are not equal to $\bar b$ where $\bar
b$ is the image of $b$ and $b-$. For $\bar b$ the order is given by extending the linear order $(\In(\bar b), \prec_{\bar
b}^{\del_e(\t)}) =(\In(b-)\amalg\In(b), \prec^{\t}_{b-}\amalg
\prec^{\t}_{b}) $ —the usual order on the union of totally ordered sets– to $E(\bar b)$ by declaring the image of $e$ and $e-$ to be the minimal element.

We define the operator $\del$ on the space $\wlbptree$ to be given by the following formula: $\del(\t) := \sum_{e\in E_{angle}}
(-1)^{\num_E(e)-1} \del_e (\tau) $.

### The Cell Complex

We define $\wlbptree(n)^k$ to be the elements of $\wlbptree(n)$ with $|E_w|=k$.

For $\t \in \wlbptree$ we define $\D(\t):=\times_{v \in V_w(\tau)}\D^{|v|}$. We define $C(\t)=|\D(\t)|$. Notice that $\dim(C(\t))=|E_w(\t)|$.

Given $\D(\t)$ and a vertex $x$ of any of the constituting simplices of $\D(\t)$ we define the $x$-th face of $C(\t)$ to be the subset of $|\D(\t)|$ whose points have the $x$-th coordinate equal to zero.

We let $\CWcact(n)$ be the CW complex whose k-cells are indexed by $\t \in \wlbptree(n)^k$ with the cell $C(\t)=|\D(\t)|$ and the attaching maps $e_{\t}$ defined as follows. We identify the $x$-th face of $C(\t)$ with $C(\t')$ where $\t'=\del_x(\t)$. This corresponds to contracting an edge of the cactus if 