We use discrete Morse theory to give a new proof of Hatcher and Vogtmann's Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_n,k using the degree of a graph, and the Degree Theorem says that each A_n,k is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature. Here we give a proof that only uses local considerations, and lends itself more readily to generalization.

The second named author gratefully acknowledges support from the SFB 701 of the DFG