Homotopy decomposition of diagonal arrangements

Abstract

Given a space $X$ and a simplicial complex $K$ with $m$-vertices, the arrangement of partially diagonal subspaces of $X^m$, called the dragonal arrangement, is defined. We decompose the suspension of the diagonal arrangement when $2(dim K + 1) < m$, which generalizes the result of Labassi. As a corollary, we calculate the Euler characteristic of the complement when $X$ is a closed connected manifold.