Marco Golla, Kyle Larson
We produce a rational homology 3-sphere that does not smoothly bound either a positive or negative definite 4-manifold. Such a 3-manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3-manifold obtained by Dehn surgery on a knot. The proof requires an analysis of short characteristic covectors in bimodular lattices.
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A survey on embeddings of 3-manifolds in definite 4-manifolds
该综述梳理定4-流形中3-流形嵌入领域的最新进展，重点阐述Donaldson对角化定理与整格组合学的关键作用，并补充了双向透镜空间在负定流形中嵌入的新成果。
Given a knot K in the three-sphere, we address the question: which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant, and compute it for torus knots.
Exotic definite four-manifolds with non-cyclic fundamental group
本文在基本群为 $\mathbb{Z}/2\times \mathbb{Z}/2$ 的定四维流形上构造了无穷多个两两非微分同胚的光滑结构，揭示了该流形上光滑结构的丰富性。
Let $X$ be a smooth simply connected closed 4-manifold with definite intersection form. We show that any automorphism of the intersection form of $X$ is realized by a diffeomorphism of $X \mathbin{\#} S^2 \times S^2$. This extends and completes Wall's foundational result from 1964.
We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite 4-manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the the knot concordance group.
Smooth structures on definite four-manifolds with infinite fundamental group
本文构造了无限多个非微分同构的光滑结构，其阿贝尔化为Z/2pZ×Z/2Z，针对具有无限基本群的定義四维流形。
Two criteria for a closed connected definite 4-manifold with infinite cyclic fundamental group to be TOP-split are given. One criterion extends a sufficient condition made in a previous paper. The result is equivalent to a purely algebraic result on the question asking when a positive definite Hermitian form over the ring of integral one-variable Laurent polynomials is represented by an integer matrix. As an application, an infinite family of orthogonally indecomposable unimodular odd definite symmetric $Z$-forms is produced.
Exotic definite four-manifolds with non-trivial fundamental group
本文受Levine-Lidman-Piccirillo结果启发，在基本群为Z/2Z且具有定交形式的闭四维流形上构造了无穷多个奇异光滑结构，并推广到更多含偶数b_2^+的情形。
On submersions with definite folds of manifolds with boundary into Euclidean spaces
研究带边流形到欧几里得空间的具有确定折叠的子mersions的微分拓扑性质，通过边界限制为圆折叠或像简单折叠映射，探讨流形的微分同胚类型和欧拉示性数。
We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite 4-manifold with trivial first homology and bounded second betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.
Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.
The possibilities for new or unusual kinds of topological, locally linear periodic maps of non-prime order on closed, simply connected 4-manifolds with positive definite intersection pairings are explored. On the one hand, certain permutation representations on homology are ruled out under appropriate hypotheses. On the other hand, an interesting homologically nontrivial, pseudofree, action of the cyclic group of order 25 on a connected sum of ten copies of the complex projective plane is constructed.
We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.
We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over $S^2$ can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson's theorem and Fintushel-Stern's $R$-invariant. We also give a short proof of the characterisation of Seifert fibered spaces which smoothly bound rational homology $S^1 \times D^3$'s.
We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.
In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algorithmic: the procedures enable us to construct various explicit examples. We also study simple stable maps of $3$--manifolds into the $2$--sphere without definite fold. Furthermore, we prove the non-existence of singular Legendre fibrations on $3$--manifolds, answering negatively t…
Definite four-manifolds with exotic smooth structures
本文研究了具有基本群Z_2和定号相交形式的闭定向4-流形的光滑结构，构造了无限多个第二Betti数为1和2的不可约定号四维流形，并证明了此类正第二Betti数流形若容许光滑结构则必容许无限多个。
Plumbed 3-Manifolds and Neumann Moves
本文通过Neumann移动将弱负定的管状树转化为负定树，利用Duchon等人的对角化算法提取框架矩阵的特征值，系统消除正特征值。
For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,q$, we obtain an infinite family of irreducible rational homology 3-spheres which are homology cobordant to the lens space $L(p,q)$ but cannot obtained by a knot surgery.
We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Frøyshov, and Ozsváth and Szabó. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.
We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.
Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition $X \cong X_1 # X_2$ then both $b_2(X_1)$ and $b_2(X_2)$ are divisible by four; this rules out a previously natural appearing source of 4-manifolds with non-empty moduli space. We give in some detail a constru…
The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsvath and Szabo for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type invariants of links of normal surface singularities.
From zero surgeries to candidates for exotic definite four-manifolds
本文研究通过0-手术同胚构造具有相同零手术的结对，以系统生成奇异四维流形候选者，并通过计算机实验找到5个拓扑切片结，若其中任一切片则可得到奇异四维球面。
We derive a cut-and-paste surgery formula of Seiberg--Witten invariants for negative definite plumbed rational homology 3-spheres. It is similar to (and motivated by) Okuma's recursion formula [arXiv:math.AG/0610464, 4.5] targeting analytic invariants of splice quotient singularities. The two formulas combined provide automatically a proof of the equivariant version [arXiv:math.AG/0310084, 5.2(b)] of the `Seiberg--Witten invariant conjecture' [arXiv:math.AG/0111298] for these singularities.
Tautological classes of definite 4-manifolds
本文证明了紧致光滑单连通定号4-流形的同义反复类对角化定理，利用Seiberg-Witten理论的族版本方法，完全确定了$\mathbb{CP}^2$等流形的同义反复环并推导出普遍成立的线性关系。
On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory
本文利用瞬子 Floer 理论推广 Froyshov 方法，分类了以特定同调三维球面为边界的定号格，并证明 (2,5) 环面结 +1 手术情形下非对角格仅为 E8 和秩 12 的不可分解幺模定号格。
We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. As an example we show that small positive surgeries on torus knots do not bound negative-definite four-manifolds.
Slicing knots in definite 4-manifolds
本文研究纽结的$\mathbb{CP}^2$-切片数，基于二重分支覆盖和Seifert形式分别给出光滑与拓扑切片数的下界和上界，并发现了两者均有限、非零且不同的纽结。
An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-mani…
In this note, we present a proof of Donaldson's Diagonalization Theorem via an abelian gauge-theoretic variant of the Seiberg-Witten equations for multiple spinors. Like the other proof of Donaldson's theorem using the standard Seiberg-Witten theory, Elkies' theorem also plays a key role in our argument.
On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with $b_{1}=1$
本文针对$b_1=1$的负半定配边3-流形定义对合Heegaard Floer版本的$d$型不变量，证明其自旋同调配边不变性，建立相关约束与障碍，计算实例并推广了纽结0-手术相关的不可实现性结果。
Weights of finite cyclic actions on definite $4$-manifolds
本文用等变Seiberg–Witten理论重新证明了正定4-流形上有限循环群作用的不动点集和切向表示与线性作用的等变连通和一致，并确定了所有等变线丛的权重。
We prove an existence theorem for gauge invariant $L^2$-normal neighborhoods of the reduction loci in the space ${\cal A}_a(E)$ of oriented connections on a fixed Hermitian 2-bundle $E$. We use this to obtain results on the topology of the moduli space ${\cal B}_a(E)$ of (non-necessarily irreducible) oriented connections, and to study the Donaldson $μ$-classes globally around the reduction loci. In this part of the article we use essentially the concept of harmonic section in a sphere bundle with respect to an Euclidean connection. Second, we concentrate on moduli spaces of instantons on defin…
In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to $-E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and the maximal rank of even definite filling is arbitrarily large.
Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of Cochran-Harvey-Horn. Such a 4-manifold is necessarily homeomorphic to a (punctured) connected sum of CP(2)'s. We characterize all links that are slice in a (punctured) connected sum of CP(2)'s in terms of ribbon moves and an operation which we call adding a generalized positive crossing. We find obstructions in the form of the Levine-Tristram signature function, …
Handle decomposition complexity and representation spaces
本文证明存在同调三维球面可界定定号四维流形，但任何这样的流形必须包含大量把手，论证基于瞬子Floer同调中的不变量Γ。
We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.
A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences
研究具有半定截面曲率的4维爱因斯坦度量，通过刻画满足特定逐点不等式的此类度量，得出一系列推论，包括在非正曲率闭爱因斯坦及凯勒 - 爱因斯坦4维流形研究中的应用，还得到了凯勒 - 爱因斯坦情形下的尖锐不等式。
3-manifolds without any embedding in symplectic 4-manifolds
本文利用Heegaard Floer修正项与instanton模空间构造不能界定正定或负定流形的L-空间，证明存在无穷多个无法嵌入闭辛4-流形的闭3-流形，并反驳相关猜想。
We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of s…
Knot lattice homology and $q$-series invariants for plumbed knot complements
本研究提出负定管道纽结补空间的统一不变量，整合了纽性格点同调与BPS $q$-级数，是加权分次根的推广，并证明了关联该不变量与术后流形加权分次根的手术公式。
Mazur manifolds and symplectic structures
利用Heegaard Floer同调的cobordism映射，阻碍了Akbulut-Kirby流形上存在辛结构的可能性，并表明存在具有确定交形式的exotic对。
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