Rational systole of a manifold Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ is given by
$$\operatorname{sys}_k(M,g) = \inf \{ \mathcal{H}^k(\Sigma) : \Sigma^k \in \mathcal{S} \text{ and } [\Sigma] \neq 0 \in H_k(M; \mathbb{Z}) \},$$
where $\mathcal{S}$ denotes the set of all closed $k$-dimensional submanifolds of $M$ and $\mathcal{H}^k$ denotes the $k$-Hausdorff measure.
Is it possible to make sense of a "rational $k$-systole" by changing the homology coefficients to be $\mathbb{Q}$? I worry because if $\Sigma \in \mathcal{S}$, then $H_k(\Sigma; \mathbb{Q}) \cong \mathbb{Q}$ is not a finitely generated abelian group, and we cannot pushforward a "fundamental class" by the inclusion $i : \Sigma \to M$ to impose $[\Sigma] \neq 0 \in H_k(M; \mathbb{Q})$.
 A: Ok, let's try to make sense of rational systoles. First, one needs to define mass on rational chains. Let $r\otimes \sigma$ be a smooth rational singular $k$-simplex, where $r\in {\mathbb Q}$. Then only meaningful definition of the mass would be $|r| \cdot |\sigma|$ where $|\sigma|$ is the $k$-volume of the simplex $\sigma(\Delta^k)$ (one can be more creative here, but the end-result will be the same). Then for a $k$-chain
$$
c= \sum_i a_i \sigma_i
$$
we get $|c|= \sum_i |a_i|\cdot |\sigma_i|$. This definition will have the property that for an integer chain we get exactly Gromov's definition, but also
$$
|r \cdot c|= |r|\cdot |c|
$$
for every $r\in {\mathbb Q}$. Now, you can define
$$
sys_{k, {\mathbb Q}}(M)= \inf_{c} |c|,
$$
where the infimum is taken over all cycles $c\in Z_k(M,  {\mathbb Q})$ which are not null-homologous. But then, whenever
there exists one such cycle $c$, you get
$$
\lim_{m\to\infty} |\frac{1}{m} \cdot c|= \lim_{m\to\infty} \frac{1}{m} |c|  =0. 
$$
Hence, $sys_{k, {\mathbb Q}}(M)=0$.

One can modify this definition as follows: Use integer cycles only, but impose a different nonvanishing condition: Require that $[c]\ne 0$ in $H_k(M, {\mathbb Q})$. In other words, you want $c$'s which aare non-torsion classes:
$$
sys_{k, {\mathbb Q}}(M)= \inf_{c} |c|,
$$
where the infimum is taken over all $c\in Z_k(M,  {\mathbb Z})$ whose images in $Z_k(M,  {\mathbb Q})$ represent nonzero elements of $H_k(M,  {\mathbb Q})$.
Such systolic quantity will be, in general, nonzero. Maybe this is what you want.
