$\mathbb{Z}_{2}$-coefficient homology $(1, 2)$-systolic freedom of $\mathbb{R}\mathbb{P}^{3}$ # $\mathbb{R}\mathbb{P}^{3}$

Abstract

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.