剪刀自同构群 I：同调稳定性与 K-理论
Scissors automorphism groups I: Homological stability and K-theory
AI总结：
本文证明在温和条件下剪刀自同构群的同调独立于对象，可由剪刀同余 K-理论谱表达，提供群论解释与计算方法，并在经典情形及后续中得出同调计算与结构结果。
摘要：
在任何具有合理覆盖概念的范畴中，每个对象都有一组剪刀自同构。我们证明，在温和条件下，该群的同调与对象无关，并且可以用 Zakharevich 定义的剪刀同余 K-理论谱来表示。因此，我们既获得了 Zakharevich 高阶剪刀同余 K-理论的群论解释，也获得了一种计算剪刀自同构群同调的方法。在经典的凸多胞形剪刀同余情形中，这导致了剪刀自同构群同调的计算和结构结果。在两篇后续文章中，我们表明更一般地，我们的方法导致群论和动力学中出现的各种群族的同调计算，恢复了 Szymik-Wahl、Li 和 Tanner 的结果。
英文摘要：
In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.
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