Topology of tensor ranks

Abstract

We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over C, the set of rank-r tensors and the set of symmetric rank-r symmetric tensors are both path-connected if r is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over C. Over R, the set of rank-r tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-r symmetric d-tensors depends on the order d: connected when d is odd but not when d is even. Border rank and symmetric border rank over R have essentially the same path-connectedness properties as rank and symmetric rank over R. When r is greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in R2⊗R2⊗R2 fall into four connected components. For multilinear rank, the manifold of d-tensors of multilinear rank (r1,…,rd) in Cn1⊗…⊗Cnd is always path-connected, and the same is true in Rn1⊗…⊗Rnd unless ni=ri=∏j≠irj for some i∈{1,…,d}. Beyond path-connectedness, we determine, over both R and C, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank.