  The interplay of electronic and structural degrees of freedom in solids is a
topic of intense research. Experience and intuition suggest that structural
changes drive conduction electron behavior, because the large number of valence
electrons dominate the structural properties. As part of a seminal paper
written over sixty years ago, Lifshitz discussed an alternative possibility:
lattice softening driven by conduction electrons at topological Fermi surface
transitions. The effect he predicted, however, was small, and has not been
convincingly observed. Using measurements of the stress-strain relationship in
the ultra-clean metal Sr$_{2}$RuO$_{4}$, we reveal a huge softening of the
Young's modulus at a Lifshitz transition of a two-dimensional Fermi surface,
and show that it is indeed entirely driven by the conduction electrons of the
relevant energy band.
