We define an index of the fermionic signature operator on even-dimensional
globally hyperbolic spin manifolds of finite lifetime. The invariance of the
index under homotopies is studied. The definition is generalized to causal
fermion systems with a chiral grading. We give examples of space-times and
Dirac operators thereon for which our index is non-trivial.Comment: 21 pages, LaTeX, 3 figures, minor corrections (published version
