Washington Mio, Andrew Ranicki
An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger construction of 2n-dimensional exotic homology manifolds.
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Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms.
End Khovanov homology and exotic Lagrangian planes
本研究定义了四维空间真嵌入曲面的环境微分同胚不变量——端Khovanov同调，并借助该不变量首次发现了四维空间中的异乎寻常拉格朗日平面与辛平面实例。
Exotic $\mathbb{R}^4$'s, RBG Links, and End Floer Homology
研究通过将微分同胚卡森柄附于微分同胚圆盘补集构造非微分同胚奇异\(\mathbb{R}^4\)，利用 RBG 链环构造找特殊切片纽结，借助端弗洛尔同调区分奇异\(\mathbb{R}^4\)。
Distinguishing exotic $\mathbb{R}^4$'s with Heegaard Floer homology
通过Heegaard Floer同调区分exotic $\mathbb{R}^4$'s，证明不同切结绳生成的exotic $\mathbb{R}^4$'s不相抵，并重新证明Y×$\mathbb{R}$具有无限多个光滑结构。
Khovanov homology can distinguish exotic Mazur manifolds
本文提出利用 Khovanov 同调区分 exotic Mazur 流形的新方法，无需依赖 skein lasagna 模块，并首次给出了 exotic Mazur 流形存在的分析自由证明。
Khovanov homology and exotic $4$-manifolds
通过Khovanov-Rozansky gl_2 skein lasagna模区分怪异结迹，首次证明了怪异紧致定向4维流形的存在，并提出了无法通过规范/弗洛尔理论方法恢复的怪异结迹家族。
Exotic disks and singular instanton Floer homology
利用带Chern-Simons滤波的奇异瞬子Floer同调构造奇异切片圆盘对，并证明存在强可逆Z-切片纽结，其任意对称Z-圆盘对均为奇异且稳定后仍奇异。
A spanning tree model for Khovanov homology, Rasmussen's s-invariant and exotic discs in the $4$-ball
本文为Khovanov同调的生成树复形明确定义了微分形式，描述了Rasmussen的s-不变量，并据此发现了一个在4-球中界定奇异圆盘的无限纽结族。
Wall-type stabilization problems investigate the collapse of exotic 4-dimensional phenomena under stabilization operations (e.g., taking connected sums with $S^2 \times S^2$). We propose an elementary approach to these problems, providing a construction of exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization -- including examples in the setting of closed, simply connected 4-manifolds. As a proof of concept, we show this construction yields exotic surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps …
A list of problems prepared for the proceedings of the Workshop on Exotic Homology Manifolds, Oberwolfach June 29-July 5 2003.
Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way, we show that the cobordism map induced by an ascending surface in a Weinstein cobordism preserves the transverse invariant in knot Floer homology.
We show that a certain involution on the well known homology sphere (the boundary of the Mazur manifold) induces a nontrivial homomorphism on its Heegard-Floer homology groups (recently defined by Ozsvath and Szabo). We discuss a possible application of this to constructing exotic smooth structures on 4-manifolds.
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $ν$ is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants $τ$ and $ε$ are not in…
Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$
该研究确定了维数非4的奇异环面中保向微分同痕类群到SL_d(Z)的同态为分裂满射的情形，计算了一类奇异环面的映射类群，证明了其SL_d(Z)作用的同调性质，并得出许多奇异环面无非平凡SL_d(Z)微分作用的结论。
The rational homology balls $B_n$ appeared in Fintushel and Stern's rational blow-down construction [FS] and were subsequently used (e.g. Fintushel-Stern[FS4], Park[Pa2]) to construct exotic smooth manifolds with small Euler numbers. We show that a large class of smooth 4-manifolds have all of the $B_n$'s for odd $n \geq 3$ embedded in them. In particular, we give explicit examples, using Kirby calculus, of several families of smooth embeddings of the rational homology balls $B_n$.
From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. We also discuss the induced contact structures on their boundaries. Furthermore, for any smooth 4-manifold pair (Z,Y) such that the complement Z-intY is a handlebody without 3- and 4-handles and with b_2>0, we construct arbitrary many…
Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure σon M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,σ), while the other one can be defined using the intersection pairing of an appropriate compact oriented 4-manifold with boundary M. In this paper, using surgery presentations of the manifold M, we prove that those two quadratic functions coincide. Our proof relies on the comparison between two distinct combinatorial descriptions of Spin^…
New constructions and invariants of closed exotic 4-manifolds
该研究借助Heegaard Floer同调提出可在覆盖中保持区分性的新型闭四维流形不变量，并构造多类闭异质4-流形。
In this paper we present the full details of the construction of a Morse-Floer type homology related to the super-quadratic perturbation of the Dirac-geodesic model. This homology is computed explicitly using a Leray-Serre type spectral sequence and this computation leads us to several existence results of Dirac-geodesics.
On localizing groups of exotic diffeomorphisms of 4-manifolds
该研究围绕4-流形外徽微分同胚群的无限生成能否局部化到更小嵌入子流形的问题展开，通过Seiberg-Witten理论等得到多项相关结论，并指出4-流形映射类群也有类似结果。
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, {\em absolute} means that the exotic structure is not relative to a particular parameterization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible ho…
Bar-Natan skein lasagna modules and exotic surfaces in 4-manifolds
本文利用 Bar-Natan Khovanov 同调构造绞合 lasagna 模，并通过连通和粘合映射在 4 维流形中生成带边界的奇异曲面，证明一次内部稳定化通常不足以消除其奇异性。
Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions
本研究证明纽结Floer同调的数值不变量$ν$和$|ε|$是任意整数$n$的纽结$n$-迹不变量，拓展后可构造新怪异对族，还通过新不变量给出4-流形作为边界切触结构强辛填充的拓扑阻碍。
In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homology of these manifolds adapted to their construction as fibre sums. We also calculate the canonical classes of the symplectic structures. This gives a new proof for a formula derived by A. Akhmedov, R. I. Baykur and B. D. Park.
In this addendum, we give a differential form interpretation of the proof of the main theorem of arXiv:1812.02448, which gives lower bounds of the dimensions of $π_k(B\mathrm{Diff}(D^4,\partial))\otimes\mathbb{Q}$ in terms of the dimensions of Kontsevich's graph homology, and explain why it can be extended to arbitrary even dimensions $d\geq 4$. We attempted to make the proof accessible to more readers. Thus we do not assume familiarity with configuration space integrals nor knowledge of finite type invariants. Part of this addendum might be joined to the original article when it will be re-su…
Insensitivity of Khovanov homology under rim surgery
研究\(B^4\)中光滑定向恰当嵌入曲面边缘手术对考夫曼同调的影响，通过局部环面替换操作，结合已有工作得出奇异曲面，利用考夫曼缠结千层饼模给出两个证明。
Exotic S^2 x S^2 candidates that are just fake
本文证明两个无限族的同调 S^2×S^2 流形是非球面的，从而排除它们是奇异流形，方法是从非球面构建块重建，并推广到其他单连通候选者。
Exotic families of embeddings
本文构造了3-流形在4-流形中的拓扑平凡但光滑非平凡的嵌入族，包括$S^4$中同调球的不同痕嵌入及爆破K3曲面中任意3-流形的嵌入族，并给出将非平凡嵌入族转化为非平凡子流形族的一般技术。
We conjecture two generalisations of Elkies' theorem on unimodular quadratic forms to non-unimodular forms. We give some evidence for these conjectures including a result for determinant 3. These conjectures, when combined with results of Froyshov and of Ozsvath and Szabo, would give a simple test of whether a rational homology 3-sphere may bound a negative-definite four-manifold. We verify some predictions using Donaldson's theorem. Based on this we compute the four-ball genus of some Montesinos knots.
From Link Homology to Topological Quantum Field Theories
该综述介绍连接链环同调理论与光滑4-流形不变量及扩展拓扑量子场论的进展，讲述满足额外条件的函子链环同调的相关性质及扩展，还提及链模的作用、计算方法、结构性质、应用等，并将其置于扩展拓扑量子场论框架中。
Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a generator of $H_3(X)$, a suitable version of the Heegaard Floer $d$ invariant of $Y$, defined using twisted coefficients, is a diffeomorphism invariant of $X$. We show how this invariant can be used to obstruct embeddings of certain types of $3$-manifolds, including those obtained as a connected sum of a rational homology $3$-sphere and any number of copies of $S^1 …
We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with much more control of these genus functions than the compact setting seems to allow. As an application, we expand the range of 4-manifolds known to have exotic smoothings (up to diffeomorphism). For example, every 2-handlebody interior (possibly infinite or nonorientable) has an e…
This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q vanishes on the kernel of the map in homology to a bounding 3-manifold. This is used by Kreck and Puppe (arXiv:0707.1599 [math.AT]) who refer for a proof to an email of the author to Kreck. A more polished and public proof seems desirable. In [KT], section 6, a Pin^- structure is …
We generalize a result of the author about the classification of 1-connected 7-manifolds and demonstrate its use by two concrete applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-manifolds with the cohomology ring of $S^2 \times S^5 \sharp S^3 \times S^4$. The answer is in terms of generalized Kreck-Stolz invariants, which in the case of 2-connected 7-manifolds is equivalent to a quadratic refinement of the linking form and a generalized Eells-Kuiper inva…
Exotic Dehn twists on sums of two contact 3-manifolds
本文利用推广的 Kronheimer–Mrowka 接触不变量与凸球面 h-原理，证明两个闭接触3-流形连通和上的 Dehn 扭转可形成无限阶奇异接触映射类，并给出 Colin 分解定理的参数化版本及过扭转情形的新现象。
Triple torsion, triple cup products, and embedding obstructions for rational homology 3-spheres
本文通过挠-链环对偶将Freedman-Krushkal的三重挠形式与模p三重杯积等同，利用Milnor不变量给出符号公式，并构造具有任意模p三重杯张量的有理同调3-球面，最终证明存在无局部平坦嵌入S^4的有理同调3-球面。
Exotic knottings and symmetries of surfaces in 4-manifolds
该研究通过环境对称性，结合rim surgery、凸几何与双曲几何，构造4维流形中的射影刚性、拓扑灵活曲面及全测地曲面，揭示奇异纽结与精细对称性现象。
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold. The conjectures we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL_2 of an imaginary quadratic field. The proof of this theorem involves ruling out the existence of an irreducible two …
Protocorks and monopole Floer homology
本文引入 protocorks 类流形，证明其扭转与 Seiberg-Witten 不变量变化及 Floer 同调分裂定理，改进 Morgan-Szabó 结果并推广约化 Floer 同调贡献的结论。
We study the nilpotent part $N'$ of a pseudo-periodic automorphism $h$ of a real oriented surface with boundary $Σ$. We associate a quadratic form $Q$ defined on the first homology group (relative to the boundary) of the surface $Σ$. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of $\tilde Q$ to the absolute homology group of $Σ$ is even whenever the quotient of the …
We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold and we use it to give …
Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof
本文研究同胚但不同胚于 CP^2 的辛 4-流形及其有限阶辛同胚的存在性，将其归结为 12 个接触管道流形的辛填充问题，并确定了伪自由辛 Z_p-作用的阶与局部表示，提出潜在构造或不存在性证明。
A monopole invariant for families of contact structures
本文利用monopole Floer homology将Kronheimer--Mrowka--Ozsváth--Szabó接触不变量推广至接触结构族，研究3-流形上接触结构空间的拓扑，并以此阻碍自然纤维化的截面存在性，从而检测出具有凸球面边界的接触3-流形上的奇异环。
A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k$ surgery on $(2,q)$ torus knots are made and compared with a different generalization of Casson's invariant to SU(3) by Boden and Herald. For those cases computed, the invariant defined here is a quadratic polynomial in k for k>0 and a quadratic polynomial for k<0.
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