${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$-coefficient homology $(1,2)$-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$-coefficient homology $1$-systole and ${\mathbb{Z}}_2$-coefficient homology $2$-systole.

23 pages

References in corpus (2)

• Metaphors in systolic geometry
• Systoles and Dehn surgery for hyperbolic 3-manifolds