arXiv subjects
Publications and source records attributed to Walid Taamallah.
For a topological space $X$, the space of finite subsets $\mathrm{Sub}_n X$ consists of non-empty subsets of $X$ of cardinality at most $n$. We compute the Euler characteristic of $\mathrm{Sub}_n X$ for any space $X$ having the homotopy type of a finite CW complex. We obtain the explicit formula $\chi(\mathrm{Sub}_n X) = \sum_{i=1}^n \binom{\chi(X)}{i}$.
The notion of Kan extendable subcategories was initially introduced to define the category of compactly generated fibrewise topological spaces over a T1 base space and to establish its cartesian closure. In this paper, we show that the same framework can likewise be applied to define the category of compactly generated strongly Hausdorff locales and to prove that it, too, is cartesian closed
Permutation products and their various "fat diagonal" subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $π_1(X)$ and $H_1(X;\bbz)$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\bbz)$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.