We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.

翻译：我们引入并研究了交换协作上同调扩张——一种单纯复形的新扩张性质。我们证明了球面建筑的交换协作上边界扩张下界，并利用这些下界推导了LSV拉马努金复形的交换协作上同调扩张下界。本研究的动机源于近期工作（在姊妹论文中）表明交换协作上同调扩张蕴含一致性定理。这两项工作共同表明这些复形在低接受率机制下支持一致性检验。交换协作上同调扩张的定义基于对给定复形 $X$ 考虑其面复形 $F^r X$：该复形的顶点为 $X$ 的 $r$ 维面，且两个顶点相连当且仅当它们的不交并也是 $X$ 中的一个面。面复形 $F^r X$ 是 $X$ 自身 $r$ 次乘积的去随机化版本，其底层图即 $X$ 的交换游走——已知该图具有卓越的谱扩张性质。$X$ 的交换协作上同调扩张定义为 $F^r X$ 的上同调扩张。我们的主要结果是：球面建筑的交换协作上边界扩张下界为 $\exp(-O(\sqrt r))$，LSV复形的交换协作上同调扩张下界同样为 $\exp(-O(\sqrt r))$。对于更一般的上边界扩张子，我们证明了一个较弱的 $\exp(-O(r))$ 下界。