Let X be a finite 2-complex with unfree fundamental group. We prove lower
bounds for the area of a metric on X, in terms of the square of the least
length of a noncontractible loop in X. We thus establish a uniform systolic
inequality for all unfree 2-complexes. Our inequality improves the constant in
M. Gromov's inequality in this dimension. The argument relies on the Reeb graph
and the coarea formula, combined with an induction on the number of freely
indecomposable factors in Grushko's decomposition of the fundamental group.
More specifically, we construct a kind of a Reeb space ``minimal model'' for X,
reminiscent of the ``chopping off long fingers'' construction used by Gromov in
the context of surfaces. As a consequence, we prove the agreement of the
Lusternik-Schnirelmann and systolic categories of a 2-complex.Comment: 29 pages; to appear in Int. Math. Res. Notice
