Description

The above applet visualizes a so-called Morse Matching on the simplicial complex of the three letters "BAT" using JavaView [5].

The simplicial complex consists of the collection of triangles, edges, and vertices you see in the picture. The elements of the simplicial complex are called faces. The Morse Matching is a matching of vertices to edges and edges to triangles such that each element of the complex is matched with at most one other element. Furthermore there is no directed cycle in the pictures, i.e., when you follow the arrows you cannot come back to the face you started from.

Morse matchings capture the main structure of discrete Morse Functions which were introduced by Robin Forman in 1998 [2]. The concept of Morse matchings was introduced by Manoj Chari [1]. For more details on computing Morse matchings with as few critical faces as possible see reference [4].

In the example, the faces which are not matched are called critical and are shown in red. (Unfortunately the critical vertices cannot be visualized with the current JavaView version.)

• For the "T" there are one critical triangle and one critical vertex. Topologically, the "T" is (homotopy equivalent to) a sphere. Indeed, there is a theorem by Robin Forman [3] which states that a combinatorial manifold (without boundary) with two critical faces is homeomorphic to a sphere.
• The "A" has one critical triangle, two critical edges, and one critical vertex. It can be deformed to a torus.
• Finally, the "B" contains one critical triangle, four critical edges, and one critical vertex. You can see that it can be deformed to a double-torus.

For more information see the following references.

1. Manoj K. Chari, On discrete Morse functions and combinatorial decompositions
   Discrete Math. 217 (2000), No. 1-3, pp. 101-113.
2. Robin Forman, Morse Theory for Cell-Complexes
   Advances in Math. 134 (1998), pp. 90-145.
3. Robin Forman, Combinatorial Differential Topology and Geometry
   New Perspectives in Geometric Combinatorics, L. Billera et al. (eds.), Cambridge University Press, Math. Sci. Res. Inst. Publ. 38, 177-206 (1999).
4. Michael Joswig and Marc E. Pfetsch, Computing Optimal Morse Matchings
   SIAM J. Discrete Math. 20, no. 11 (2006), 11-25
5. Konrad Polthier et al. JavaView