We introduce a Morse theory for posets of Bestvina–Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.

In this paper, we determine the homotopy types of the Morse complexes of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if K contains two leaves that share a common vertex, then its Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. (2008, Topology and Its Applications 155, 2084–2089). In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$ . This result is used to compute the homotopy type of the Morse complex of some families of graphs, including Caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.

Let $\left| K \right|$ be the metric polyhedron of a simplicial complex $K$. In this paper, we characterize a simplicial subdivision ${{K}^{\prime }}$ of $K$ preserving the metric topology for $\left| K \right|$ as the one such that the set ${{K}^{\prime }}(0)$ of vertices of ${{K}^{\prime }}$ is discrete in $\left| K \right|$. We also prove that two such subdivisions of $K$ have such a common subdivision.

Let $F$ be a non-separable $\text{LF}$-space homeomorphic to the direct sum ${{\sum }_{n\in \text{N}}}\,{{\ell }_{2}}\left( {{\tau }_{n}} \right)$, where ${{\aleph }_{0}}<{{\tau }_{1}}<{{\tau }_{2}}<\cdot \cdot \cdot $. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $\left| K \right|\,\times \,F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v\,\in \,{{K}^{\left( 0 \right)}}$ has the star $\text{St}\left( v,\,K \right)$ with card $\text{St}{{\left( v,K \right)}^{\left( 0 \right)}}<\tau =\sup {{\tau }_{n}}$ (and card ${{K}^{\left( 0 \right)}}\le \tau $), and, conversely, if $K$ is such a simplicial complex, then the product $\left| K \right|\,\times \,F$ can be embedded in $F$ as an open set, where $\left| K \right|$ is the polyhedron of $K$ with the metric topology.