Every boundary simplex appears even number of times, but not a cycle?

Question

If we have a set $S$ of $n$-simplices in a simplicial complex $X$, is it possible that for each $(n-1)$-simplex $\Delta$ in $X$, there is an even number of $n$-simplices in $S$ which have $\Delta$ as a boundary face, yet no $f : S \to \{-1, 1\}$ makes $\sum_{s \in S} f(s)s$ a cycle?

Answer 1

Yes, if you're working with integer coefficients. Consider a triangulation of the Klein bottle or the real projective plane, for example: every 1-simplex appears in the boundary of exactly two 2-simplices, but there are no nonzero 2-cycles.