Simplicial complexes are useful in applications such as geometric modeling and computer graphics for computing the topology of models. Due to the complicated topological spaces, implicit, nonlinear models, and decomposition algorithms that arise in motion planning, they are insufficient for the most general problems. A singular complex is a generalization of the simplicial complex. Instead of being limited to , a singular complex can be defined on any manifold, (it can even be defined on any Hausdorff topological space). The main difference is that, for a simplicial complex, each simplex is a subset of ; however, for a singular complex, each singular simplex is actually a homeomorphism from a (simplicial) simplex in to a subset of .
To help understand the idea, first consider a 1D singular complex, which happens to be a topological graph (as introduced in Example 4.6). The interval is a -simplex, and a continuous path is a singular -simplex because it is a homeomorphism of to the image of in . Suppose is a topological graph. The cells are subsets of that are defined as follows. Each point is a 0-cell in . To follow the formalism, each is considered as the image of a function , which makes it a singular 0-simplex, because is a 0-simplex. For each path , the corresponding -cell is
These principles will now be generalized to higher dimensions. Since all balls and simplexes of the same dimension are homeomorphic, balls can be used instead of a simplex in the definition of a singular simplex. Let denote a closed, -dimensional unit ball,
A simplicial complex requires that the simplexes fit together nicely. The same concept is applied here, but topological concepts are used instead because they are more general. Let be a set of singular simplexes of varying dimensions. Let denote the union of the images of all singular -simplexes for all .
A collection of singular simplexes that map into a topological space is called a singular complex if:
One way to avoid some of these strange conclusions from the topology
is to build the vertical decomposition in
, the closure of
. This can be obtained by
starting with the previously defined vertical decomposition and
adding a new -cell for every edge of
and a 0-cell for
every vertex of
, which is closed in
. Likewise, , , and , are closed in the usual
way. Each of the individual -dimensional cells, however, is open
in the topological space . The only strange case is that the
0-cells are considered open, but this is true in the discrete
topological space .
Steven M LaValle 2012-04-20