Equivalence of definitions of simplicial manifolds, and which ones imply "no branching"

Question

I've found a couple of different definitions of simplicial manifolds with boundary:

1. A pure abstract simplicial $n$-complex such that the (geometric realization of the) link of every simplex $\sigma$ of dimension $k$ is homeomorphic to a sphere or ball of dimension $n - 1 - k$. (e.g., these notes or these notes.)

2. A pure abstract simplicial $n$-complex such that the (geometric realization of the) link of every vertex $v$ is homeomorphic to a sphere or ball of dimension $n - 1$. (i.e., #1 but just for $k=0$). (e.g., these notes.)

#1 obviously implies #2, but I'm wondering if #2 implies #1.

In particular, I'm interested in showing that a simplicial manifold obeys the "no branching" condition (part of the definition of a pseudomanifold), which is that every $(n-1)$-simplex is the proper face of 1 or 2 $n$-simplices. #1 implies no branching immediately, and if #2 implied #1, then #2 would also imply no branching. But failing that, does #2 imply no branching also?

Answer 1

So I'm not aware of "simplicial manifold" as a thing. On the other hand the category of "PL manifolds" is a thing, and its definition is slightly more intricate than what you are saying, namely a PL version of 2: A simpicial $n$-complex is a PL-manifold if the link of every vertex is a PL-manifold of dimension $n-1$ that is PL-homeomorphic to the standard PL manifold structure on the sphere of dimension $n-1$ (e.g. the boundary of the $n$-simplex). And then this definition does indeed imply the correponding version of 1. Here's a quick proof, by induction on $n$.

Consider a PL manifold $M$ and a $k$-simplex $\sigma \subset M$, with $k \ge 1$. Pick a $0$-simplex $v \in \sigma$ and therefore $S = \text{Link}_M(v)$ is an $n-1$-sphere with the standard PL structure. The intersection $\tau = \sigma \cap S$ is a $k-1$ simplex in $S$. The key observation is that $\text{Link}_M(\sigma)$ is simplicially isomorphic to $\text{Link}_S(\tau)$, and so it follows by induction on $n$ that this link is a standard PL sphere of dimension $n-k-1$.

But in general, 2 does not imply 1. A counterexample is given by the double suspension theorem of Cannon and Edwards, which produces a simplicial structure on $S^5$ that satisfies 2, and a 1-simplex in that simplicial structure whose link is a 3-dimensional manifold that is not even homeomorphic to $S^3$.