In the paper **"Triangulations of Surfaces"** Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes each boundary curve to have exactly one end point and constructed a flow (Hatcher flow) to show it is contractible. In the second part he used induction and showed that for multiple points in the boundary the arc complex is the suspension of the of the previous arc complex with one less boundary point. My question is **Why can't we use the flow for multiple endpoints directly?** The definition works perfectly well for the flow. Will there be any problem with the continuty of the flow?


Thanks in advance. 



A query about Hatcher flow on arc complex
The flow consists of a sequence of surgeries using one fixed oriented arc $\alpha$ to cut (and isotope) all other arcs $\beta$ to remove one point of $\alpha\cap\beta$ at a time. Each surgery cuts one arc into two arcs. Say $\beta$ is cut into $\beta'$ and $\beta''$. It can happen that one of these two arcs, say $\beta'$, is trivial, cutting off a disk $D$ from the surface such that $D$ intersects the given set $V$ of allowable endpoints of arcs only in $\partial\beta'$. In this case one discards $\beta'$ and keeps $\beta''$. The bad situation would be that this happens for both $\beta'$ and $\beta''$, so both would be discarded. This can't happen if $V$ contains at most one point in each boundary component of the surface, but it can happen when some boundary component has two or more points in $V$. The surgery flow could then produce an empty arc system, which is not allowed.


There are two situations where the arc complex is homeomorphic to a sphere and hence not contractible, so the surgery process must fail, namely when the surface is a disk with $V$ contained in its boundary, or an annulus with $V$ contained in one of its boundary circles. The simplest case is when the surface is a disk with $V$ consisting of four boundary points. The arc complex is then $S^0$, two points, represented by two arcs connecting the two nonadjacent pairs of points in $V$. Surgering one arc by the other then produces only trivial arcs.


On my webpage is an updated version of the original 1991 paper containing the construction of this flow. The update adds the exceptional cases when the arc complex is a sphere, and it also contains a few small improvements in the exposition. The update is titled "Triangulations of surfaces" and can be found under the heading "Updates of older papers".


