## Top homology of hypergraph matching complexes, $p$-cycle
complexes and Quillen complexes of symmetric groups

**
John Shareshian and Michelle L. Wachs**
**Abstract:**
We investigate the representation of a symmetric group $S_n$ on the homology of its Quillen
complex at a prime $p$. For homology groups in small codimension, we derive an explicit formula
for this representation in terms of the representations of symmetric groups on homology groups of
$p$-uniform hypergraph matching complexes. We conjecture an explicit formula for the
representation of $S_n$ on the top homology group of the corresponding hypergraph matching complex
when $n \equiv 1 \bmod p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the
conjecture when $p=3$.

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