Title : Combinatorics of branchings in higher dimensional automata Abstract : We explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular $\omega$-category and the combinatorics of a new homology theory called the reduced negative corner homology. This latter is the homology of the quotient of the corner complex by the sub-complex generated by its thin elements. As application, we calculate the corner homology of some $\omega$-categories and we give some invariance results for the reduced corner homology. We only treat the negative side. The positive side, that is to say the case of merging areas of paths is similar and can be easily deduced from the negative side. URL : math.CT/9912059 (XXX archive http://arXiv.org/) Comments : LaTeX2e, 42 pages