论文标题
圆圈的越野式公制圆形增厚
Vietoris-Rips Metric Thickenings of the Circle
论文作者
论文摘要
先前已提出越越野式度量，以理解越野裂式的简单络合物及其持续的同源性的替代方法。最近的工作表明，对于完全有限的度量空间，越野式度量度的度量增厚具有持久的同源条形码，与越野式 - 裂片简单络合物一致，而忽略了条形的端点是打开还是封闭。将此结果与圆圈的越野斗式简单络合物的已知同质类型和条形码相结合，可以将圆圈的越野河流度量的条形码推导为端点，并且对其同质拷贝的构成构成。我们确认这些猜想是正确的，证明了圆的越野杆度量增厚是同质的，等于在预期尺度参数下等于奇数球。我们的方法是找到保留同型类型的度量增厚的商，并表明商空间可以描述为CW复合物。相对于比例参数，商图也很自然，因此可以直接证明公制增厚的持续同源性。
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the circle can be deduced up to endpoints, and conjectures have been made about their homotopy types. We confirm these conjectures are correct, proving that the Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters. Our approach is to find quotients of the metric thickenings that preserve homotopy type and show that the quotient spaces can be described as CW complexes. The quotient maps are also natural with respect to the scale parameter and thus provide a direct proof of the persistent homology of the metric thickenings.
