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

Lizhi Chen

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

Differential Geometry 2014-12-02 v3

Authors: Lizhi Chen

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.

Keywords

systolic freedom homology systole Riemannian metric connected sum real projective space Z2-coefficient

Cite

@article{arxiv.1402.4504,
  title  = {$\mathbb{Z}_{2}$-coefficient homology $(1, 2)$-systolic freedom of $\mathbb{R}\mathbb{P}^{3}$ # $\mathbb{R}\mathbb{P}^{3}$},
  author = {Lizhi Chen},
  journal= {arXiv preprint arXiv:1402.4504},
  year   = {2014}
}

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23 pages

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

Abstract

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