Hokuto Konno. Characteristic classes via 4-dimensional gauge theory. Geom. Topol. , 25(2):711--773, 2021. https://doi.org/10.2140/gt.2021.25.711 doi:10.2140/gt.2021.25.711

    2021 doi

  49. [58]

    The homology of moduli spaces of 4-manifolds may be infinitely generated

    Hokuto Konno. The homology of moduli spaces of 4-manifolds may be infinitely generated. Forum Math. Pi , 12:Paper No. e25, 2024. https://doi.org/10.1017/fmp.2024.26 doi:10.1017/fmp.2024.26

    2024 doi

  50. [59]

    M. Kreck. Isotopy classes of diffeomorphisms of (k-1) -connected almost-parallelizable 2k -manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978) , pages 643--663. Springer, Berlin, 1979

    1978

  51. [60]

    Differential algebraic topology , volume 110 of Graduate Studies in Mathematics

    Matthias Kreck. Differential algebraic topology , volume 110 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2010. From stratifolds to exotic spheres. https://doi.org/10.1090/gsm/110 doi:10.1090/gsm/110

    2010 doi

  52. [61]

    Hilbert stratifolds and a Q uillen type geometric description of cohomology for H ilbert manifolds

    Matthias Kreck and Haggai Tene. Hilbert stratifolds and a Q uillen type geometric description of cohomology for H ilbert manifolds. Forum Math. Sigma , 6:Paper No. e4, 33, 2018. https://doi.org/10.1017/fms.2018.1 doi:10.1017/fms.2018.1

    2018 doi

  53. [62]

    The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary

    Hokuto Konno and Masaki Taniguchi. The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary. Adv. Math. , 409:Paper No. 108627, 58, 2022. https://doi.org/10.1016/j.aim.2022.108627 doi:10.1016/j.aim.2022.108627

    2022

  54. [63]

    Lectures on diffeomorphism groups, 2019

    Alexander Kupers. Lectures on diffeomorphism groups, 2019. https://people.math.harvard.edu/ kupers/teaching/272x/book.pdf

    2019

  55. [64]

    John M. Lee. Introduction to smooth manifolds , volume 218 of Graduate Texts in Mathematics . Springer, New York, second edition, 2013

    2013

  56. [65]

    The family Seiberg-Witten invariant and nonsymplectic loops of diffeomorphisms, 2022

    Jianfeng Lin. The family Seiberg-Witten invariant and nonsymplectic loops of diffeomorphisms, 2022. http://arxiv.org/abs/2208.12082 arXiv:2208.12082

    2022 arXiv

  57. [66]

    Isotopy of the D ehn twist on K3\, \#\, K3 after a single stabilization

    Jianfeng Lin. Isotopy of the D ehn twist on K3\, \#\, K3 after a single stabilization. Geom. Topol. , 27(5):1987--2012, 2023. https://doi.org/10.2140/gt.2023.27.1987 doi:10.2140/gt.2023.27.1987

    1987 doi

  58. [67]

    Family blowup formula, admissible graphs and the enumeration of singular curves

    Ai-Ko Liu. Family blowup formula, admissible graphs and the enumeration of singular curves. I . J. Differential Geom. , 56(3):381--579, 2000. URL: http://projecteuclid.org/euclid.jdg/1090347696

    2000

  59. [68]

    Family S eiberg- W itten invariants and wall crossing formulas

    Tian-Jun Li and Ai-Ko Liu. Family S eiberg- W itten invariants and wall crossing formulas. Comm. Anal. Geom. , 9(4):777--823, 2001

    2001

  60. [69]

    Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture , 2021

    Jianfeng Lin and Anubhav Mukherjee. Family Bauer--Furuta invariant, Exotic Surfaces and Smale conjecture , 2021. http://arxiv.org/abs/2110.09686 arXiv:2110.09686

    2021 arXiv

  61. [70]

    Decomposing analytic surfaces

    Richard Mandelbaum. Decomposing analytic surfaces. In Geometric topology ( P roc. G eorgia T opology C onf., A thens, G a., 1977) , pages 147--217. Academic Press, New York-London, 1979

    1977

  62. [71]

    Irrational connected sums and the topology of algebraic surfaces

    Richard Mandelbaum. Irrational connected sums and the topology of algebraic surfaces. Trans. Amer. Math. Soc. , 247:137--156, 1979. https://doi.org/10.2307/1998778 doi:10.2307/1998778

    1979 doi

  63. [72]

    A monopole invariant for families of contact structures

    Juan Muñoz-Echániz. A monopole invariant for families of contact structures. Advances in Mathematics , 439:109483, March 2024. URL: http://dx.doi.org/10.1016/j.aim.2023.109483, https://doi.org/10.1016/j.aim.2023.109483 doi:10.1016/j.aim.2023.109483

    2024

  64. [74]

    On the topology of simply connected algebraic surfaces

    Richard Mandelbaum and Boris Moishezon. On the topology of simply connected algebraic surfaces. Trans. Amer. Math. Soc. , 260(1):195--222, 1980

    1980

  65. [75]

    Morgan, Tomasz S

    John W. Morgan, Tomasz S. Mrowka, and Zolt \'a n Szab \'o . Product formulas along T 3 for S eiberg- W itten invariants. Math. Res. Lett. , 4:915--929, 1997

    1997

  66. [76]

    Complex surfaces and connected sums of complex projective planes

    Boris Moishezon. Complex surfaces and connected sums of complex projective planes . Springer-Verlag, Berlin, 1977. With an appendix by R. Livne, Lecture Notes in Mathematics, Vol. 603

    1977

  67. [77]

    S. Morita. Characteristic classes of surface bundles. Invent. Math. , 90:551--578, 1987. URL: http://eudml.org/doc/143521

    1987

  68. [78]

    John W. Morgan. The S eiberg- W itten equations and applications to the topology of smooth four-manifolds , volume 44 of Mathematical Notes . Princeton University Press, Princeton, NJ, 1996

    1996

  69. [79]

    Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant

    Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant . J. Differential Geom. , 88:333--377, 2011. https://doi.org/10.2140/agt.2014.14.3689 doi:10.2140/agt.2014.14.3689

    2011 doi

  70. [80]

    An index theorem for end-periodic operators

    Tomasz Mrowka, Daniel Ruberman, and Nikolai Saveliev. An index theorem for end-periodic operators. Compos. Math. , 152(2):399--444, 2016. https://doi.org/10.1112/S0010437X15007502 doi:10.1112/S0010437X15007502

    2016 doi

  71. [81]

    Towards an enumerative geometry of the moduli space of curves

    David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, V ol. II , volume 36 of Progr. Math. , pages 271--328. Birkh\" a user Boston, Boston, 1983

    1983

  72. [82]

    The stable moduli space of R iemann surfaces: M umford's conjecture

    Ib Madsen and Michael Weiss. The stable moduli space of R iemann surfaces: M umford's conjecture. Ann.Math. , 165(3):843--941, 2007. https://doi.org/10.4007/annals.2007.165.843 doi:10.4007/annals.2007.165.843

    2007 doi

  73. [83]

    Nicolaescu

    Liviu I. Nicolaescu. Notes on S eiberg- W itten theory . American Mathematical Society, Providence, RI, 2000

    2000

  74. [84]

    Nicolaescu

    Liviu I. Nicolaescu. On the C appell- L ee- M iller gluing theorem. Pacific J. Math. , 206(1):159--185, 2002. https://doi.org/10.2140/pjm.2002.206.159 doi:10.2140/pjm.2002.206.159

    2002 doi

  75. [85]

    Mapping class groups for simply-connected 4-manifolds with boundary, 2022

    Patrick Orson and Mark Powell. Mapping class groups for simply-connected 4-manifolds with boundary, 2022. J . D iff. G eom., to appear. http://arxiv.org/abs/2207.05986 arXiv:2207.05986

    2022 arXiv

  76. [86]

    Ozsv\' a th, Andr\' a s I

    Peter S. Ozsv\' a th, Andr\' a s I. Stipsicz, and Zolt\' a n Szab\' o . Concordance homomorphisms from knot F loer homology. Adv. Math. , 315:366--426, 2017. https://doi.org/10.1016/j.aim.2017.05.017 doi:10.1016/j.aim.2017.05.017

    2017 doi

  77. [87]

    The geography of S pin symplectic 4-manifolds

    Jongil Park. The geography of S pin symplectic 4-manifolds. Math. Zeitschrift , 240(2):405--421, 2002

    2002

  78. [88]

    Pseudo-isotopies et isotopies en dimension quatre dans la cat\'egorie topologique

    Bernard Perron. Pseudo-isotopies et isotopies en dimension quatre dans la cat\'egorie topologique. Topology , 25:381--397, 1986

    1986

  79. [89]

    Perelman

    G. Perelman. Construction of manifolds of positive R icci curvature with big volume and large B etti numbers. In Comparison geometry ( B erkeley, CA , 1993--94) , volume 30 of Math. Sci. Res. Inst. Publ. , pages 157--163. Cambridge Univ. Press, Cambridge, 1997

    1993

  80. [90]

    Exotic diffeomorphisms on 4 -manifolds with b_2^+ = 2 , 2024

    Haochen Qiu. Exotic diffeomorphisms on 4 -manifolds with b_2^+ = 2 , 2024. http://arxiv.org/abs/2409.07009 arXiv:2409.07009

    2024 arXiv

  81. [91]

    Isotopy of 4 -manifolds

    Frank Quinn. Isotopy of 4 -manifolds. J. Differential Geom. , 24(3):343--372, 1986. URL: http://projecteuclid.org/euclid.jdg/1214440552

    1986

  82. [92]

    Manifolds of positive scalar curvature: a progress report

    Jonathan Rosenberg. Manifolds of positive scalar curvature: a progress report. In Surveys in differential geometry. V ol. XI , volume 11 of Surv. Differ. Geom. , pages 259--294. Int. Press, Somerville, MA, 2007. https://doi.org/10.4310/SDG.2006.v11.n1.a9 doi:10.4310/SDG.2006.v11.n1.a9

    2007 doi

  83. [93]

    Metrics of positive scalar curvature and connections with surgery

    Jonathan Rosenberg and Stephan Stolz. Metrics of positive scalar curvature and connections with surgery. In Surveys on surgery theory, Vol. 2 , pages 353--386. Princeton Univ. Press, Princeton, NJ, 2001

    2001

  84. [94]

    An obstruction to smooth isotopy in dimension 4

    Daniel Ruberman. An obstruction to smooth isotopy in dimension 4 . Math. Res. Lett. , 5(6):743--758, 1998. https://doi.org/10.4310/MRL.1998.v5.n6.a5 doi:10.4310/MRL.1998.v5.n6.a5

    1998 doi

  85. [95]

    A polynomial invariant of diffeomorphisms of 4-manifolds

    Daniel Ruberman. A polynomial invariant of diffeomorphisms of 4-manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998) , pages 473--488 (electronic). Geom. Topol., Coventry, 1999. https://doi.org/10.2140/gtm.1999.2.473 doi:10.2140/gtm.1999.2.473

    1998 doi

  86. [96]

    Positive scalar curvature, diffeomorphisms and the S eiberg- W itten invariants

    Daniel Ruberman. Positive scalar curvature, diffeomorphisms and the S eiberg- W itten invariants. Geom. Topol. , 5:895--924 (electronic), 2001. https://doi.org/10.2140/gt.2001.5.895 doi:10.2140/gt.2001.5.895

    2001 doi

  87. [97]

    Homotopy groups of Diff(X) and Homeo(X)

    Danny Ruberman. Homotopy groups of Diff(X) and Homeo(X) . MathOverflow, 2020. http://arxiv.org/abs/https://mathoverflow.net/q/373503 arXiv:https://mathoverflow.net/q/373503

    2020

  88. [98]

    Diffeomorphisms of discs , 2020

    Oscar Randal-Williams. Diffeomorphisms of discs , 2020. Slides from lecture at MIT . https://www.dpmms.cam.ac.uk/\

    2020

  89. [99]

    S. Smale. An infinite dimensional version of S ard's theorem. Amer. J. Math. , 87:861--866, 1965. https://doi.org/10.2307/2373250 doi:10.2307/2373250

    1965 doi

  90. [100]

    From flops to diffeomorphism groups

    Gleb Smirnov. From flops to diffeomorphism groups. Geom. Topol. , 26(2):875--898, 2022. https://doi.org/10.2140/gt.2022.26.875 doi:10.2140/gt.2022.26.875

    2022 doi

  91. [101]

    On the structure of manifolds with positive scalar curvature

    Richard Schoen and Shing Tung Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math. , 28(1-3):159--183, 1979

    1979

  92. [102]

    Positive R icci curvature on the connected sums of S^n S^m

    Ji-Ping Sha and DaGang Yang. Positive R icci curvature on the connected sums of S^n S^m . J. Differential Geom. , 33(1):127--137, 1991. URL: http://projecteuclid.org/euclid.jdg/1214446032

    1991

  93. [103]

    Positive R icci curvature on compact simply connected 4 -manifolds

    Ji-Ping Sha and DaGang Yang. Positive R icci curvature on compact simply connected 4 -manifolds. In Differential geometry: R iemannian geometry ( L os A ngeles, CA , 1990) , volume 54, Part 3 of Proc. Sympos. Pure Math. , pages 529--538. Amer. Math. Soc., Providence, RI, 1993....

    1990 doi

  94. [104]

    Characteristic cohomotopy classes for families of 4-manifolds

    Markus Szymik. Characteristic cohomotopy classes for families of 4-manifolds. Forum Math. , 22(3):509--523, 2010. https://doi.org/10.1515/FORUM.2010.027 doi:10.1515/FORUM.2010.027

    2010 doi

  95. [105]

    C. Taubes. Self-dual connections on non-self-dual manifolds. J. Differential Geom. , 17:139--170, 1982

    1982

  96. [106]

    C. Taubes. L^2 --moduli spaces on 4 --manifolds with cylindrical ends , volume 1 of Monographs in Geometry and Topology . International Press, 1994

    1994

  97. [107]

    The S eiberg- W itten invariants and symplectic forms

    Clifford Henry Taubes. The S eiberg- W itten invariants and symplectic forms. Math. Res. Lett. , 1(6):809--822, 1994

    1994

  98. [108]

    C. T. C. Wall. Diffeomorphisms of 4 -manifolds. J. London Math. Soc. , 39:131--140, 1964

    1964

  99. [109]

    Theta-graph and diffeomorphisms of some 4-manifolds, 2020

    Tadayuki Watanabe. Theta-graph and diffeomorphisms of some 4-manifolds, 2020. http://arxiv.org/abs/2005.09545 arXiv:2005.09545

    2020 arXiv

  100. [110]

    E. Witten. Monopoles and four-manifolds. Math.\ Res.\ Lett. , 1(6):769--796, 1994

    1994

  101. [111]

    The B auer- F uruta invariant and a cohomotopy refined R uberman invariant

    Ming Xu. The B auer- F uruta invariant and a cohomotopy refined R uberman invariant . ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)--State University of New York at Stony Brook. URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:d...

    2004

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