We prove the $3$-manifold $\RP^3 \# \RP^3$ is of $\Z_{ 2}$-coefficient homology $(1, 2)$-systolic freedom. Given a Riemannian metric on $\RP^{ 3}\# \RP^{ 3}$, we define $\Z_{ 2}$-coefficient homology $1$-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{ 1}(\RP^3\#\RP^3; \Z_{ 2})$. The $\Z_{ 2}$-coefficient homology $2$-systole is defined to be the infimum of areas of all nonseparating surfaces representing nontrivial classes in $H_{ 2}(\RP^{ 3}\#\RP^{ 3}; \Z_2)$. In the paper we show that there exists a sequence of Riemannian metrics on $\RP^{ 3} \# \RP^{ 3}$ such that the volume of $\RP^3 \# \RP^3$ cannot be bounded below in terms of the product of $\Z_{ 2}$-coefficient homology $1$-systole and $\Z_{ 2}$-coefficient homology $2$-systole.
Categories: math.DG