in two pairs of $V$.

If $V,W$ are two gradient vector fields, write $V\leq W$ whenever the regular pairs of $V$ are also regular pairs of $W$. In general, we say that a collection of primitive vector fields $V_0, V_1, \ldots, V_n$ is **compatible** if there exists a gradient vector field $V$ such that $V_i\leq V$ for all $0\leq i\leq n$.

\[MorseComplexDef2\] The **Morse complex** of $K$, denote $\mathcal{M}(K)$, is the simplicial complex whose vertices are given by primitive gradient vector fields and whose $n$-simplices are given by gradient vector fields with $n+1$ regular pairs. A gradient vector field $V$ is then associated with all primitive gradient vector fields $V:=\{V_0, \ldots, V_n\}$ with $V_i\leq V$ for all $0\leq i\leq n$.

Let $K={\partial\Delta}^1=C_3$ be the simplicial complex given by

$$\begin{tikzpicture}[scale=1]

\node[inner sep=1pt, circle, fill=black](a) at (0,2) {};
\node[inner sep=1pt, circle, fill=black](b) at (1,0) {};
\node[inner sep=1pt, circle, fill=black](c) at (-1,0) {};


\draw[-]  (a)--(b);
\draw[-]  (a)--(c);
\draw[-]  (b)--(c);

\node[anchor = south]  at (a) {\small{$a$}};
\node[anchor =  west]  at (b) {\small{$b$}};
\node[anchor = east]  at (c) {\small{$c$}};

\end{tikzpicture}$$

Here we adopt the convention that if $(x,xy)$ is a primitive vector field consisting of a vertex and edge, we denote this by $xy$. Note 