Computing cohomology of configuration spaces
Computing cohomology of configuration spaces
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计算配置空间的上同调
DOI：
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发表时间：
2016
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通讯作者：
Derek Francour
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作者：
M. Maguire;W. M. Christie;Derek Francour
本文给出了一个具体的方法来显式地计算连通、定向、闭、偶维有限型流形的无序构形空间的有理上同调，该方法已在Sage [S+09]中实现.作为应用，我们给出了无序构形空间的稳定和不稳定有理上同调的完整计算，包括$mathbb{CP}^3$和亏格为1的Riemann曲面，它等价于椭圆辫群的同调.在附录中，我们还给出了无序位形空间的不稳定和稳定Betti数的大表。由此，我们经验地观察到一些流形的无序构形空间的不稳定上同调的稳定性现象，其中一些我们证明了，其中一些我们陈述为猜想。
We give a concrete method to explicitly compute the rational cohomology of the unordered configuration spaces of connected, oriented, closed, even-dimensional manifolds of finite type which we have implemented in Sage [S+09]. As an application, we give acomplete computation of the stable and unstable rational cohomology of unordered configuration spaces in some cases, including that of $mathbb{CP}^3$ and a genus 1 Riemann surface, which is equivalently the homology of the elliptic braid group. In an appendix, we also give large tables of unstable and stable Betti numbers of unordered configuration spaces. From these, we empirically observe stability phenomenon in the unstable cohomology of unordered configuration spaces of some manifolds, some of which we prove and some of which we state as conjecture.