The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$. The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial complex formed by the $2^n$ standard simplices $\Delta^{n-1}$ and all of its faces. I want to investigate subcomplexes of $S^{n-1}_1$ made by subsets of these $2^n$ simplices, and also their relations with their restrictions to subspaces of the form $\{v: v_{i_1} = v_{i_2} = \cdots = v_{i_k} = 0\}$. For example, when $n=3$, one can choose 4 of the 8 simplices such that one only intersects another at a single point. This space is homotopy equivalent to a sphere with 4 holes in it. When this complex is restricted to any of the 3 coordinate planes, we get back circles.

I'm particularly interested in some kind of characterization of their homologies (or maybe even homotopies, but seeing as we can't even compute all the sphere homotopy groups...). This doesn't seem like too exotic a question, so I'm guessing there are prior work. However, I haven't been able to find any such work by googling the terms that come to me, so I'm hoping someone can perhaps give some references on papers that are relevant.


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