On Vietoris–Rips complexes of hypercube graphs

We describe the homotopy types of Vietoris–Rips complexes of hypercube graphs at small scale parameters. In more detail, let Q_n be the vertex set of the hypercube graph with 2^n vertices, equipped with the shortest path metric. Equivalently, Q_n is the set of all binary strings of length n , equipped with the Hamming distance. The Vietoris–Rips complex of Q_n at scale parameter zero is 2^n points, and the Vietoris–Rips complex of Q_n at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris–Rips complex of Q_n at scale parameter two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres. Many questions about the Vietoris–Rips complexes of Q_n at larger scale parameters remain open.