We say that a simplicial complex $K$ is acyclic if it's integral reduced simplicial homology groups are trivial in all dimensions.


For a vertex ${v} \in K$, we define the link


$$lk(v) :=\{\sigma \in K \; | \; \sigma \cup \{v\} \in K, \sigma \cap \{v\} = \emptyset\}.$$


A simplicial complex is an integral generalised homology $n$-sphere if it has the homology of $S^n$, and has the same integral local homology groups as $\mathbb{R}^n$.


There exist acyclic finite simplicial complexes such that the link of every vertex is non-acyclic. Does there exist a finite acyclic simplicial complex for which the link of every vertex is itself an integral generalised homology sphere? Alternatively, such that the link of every vertex has the homology of a sphere?



An acyclic simplicial complex where all links are generalised homology spheres
If all vertex links in a finite simplicial complex $K$ are homology $n-1$-spheres (i.e., homeomorphic to $n-1$-manifolds with the same homology as an $n-1$-sphere), then the simplicial complex $K$ is a closed homology $n$-manifold. As such it has a mod-2 fundamental class: $H\_n(K;\mathbb{F}\_2)\cong \mathbb{F}\_2$. Applying the universal coefficient theorem, it follows that $K$ cannot be integrally acyclic.


