A full subcomplex makes $\\mathcal Z_K$ a retract of $\\mathcal Z_L$: homology embedding in every degree

Let $K$ be a finite abstract simplicial complex on $s$ vertices and $L$ one on $m$ vertices, and let $e:V(K)\hookrightarrow V(L)$ be an injection of vertex sets which identifies $K$ with the full subcomplex of $L$ spanned by $e(V(K))$; that is, a finite set $\sigma$ of vertices of $K$ is a face of $K$ if and only if $e(\sigma)$ is a face of $L$. Then for every degree $q$ there is an injective additive homomorphism

The mechanism is the standard one for full subcomplexes: the coordinate map that extends a point of $\mathcal Z_K$ by the value $1$ in the coordinates outside $e(V(K))$ lands in $\mathcal Z_L$, and the coordinate projection $\mathcal Z_L\to\mathcal Z_K$, $w\mapsto (w_{e(i)})_{i\in V(K)}$, is a left inverse of it. Fullness is exactly what makes the projection land in $\mathcal Z_K$: the preimage of a face of $L$ is a face of $K$. Applying a homology functor to a retraction turns it into a split injection of abelian groups.

Formalization Note The moment-angle space is the disk-circle polyhedral product of the ambient interface file, homology is integral singular homology in the fixed degree $q$, and the conclusion is an explicit injective additive homomorphism. No sphere, dimension or neighbourliness hypothesis is imposed on either complex.