The Euler Characteristic of Finite Subset Spaces

Walid Taamallah

The Euler Characteristic of Finite Subset Spaces

Algebraic Topology 2026-07-20 v1

Authors: Walid Taamallah

Abstract

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 $\chi(\mathrm{Sub}_n X) = \sum_{i=1}^n \binom{\chi(X)}{i}$ .

Keywords

Euler characteristic finite subset spaces CW complex homotopy type combinatorial formula

Cite

@article{arxiv.2607.17830,
  title  = {The Euler Characteristic of Finite Subset Spaces},
  author = {Walid Taamallah},
  journal= {arXiv preprint arXiv:2607.17830},
  year   = {2026}
}