Paolo Lisca
We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.
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Genus three Goeritz groups of connected sums of two lens spaces
该研究证明两个透镜空间连通和的亏格3 Heegaard分解的映射类群有限生成，对应约化球面复形均连通，为低维拓扑中透镜空间连通和的Heegaard分解结构研究提供关键结果。
Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions
本文用特定 Morse-Bott 函数刻画表示为 Lens 空间、$S^2 \ imes S^1$ 和环面在圆上的丛的连通和的 $3$ 维流形，强化并明确了 Saeki 2006 年的相关结果。
We determine when there exists a ribbon rational homology cobordism between two connected sums of lens spaces, i.e. one without $3$-handles. In particular, we show that if a lens space $L$ admits a ribbon rational homology cobordism to a different lens space, then $L$ must be homeomorphic to $L(n,1)$, up to orientation-reversal. As an application, we classify ribbon $χ$-concordances between connected sums of $2$-bridge links. Our work builds on Lisca's work on embeddings of linear lattices.
On lens spaces bounding smooth 4-manifolds with $\boldsymbol{b_2=1}$
该文研究不同拓扑下可界定b₂=1的光滑4-流形的透镜空间，构造出两类无穷族透镜空间并证明其拓扑性质，研究受有理同调射影平面相关问题启发。
$π_1$-injective bounding and application to 3- and 4-manifolds
本文研究闭流形的$π_1$-单射入界，证明基本群的剩余有限性与有限性可提升至入界流形，并给出其在透镜空间配边、有限群作用及最小入界指数中的应用。
We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casson-Walker invariant of a lens space equals the average of its Heegaard-Floer correction terms. Additionally, we find an obstruction for the equality and equality with opposite sign, of two correction terms of the same lens space. Using this obstruction we are able to derive an opt…
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.
Cubiquitous Lattices and Branched Covers bounding rational balls
本文研究哪些子晶格是处处存在的，并利用Wu障碍物对它们进行分类，应用于双分支覆盖和有理球边界问题。
In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein manifold. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.
The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces, and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$…
On lens space surgeries from the Poincaré homology sphere
本文基于Greene的changemaker格发展格嵌入障碍，确定庞加莱同调球面上纽结手术产生透镜空间的条件，证明半整数手术仅有两个特例，并得出双曲纽结的透镜空间手术必为整数的结论。
A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spaces or $S^2 \times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precis…
It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an alternative definition of rational balls, $A_{m,n}$, bounding $L(p^2,pq-1)$ by their handlebody decompositions alone. We show that these two families coincide - answering a question of Kadokami and Yamada. To that end, we show that each $A_{m,n}$ admits a Stein filling of the "st…
By taking the complements of embeddings of sphere plumbings in connected sums of $\mathbb{C} P^2$, we construct examples of simply connected four-manifolds with lens space boundary and $b_2 = 1$. The resulting boundaries include many lens spaces that cannot come from integer surgery on any knot in $S^3$, so the corresponding four-manifolds cannot be built by attaching a single two-handle to $B^4$. Using similar constructions, we give an example of an embedded sphere in $\#^4 \mathbb{C} P^2$ with self-intersection number $20$, and conjecture that this is the maximum possible.
We study a class of 3-manifolds called strong L-spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L-space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L-space admitting a strong Heegaard diagram of genus two via an explicit classification. We also show that there exist finitely many strong L-spaces with bounded order of first homology; for instance, through order eig…
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