On the topology of complexes of injective words

Wojtek Chacholski, Ran Levi, Roy Meshulam (Corresponding Author)

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An injective word over a finite alphabet V is a sequence w=v1v2⋯vt of distinct elements of V. The set Inj(V) of injective words on V is partially ordered by inclusion. A complex of injective words is the order complex Δ(W) of a subposet W⊂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 Δ(W) , where W is the downward closed subposet of Inj(V) generated by a set of permutations of V. In particular, we determine the homotopy type of Δ(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 Γ(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.
Original languageEnglish
Number of pages16
JournalJournal of applied and computational topology
Early online date21 Aug 2019
Publication statusE-pub ahead of print - 21 Aug 2019



  • Homology
  • Random complexes
  • order dimension

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