its weight goes to zero (see Remark \[arctoedge\]) so that $\Delta(\del \t)$ is identified with $\del (\Delta(\t))$.

We define the topology of $\cact^1(n)$ to be that induced by the bijection with $\CWcact(n)$. Via Remark \[setrem\] this gives a topology to the spaces $\Cact(n),\cacti(n)$ and $\cacti^1(n)$.

The (quasi)-operad structure
----------------------------

### The operad of cacti

The gluing maps for cacti $$\circ_i:\cacti(n)\otimes \cacti(m)\rightarrow \cacti(n+m-1)$$ are defined on elements $(c,c')\mapsto c\circ_i c'$ as follows

-   Scaling the weight function $w'$ of $c'$ by the length $\frac{r_i}{R}$ where $r_i$ is the length of the cycle $c_i$ of the cactus $c$ and $R$ is the length of the cycle $c_0$ of $c'$.

-   Identifying the realization of the cycle $c_0$ of $c'$ with the cycle $c_i$ of $c$ via the maps $\phi_0(c')$ and $\phi_i(c)$, with the orientation on the second $S^1$ reversed, as usual.

These maps together with the $\Sn$ action permuting the labels turn the collection $\{\cacti(n)\}$ into an operad $\cacti$. The collection $\{\cact(n)\}$ forms the suboperad $\cact$.

### The quasi-operad of normalized cacti

We recall from [@cact] that a quasi-operad is the generalization of a (pseudo)-operad in which the axiom of associativity is omitted and the others are kept.

The gluing maps for 