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 $\mathbb{C}$, 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 $\mathbb{C}$. Over $\mathbb{R}$, 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 $\mathbb{R}$ have essentially the same path-connectedness properties as rank and symmetric rank over $\mathbb{R}$. 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 $\mathbb{R}^2 \otimes \mathbb{R}^2 \otimes \mathbb{R}^2$ fall into four connected components. For multilinear rank, the manifold of $d$-tensors of multilinear rank $(r_1,\dots,r_d)$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ is always path-connected, and the same is true in $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ unless $n_i = r_i = \prod_{j \ne i} r_j$ for some $i\in\{1, \dots, d\}$. Beyond path-connectedness, we determine, over both $\mathbb{R}$ and $\mathbb{C}$, 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.

中文翻译：

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张量秩的拓扑

我们研究了由张量秩、边界秩、多线性秩定义的张量集的路径连通性和同伦群，以及它们的对称张量的对称对应物。我们表明，在 $\mathbb{C}$ 上，如果 $r$ 不超过复杂的通用秩，秩-$r$ 张量集合和对称秩-$r$ 对称张量集合都是路径连通的; 这些结果也扩展到 $\mathbb{C}$ 上的边界秩和对称边界秩。在 $\mathbb{R}$ 上，如果 rank-$r$ 张量集具有预期的维度，则它是路径连接的，但对称 rank-$r$ 对称 $d$-张量的相应结果取决于顺序 $ d$：当 $d$ 为奇数时连接，但当 $d$ 为偶数时不连接。$\mathbb{R}$ 上的边界秩和对称边界秩与 $\mathbb{R}$ 上的秩和对称秩具有基本相同的路径连通性属性。当 $r$ 大于复杂的通用等级时，我们无法辨别任何一般模式：例如，我们显示 $\mathbb{R}^2 \otimes \mathbb{R}^ 中的边界等级三张量2 \otimes \mathbb{R}^2$ 分为四个连通分量。对于多重线性秩，$\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d 中多重线性秩 $(r_1,\dots,r_d)$ 的 $d$-张量的流形}$ 总是路径连接的，在 $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ 中也是如此，除非 $n_i = r_i = \prod_{j \ne i} r_j$ 表示一些 $i\in\{1, \dots, d\}$。除了路径连通性之外，我们在 $\mathbb{R}$ 和 $\mathbb{C}$ 上确定，固定小阶张量集合的基本和更高同伦群，以及利用 Bott 周期性，固定多线性阶张量流形的基群。对于固定对称秩或固定对称多重线性秩的对称张量，我们还获得了这些结果的类似物。