4-manifolds, surgery on loops and geometric realization of Tietze transformations

The lens space realization problem

We determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere. Specifically, if surgery along a knot produces a lens space, then there exists an equivalent surgery along a Berge knot with the same knot Floer homology groups. This leads to sharp information about the genus of such a knot. The arguments rely on tools from Floer homology and lattice theory, and are primarily combinatorial in nature.

手术谱序列和分层流形

Surgery spectral sequence and stratified manifolds

Cappell和Shaneson在1978年指出了Browder - Livesay不变量的有趣性质，这些不变量类似于某些谱序列中的微分。这种谱序列是由Hambleton和Kharshiladze在1991年构造的。这个谱序列与用闭流形的法线映射实现沃尔群的元素的问题密切相关。构建谱序列的主要步骤是对谱进行无限滤波，其中只有前两个谱具有明确的几何意义。第一个是用于歧管$X$的手术阻塞组的谱$\Bbb L(π_1(X))$，第二个$\Bbb LP_*(F)$是用于Browder-Livesay歧管对$Y\subset X$上的手术的谱。过滤的第三项的几何意义由Muranov，Repoveli和Spaggiari在2002年解释。本文给出了Hambleton和Kharshiladze构造中所有滤过谱的几何解释。我们引入组的障碍手术系统的embeddedd流形，并证明了频谱，实现这些群体符合频谱的过滤Hambleton和Kharshiladze。我们描述了代数和几何性质的引进障碍群和它们的关系，经典的手术理论。证明了层流形的Browder-Quinn $L$ -群与引入群之间的同构。给出了结果在闭流形手术问题和迭代Browder-Livesay不变量中的应用。

The Bryant-Ferry-Mio-Weinberger construction of generalized manifolds

Following Bryant, Ferry, Mio and Weinberger we construct generalized manifolds as limits of controlled sequences p_i: X_i --> X_{i-1} : i = 1,2,... of controlled Poincaré spaces. The basic ingredient is the epsilon-delta-surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only in cases when the target is a manifold, but a controlled Poincaré complex, we explain this issue very roughly. Specifically, it is applied in the inductive step to construct the desired controlled homotopy equivalence p_{i+1}: X_{i+1} --> X_i. Our main theorem requires a suffici…

A link surgery spectral sequence in monopole Floer homology

To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the delta-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well-adapted to Khovanov homology. More generally, we construct new bigraded invariants of a framed l…

棱柱quandle $P_7$ 在标准四维球面中的纤维实现

Fibred realizations of the prism quandle $P_7$ in the standard four-sphere

本文证明由四元数棱柱流形构造的两个同伦四维球面均为标准，并给出棱柱quandle阶五十六的两个不等价纤维实现，解决了阶七情形。

$\mathbb{Z}$-环面外部与小结手术四流形

$\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds

本文通过切割-粘贴操作替换环面邻域，构造了与$S^2\times S^2$和$\\#_3(S^2\times S^2)$同胚的单连通四流形及光滑同伦$4$-球面，并证明相关Seiberg-Witten不变量消失，所得流形非辛。