Perfect discrete Morse functions on 2-complexes

Highlights? We get conditions under which a 2-complex admits a perfect discrete Morse function. ? We study the topology of comlpexes admitting such kind of functions. ? These results are a first step in the study of these functions on 3-manifolds. This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the ith Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties necessary for a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature (Lewiner et al., 2003), since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.