# On the topology of complexes of injective words

## TL;DR

This paper studies the topology of complexes formed by injective words, especially permutation complexes, determining their homotopy types, decompositions, and connections to other topological and combinatorial structures, with applications in neuroscience.

## Contribution

It characterizes the homotopy types of permutation complexes generated by two permutations and shows all stable homotopy types are realizable within this framework.

## Key findings

- Homotopy type of permutation complexes generated by two permutations is determined.
- Any stable homotopy type can be realized by a permutation complex.
- A homotopy decomposition for complexes associated with simplicial complexes is provided.

## Abstract

An injective word over a finite alphabet $V$ is a sequence $w=v_1v_2\cdots v_t$ of distinct elements of $V$. The set $\mathrm{inj}(V)$ of injective words on $V$ is partially ordered by inclusion. A complex of injective words is the order complex $\Delta(W)$ of a subposet $W \subset \mathrm{inj}(V)$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $\Delta(W)$, where $W$ is the downward closed subposet of $\mathrm{inj}(V)$ generated by a set of permutations of $V$. In particular, we determine the homotopy type of $\Delta(W)$ when $W$ is generated by two permutations, and prove that any stable homotopy type is realizable by a permutation complex. We describe a homotopy decomposition for the complex of injective words $\Gamma(K)$ associated with a simplicial complex $K$, and point out a connection to a result of Randal-Williams and Wahl. Finally, we discuss some probabilistic aspects of random permutation complexes.