Singular complex

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

for some | (6.4) |

Expressed differently, it is , the image of the path , except that the endpoints are removed because they are already covered by the 0-cells (the cells must form a partition).

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,

(6.5) |

in which is the Euclidean norm. A

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:

- For each dimension , the set must be closed. This means that the cells must all fit together nicely.
- Each -cell is an open set in the topological subspace . Note that 0-cells are open in , even though they are usually closed in .

One way to avoid some of these strange conclusions from the topology
restricted to
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
. Now
, 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 2020-08-14