  By examining the homology groups of a 4-manifold associated to an integral
surgery on a knot $K$ in a rational homology 3-sphere $Y$ yielding a rational
homology 3-sphere $Y^*$ with surgery dual knot $K^*$, we show that the
subgroups generated by $[K]$ and $[K^*]$ in $H_1(Y)$ and $H_1(Y^*)$,
respectively, have co-prime orders. We obtain an immediate corollary that, in
conjunction with an argument of Lidman and Moore, proves the cosmetic crossing
conjecture for knots whose branched double covers are Heegaard Floer L-spaces.
