# Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?

A triangulation of a topological manifold $$\mathcal{M}$$ possibly with boundary is an abstract simplicial complex $$\Delta$$ together with a homeomorphism $$\varphi:\vert\Delta\vert\to\mathcal{M}$$, where $$\vert\Delta\vert$$ denotes the geometric realization and where an abstract simplicial complex is a collection of simplices with the property that $$\sigma\in\Delta$$ and $$\tau\subset\sigma$$ implies $$\tau\in\Delta$$.

Now, for simplicial complexes one can define the following extra properties:

1. $$\Delta$$ is "pure", i.e. every simplex $$\sigma\in\Delta$$ of dimension $$ is the face of some $$d$$-simplex.
2. $$\Delta$$ is "non-branching", i.e. every $$(d-1)$$-simplex is face of exactly one or two $$d$$-simplices.
3. $$\Delta$$ is "strongly-connected", i.e. for every pair of $$d$$-simplices $$\sigma,\tau\in\Delta_{d}$$, there is a sequence of $$d$$-simplices $$\sigma=\sigma_{1},\sigma_{2},\dots,\sigma_{k}=\tau$$ such that the intersection $$\sigma_{l}\cap\sigma_{l+1}$$ is a $$(d-1)$$-simplex for every $$l\in\{1,\dots,k-1\}$$.

An abstract simplicial complex with these properties is usually called a "pseudomanifold (possibly with boundary)". Now, I know that every piecewise-linear manifold, i.e. a manifold with a triangulation satisfying the extra property that the links of every vertex is a sphere or ball (or equivalently a manifold with a piecewise-linear atlas), is a pseudomanifold. This is stated in many resources and is clear I think.

However, on the other side, there are examples of triangulation for dimension $$d>4$$, which are not piecewise-linear. In fact, there are even triangulations of manifolds (like for $$S^{5}$$) whose links are not even manifolds. However, do these complexes still have the properties above? I think they should at least be pure, since this property is needed in order to define the boundary of a simplicial complex and I think this should be possible for arbitrary triangulations of manifold with boundary. On the other hand, triangulations of manifolds should be non-branching, I guess, since this is needed in order to define orientability of simplicial complexes.

In some sense, I am asking if the concept of pseudomanifold generalizes the concept of PL-manifolds or of general triangulazible manifolds.

• I think it should be pure. Otherwise there is a $d'$- simplex which is not in the boundary of an higher dimensional simplex. Thus its midpoint would have a open neighborhood homeomorphic to $\mathbb{R}^{d'}$. By the well definedness of the dimension of a topological manifold, we would then get $d=d'$. Jan 5, 2022 at 19:24
• Non branching should also be clear. We now only have to exclude that a $d-1$ simplex is a face of three or more $d$-simplices. It should be possible to show that a point in the $d-1$-simplex cannot have a neighborhood homeomorphic to $R^d$. Jan 5, 2022 at 19:29
• You need to assume that $M$ is connected (and thus path connected) to obtain "strongly connected" for $\Delta$. Jan 6, 2022 at 10:00

Suppose that $$M$$ is a connected $$d$$-dimensional topological manifold without boundary. (We make the last assumption to simplify matters.) Let $$\Delta$$ be the given pseudo-triangulation. So the realisation $$|\Delta|$$ has the same local homology groups as $$M$$. These are $$H_k(M, M - x) \cong \mathbb{Z}$$ if $$k = d$$ and are zero if $$k \neq d$$.
If $$\Delta$$ is not pure then there is a $$\ell$$-simplex, say $$\sigma$$, with $$\ell < d$$ which is not the face of a $$\ell+1$$-simplex. Thus, for a point $$x$$ in the interior of $$\sigma$$ we have $$H_\ell(M, M - x) \cong \mathbb{Z}$$, a contradiction.
If $$\Delta$$ is branching then there is a $$(d-1)$$-simplex, say $$\sigma$$, which is the face of $$n > 2$$ top-dimensional simplices. Thus, for a point $$x$$ in the interior of $$\sigma$$ we have $$H_d(M, M - x) \cong \mathbb{Z}^{n-1}$$, a contradiction.
If $$\Delta$$ is not strongly connected, then there is a simplex, say $$\sigma$$, whose link has $$n > 1$$ connected components. Thus, for a point $$x$$ in the interior of $$\sigma$$ we have $$H_0(M, M - x) \cong \mathbb{Z}^{n-1}$$, a contradiction.