TY - JOUR

T1 - On the topology of complexes of injective words

AU - Chacholski, Wojtek

AU - Levi, Ran

AU - Meshulam, Roy

N1 - W. Chacholski: Supported by VR Grant 2014-04770. R. Levi: Supported by EPSRC Grant EP/P025072/1. R. Meshulam: Supported by ISF Grant 326/16.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Homology

KW - Random complexes

KW - order dimension

UR - http://www.mendeley.com/research/topology-complexes-injective-words

U2 - 10.1007/s41468-019-00039-6

DO - 10.1007/s41468-019-00039-6

M3 - Article

VL - 4

SP - 29

EP - 44

JO - Journal of applied and computational topology

JF - Journal of applied and computational topology

SN - 2367-1726

ER -