Embedding-complex invariance conjecture for statistics of topological excitations

Let $M_1$ and $M_2$ be manifolds, and let $\mathcal{C}_{p+1}$ be the collection of all finite $(p+1)$-dimensional simplicial complexes. Define

$$\mathcal{C}_{p+1}(M)=\{C\in\mathcal{C}_{p+1}\mid \text{there is an embedding }C\longrightarrow M\}.$$

For excitations of dimension at most $p$, let $T_p(M,G)$ denote the statistics group with fusion group $G$.

Embedding-complex invariance conjecture. If

$$\mathcal{C}_{p+1}(M_1)=\mathcal{C}_{p+1}(M_2),$$

then

$$T_p(M_1,G)\simeq T_p(M_2,G).$$

This proposes that statistics of excitations up to dimension $p$ are determined by which finite $(p+1)$-dimensional complexes embed in the ambient manifold. The source presents this as a conjectural form of dimensional and global-topological dependence.