Quantifying the topology of large–scale structure

We propose and investigate a new algorithm for quantifying the topological properties of cosmological density uctuations. We rst motivate this algorithm by drawing a formal distinction between two de nitions of relevant topological characteristics, based on concepts, on the one hand, from di erential topology and, on the other, from integral geometry. The former approach leads one to concentrate on properties of the contour surfaces which, in turn, leads to the algorithms CONTOUR2D and CONTOUR3D familiar to cosmologists. The other approach, which we adopt here, actually leads to much simpler algorithms in both two and three dimensions. (The 2D algorithm has already been introduced to the astronomical literature.) We discuss the 3D case in some detail and compare results obtained with it to analogous results using the CONTOUR3D algorithm.