For specific classes of smooth, projective varieties X over a field k, we compare two cycle maps on the torsion subgroup \(CH^2(X)_{\text{tors} }\) of the second Chow group. The first one goes back to work of S. Bloch (1981), the second one is Jannsen’s cycle map into continuous \(\ell \) -adic cohomology, whose injectivity properties have attracted attention in two recent papers. The comparison gives sufficient hypotheses to guarantee injectivity of Jannsen’s cycle map \(CH^2(X) \to H^4_{\mathrm {cont}}(X, \mathbb Z_{\ell }(2))\) on \(\ell \) -primary torsion. Using counterexamples to injectivity of the first map due to Sansuc and the first author (1983), we give examples of smooth, projective, geometrically rational surfaces over a rational function field in one variable over a totally imaginary number field for which Jannsen’s map for \(\ell =2\) is not injective on 2-torsion. This answers questions raised in a recent paper.