标题： 交换对合的 bordism 的普遍性质 显示英文标题

标题： The universal property of bordism of commuting involutions

摘要： 我们提出了一种形式化方法，以捕捉具有交换对合的光滑流形的等变边群环的结构。 我们引入了定向 el$_2^{RO}$-代数的概念，这是一种代数结构，为所有初等阿贝尔 2-群提供了带表示的环，这些环通过限制同态连接，包含一个预欧拉类和一个逆 Thom 类；这些数据受到一个精确性性质的约束。 除了等变边群外，定向全局环谱也产生定向 el$_2^{RO}$-代数，因此存在许多例子。 反转逆 Thom 类会产生一个全局 2-挠群定律。 从这个意义上说，我们的定向 el$_2^{RO}$-代数是全局 2-挠群定律的非局部化推广。 我们的主要结果表明，初等阿贝尔 2-群的等变边群是一个初始的定向 el$_2^{RO}$-代数。 几种其他有趣的等变同调理论也可以在初等阿贝尔 2-群上通过类似的普遍性质来表征。 我们证明，稳定的等变边群是一个带有可逆方向的初始 el$_2^{RO}$-代数； 常系数模 2 的 Bredon 同调是一个带有加法方向的初始 el$_2^{RO}$-代数；而模 2 系数的 Borel 等变同调是一个带有同时加法和可逆方向的初始 el$_2^{RO}$-代数。 显示英文摘要

摘要： We propose a formalism to capture the structure of the equivariant bordism rings of smooth manifolds with commuting involutions. We introduce the concept of an oriented el$_2^{RO}$-algebra, an algebraic structure featuring representation graded rings for all elementary abelian 2-groups, connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom class; this data is subject to one exactness property. Besides equivariant bordism, oriented global ring spectra also give rise to oriented el$_2^{RO}$-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el$_2^{RO}$-algebras are delocalized generalizations of global 2-torsion group laws. Our main result shows that equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra. Several other interesting equivariant homology theories can also be characterized, on elementary abelian 2-groups, by similar universal properties. We prove that stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el$_2^{RO}$-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el$_2^{RO}$-algebra with an orientation that is both additive and invertible.