论文标题
具有边界的4个manifolds的差异和同构的群体群
The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary
论文作者
论文摘要
我们使用Manolescu的Seiberg-inten Floer稳定同型类型对具有边界的4个模型的光滑家族进行限制，但前提是家族对边界的光纤限制是3个manifolds的琐碎家族。作为一个应用程序，我们表明，对于简单连接的紧凑型紧凑型4个manifold $ x $，边界具有界限，并在Frøyshov不变或Manolescu不变式$α，β，γ$ of $ \ partial x $的$ \ pottial x $，包含地图\ mathrm {homeo}（x，\ partial）$之间的差异性和同构的组之间的固定在边界方面固定边界并不是一个弱同质的等效性。这与维度3中的经典结果相结合，意味着包含映射$ \ mathrm {diff}（x）（x）\ hookrightArrow \ mathrm {homeo}（x）$在$ \ partial x $上的同一假设下也不是弱同质量。我们的约束将两种限制概括对Baraglia证明的封闭的4个manifolds的光滑家族和唐纳森型定理，用于最初是由于弗罗伊斯霍夫（Frøyshov）而造成的，具有边界的光滑4个manifolds。
We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold $X$ with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants $α, β, γ$ of $\partial X$, the inclusion map $\mathrm{Diff}(X,\partial) \hookrightarrow \mathrm{Homeo}(X,\partial)$ between the groups of diffeomorphisms and homeomorphisms which fix the boundary pointwise is not a weak homotopy equivalence. This combined with a classical result in dimension 3 implies that the inclusion map $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ is also not a weak homotopy equivalence under the same assumption on $\partial X$. Our constraints generalize both of constraints on smooth families of closed 4-manifolds proven by Baraglia and a Donaldson-type theorem for smooth 4-manifolds with boundary originally due to Frøyshov.
