标题： 论Bauer--Furuta构造 显示英文标题

标题： On the Bauer--Furuta construction

摘要： 使用拓扑空间上谱层的六函子形式主义，我们提供了Bauer--Furuta不变量及其族版本的新构造。这种方法避免了基于有限维子空间近似的传统论证，而是采用相对于巴拿赫空间之间弗雷德霍姆映射的博尔-莫里同调谱。这里的一个关键观察是，巴拿赫流形之间的$C^1$-可微弗雷德霍姆映射是局部紧致的，从而定义了急进函子，其对偶对象可以描述为阿蒂亚--辛格族指标的 Thom 谱。我们还概述了具有李群作用的拓扑空间上真正等变谱层的稳定同伦理论的可能候选。在此背景下，我们研究了适当的前推函子，该函子容纳了真正的等变 Bauer--Furuta 不变量。 显示英文摘要

摘要： Using the six-functor formalism for sheaves of spectra on topological spaces, we provide a novel construction of the Bauer--Furuta invariant, as well as its family version. This approach avoids the conventional arguments based on approximations by finite-dimensional subspaces, and we instead employ the Borel--Moore homology spectra relative to Fredholm maps between Banach spaces. A key observation here is that $C^1$-differentiable Fredholm maps between Banach manifolds are locally proper, thereby defining the shriek functors, whose dualizing objects may be described as the Thom spectra of the Atiyah--Singer families index. We also outline a possible candidate for the stable homotopy theory of genuine equivariant sheaves on topological spaces with Lie group actions. In this context, we investigate the proper pushforward functor, which accommodates the genuine equivariant Bauer--Furuta invariant.