Eilenberg–MacLane homology conjecture for statistics of topological excitations

Let $p$-dimensional topological excitations have a finite Abelian fusion group $G$, let $S^d$ be the $d$-sphere, and let $\partial\Delta_{d+1}$ be the boundary of the $(d+1)$-dimensional simplex. Let $K(G,d-p)$ denote an Eilenberg–MacLane space.

Eilenberg–MacLane homology conjecture. The following three groups are equivalent:

1. $T_p(S^d,G)$;
2. $T_p(\partial\Delta_{d+1},G)$;
3. $H_{d+2}(K(G,d-p),\mathbb{Z})$.

The conjecture is motivated by agreement with known anyon and fermionic loop statistics and with homology groups of Eilenberg–MacLane spaces. The equivalence of the first two groups is tied to triangulation and topological-space invariance, while the other equivalence is supported by computational results and remains conjectural.