  Higher chromatic numbers $\chi_s$ of simplicial complexes naturally
generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension
$d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily
large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only
little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$.
  A particular class of $d$-complexes are triangulations of $d$-manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high $\chi_2$ were known.
  We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
  For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere $S^3$ with face vector
$f=(167,1579,2824,1412)$ and 2-chromatic number 5.
