Cappell-Shaneson's 4-dimensional s-cobordism

Cappell-Shaneson 结绳对与其相同的Alexander多项式

Cappell-Shaneson knot pairs with the same Alexander polynomial

本文构造了无限多新的Cappell-Shaneson结绳对，并给出了具有相同Alexander多项式但不等价的结绳对例子。

无穷多族标准 Cappell-Shaneson 球面

Infinite families of standard Cappell-Shaneson spheres

本文证明了更多的 Cappell-Shaneson 同伦 4 维球面是标准的，并给出了 145 个微分同胚于标准 4 维球面的新无穷族。

Browder-Livesay filtrations and the example of Cappell and Shaneson

Let $M^3$ be a 3-dimensional manifold with fundamental group $π_1(M)$ which contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$ which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of $σ(f)\in L_5(π_1(M))$ is based on consideration of a Browder-Livesay filtration of a manifold $X$ with $π_1(X)\cong π_1(M)$. For a Browder-Livesay pair $Y^{n-1…

Augmentation Rank of Satellites with Braid Pattern

Given a knot $K$ in $S^3$, a question raised by Cappell and Shaneson asks if the meridional rank of $K$ equals the bridge number of $K$. Using augmentations in knot contact homology we consider the persistence of equality between these two invariants under satellite operations on $K$ with a braid pattern. In particular, we answer the question in the affirmative for a large class of iterated torus knots.

二歧分支覆盖的配边群

Cobordism groups of dihedral branched covers

该研究计算了 S³ 的二歧 n 重分支覆盖的配边群，证明其为循环群，由 (2,n) 环面纽结的 n 重连通二歧覆盖生成，并用推广 Cappell-Shaneson 特征纽结的配边不变量确定其同构类型。

Miraculous Cancellation and Pick's Theorem

We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber-Novikov algebra in the complex cobordism ring of a point.

Man and machine thinking about the smooth 4-dimensional Poincaré conjecture

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen's s-invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots K for which nonzero s(K) would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those K, with the computations showing that s was…