Discrete Morse theory for open complexes

Abstract

We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order complex $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. Finally, we show that the Borel-Moore homology of $K$, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on $K$ and deduce corresponding weak Morse inequalities. The gradient vector field on $K$ provides a novel alternative to compute Borel-Moore homology.

Figures (10)

• Figure 1: An open simplicial complex.
• Figure 2: The order complex of the complex from Figure \ref{['fig:runningexample']}.
• Figure 3: A discrete gradient on the interior of a solid torus. The red $2$-cell $\sigma$ and the blue $3$-cell $\tau$ are critical.
• Figure 4: A discrete Morse function on the complex from Figure \ref{['fig:runningexample']}.
• Figure 5: A justification of the modification to the procedure of Zhukova2017 on ${{\mathrm{sd}}}(\sigma)$. On the left is the result of choosing $\tau$ to be critical; this pairs $b$ and $f$ in $W$. On the right is the result of choosing $\tau'$ to be critical; this leaves $b$ and $f$ unpaired in $W$ since they are each paired with a simplex not in $S_K$.

Theorems & Definitions (33)

• Proposition 2.1: JostZhang23, Prop. 5.6
• Theorem 3.1
• proof
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Remark 3.6
• Corollary 3.7: Weak Morse inequalities
• proof