Families of short cycles on Riemannian surfaces

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a non-trivial element of the k’th homology group of the space of cycles and such that the mass of each cycle is bounded above by $C\max \{\sqrt{k},\sqrt{g}\}\sqrt{Area(M)}$. This result is optimal up to a multiplicative constant.