论文标题
同源性类中的Hypersurfaces复合体
The Complex of Hypersurfaces in a Homology Class
论文作者
论文摘要
对于紧凑的平滑$ n $ -manifold $ m $和一个编码 - $ 1 $同源类$ ϕ \ in \ perperatorname {h} _ {n-1}（n-1}（m，m，\ partial m）$，我们调查了一个简单的复杂$ \ nathcal $ \ nathcal {s}}}^\ dagger $ sulting $ suped $ sulting $ supering n off the usting y M. $ ϕ $。它的定义类似于其他经典复合物的定义，例如表面的曲线复合物或结的kakimizu复合物，其差异是高表面不被占据同位素。 我们证明$ \ Mathcal {S}^\ Dagger（m，ϕ）$已连接，并且仅在每个维度$ n $中连接。我们还显示了类似的复杂$ \ Mathcal {T}^\ Dagger（m，ϕ）$适用于$ 3 $维情况的连接性，仅考虑Thurston Norm-sorter surfaces。连接性结果被传输到复合物$ \ MATHCAL {s}（m，ϕ），\ MATHCAL {t}（m，ϕ）$，在其中将超曲面带到了同位素上，对于$ n = 2 $，简单的连接结果也随之而来。我们还简要讨论了Turaev研究的上下文的扩展，该环境中的常规图表使用$ 2 $ - 复合物代表$ 1 $二维的共同体学课程。 We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use connectedness of $\mathcal{T}^\dagger(M, ϕ)$ to define a new $\ell^2$-invariant of $2$-dimensional homology classes in irreducible and boundary-irreducible定向紧凑的连接$ 3 $ - 具有空或环边界的manifolds。
For a compact oriented smooth $n$-manifold $M$ and a codimension-$1$ homology class $ϕ\in \operatorname{H}_{n-1}(M, \partial M)$, we investigate a simplicial complex $\mathcal{S}^\dagger(M, ϕ)$ relating the properly embedded hypersurfaces in $M$ representing $ϕ$. Its definition is akin to that of other classical complexes, such as the curve complex of a surface or the Kakimizu complex of a knot, with the difference that hypersurfaces are not taken up to isotopy. We prove that $\mathcal{S}^\dagger(M, ϕ)$ is connected and simply connected in every dimension $n$. We also show connectedness of a similar complex $\mathcal{T}^\dagger(M, ϕ)$ adapted to the $3$-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(M, ϕ), \mathcal{T}(M, ϕ)$ where hypersurfaces are taken up to isotopy, and for $n=2$ the simple connectedness result carries over as well. We also briefly discuss extensions to a context studied by Turaev, where regular graphs in $2$-complexes are used to represent $1$-dimensional cohomology classes. We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use connectedness of $\mathcal{T}^\dagger(M, ϕ)$ to define a new $\ell^2$-invariant of $2$-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected $3$-manifolds with empty or toroidal boundary.
