Theorem 1.7 — Arbitrary torsion in moment-angle and loop-space homology

Let $G$ be a finitely generated abelian group. Prove that there exist a natural number $m$ and an abstract simplicial complex $L$ on Fin m whose geometric realization is homeomorphic to the unit sphere $S^4\subset\mathbb R^5$, such that the same moment-angle space $\mathcal Z_L$ admits injective additive homomorphisms

Here $\mathcal Z_L$ is the union, over faces $\sigma$ of $L$, of coordinate products having a disk $D^2$ in the coordinates of $\sigma$ and a circle $S^1$ elsewhere. The loop space is based at the all-ones point, and $H_*$ denotes the direct sum of integral singular homology over all nonnegative degrees.

The same simplicial $4$-sphere must witness both embeddings. The claim concerns additive subgroups; the embeddings need not occupy the same homological degree or preserve a graded ring structure.

Formalization Note The sphere condition is an actual homeomorphism, the moment-angle and loop spaces are concrete topological spaces, and the two subgroup claims are explicit injective additive homomorphisms.