In this article, the author defines an invariant of rational homology 3-spheres equipped with a contact structure as an element of a cohomotopy set of the Seiberg–Witten Floer spectrum as defined in Manolescu (Geometry Topol 7(2):889–932, 2003). Furthermore, in light of the equivalence established in Lidman and Manolescu (Astérisque 399:25, 2018) between the Borel equivariant homology of said spectrum and the Seiberg–Witten Floer homology of Kronheimer and Mrowka (Monopoles and three-manifolds, vol. 10, Cambridge University Press, Cambridge, 2007), the author shall show that this homotopy theoretic invariant recovers the already well known contact element in the Seiberg–Witten Floer cohomology (vid. e.g. Kronheimer et al. in Ann Math 20:457–546, 2007) in a natural fashion. Next, the behaviour of the cohomotopy invariant is considered in the presence of a finite covering. This setting naturally asks for the use of Borel cohomology equivariant with respect to the group of deck transformations. Hence, a new equivariant contact invariant is defined and its properties studied. The invariant is then computed in one concrete example, wherein the author demonstrates that it opens the possibility of considering scenarios hitherto inaccessible.

在这篇文章中，作者定义了一个具有接触结构的有理同调 3-球体的不变量作为 Seiberg-Witten Floer 谱的上同伦集的一个元素，如在 Manolescu 中定义的那样（Geometry Topol 7(2):889–932， 2003）。此外，根据 Lidman 和 Manolescu (Astérisque 399:25, 2018) 在所述光谱的 Borel 等变同源与 Kronheimer 和 Mrowka 的 Seiberg-Witten Floer 同源之间建立的等价性（单极子和三流形，第 10 卷， Cambridge University Press, Cambridge, 2007)，作者将证明这个同伦理论不变量恢复了 Seiberg–Witten Floer 上同调中众所周知的接触元素（vid. eg Kronheimer et al. in Ann Math 20:457–546, 2007 ) 以一种自然的方式。下一个，在存在有限覆盖的情况下考虑上同伦不变量的行为。此设置自然要求使用 Borel 上同调等变关于甲板变换组。因此，定义了一个新的等变接触不变量并研究了它的性质。然后在一个具体示例中计算不变量，其中作者证明它打开了考虑迄今为止无法访问的场景的可能性。