Stable norm and stable systoles

We recall the definition of the stable norm in the real homology of a polyhedron $X$ with a piecewise Riemannian metric, following [BaK03,BaK04].

Definition 4.1   The stable norm $\Vert h\Vert$ of  $h\in H_k(X,{\mathbb{R}})$ is the infimum of the volumes

$$\vol _k(c)=\Sigma_i \vert r_i\vert \vol _k(\sigma_i)$$ (4.1)

over all real Lipschitz cycles  $c=\Sigma_i r_i \sigma_i$ representing $h$.

Note that $\Vert\;\Vert$ is indeed a norm, cf.  [Fed74] and [Gro99, 4.C].

We denote by  $H_k(X,{\mathbb{Z}})_{{\mathbb{R}}}$ the image of  $H_k(X,{\mathbb{Z}})$ in $H_k(X,{\mathbb{R}})$ and by  $h_{{\mathbb{R}}}$ the image of  $h\in H_k(X,{\mathbb{Z}})$ in $H_k(X,{\mathbb{R}})$. Recall that  $H_k(X,{\mathbb{Z}})_{{\mathbb{R}}}$ is a lattice in $H_k(X,{\mathbb{R}})$. Obviously

$$\Vert h_{\mathbb{R}}\Vert \le \vol _k(h)$$ (4.2)

for all  $h\in H_k(X,{\mathbb{Z}})$, where  $\vol _k(h)$ is the infimum of volumes of all integral $k$-cycles representing $h$. Moreover, one has  $\Vert h_{\mathbb{R}}\Vert = \vol _n(h)$ if  $h\in H_n(X,{\mathbb{Z}})$. H. Federer [Fed74, 4.10, 5.8, 5.10] (see also [Gro99, 4.18 and 4.35]) investigated the relations between  $\Vert h_{\mathbb{R}}\Vert$ and  $\vol _k(h)$ and proved the following.

Proposition 4.2   If  $h\in H_k(X,{\mathbb{Z}})$,  $1\le k < n$, then

$$\Vert h_{\mathbb{R}}\Vert = \lim\limits_{i\rightarrow\infty} \frac{1}{i} \vol _k (i h).$$ (4.3)

Equation (4.3) is the origin of the term stable norm for $\Vert\;\Vert$. The stable $k$-systole of a metric  $(X,{\mathcal G})$ is defined by setting

$$\stsys _k({\mathcal G})=\lambda_1\left( H_k(X, {\mathbb{Z}})_{\mathbb{R}}^{\phantom{I}}, \Vert\;\Vert \right),$$ (4.4)

where $\lambda_1$ denotes the first successive minimum of the lattice $\left( H_k(X, {\mathbb{Z}})_{\mathbb{R}}^{\phantom{I}}, \Vert\;\Vert \right)$, i.e. the least norm of a nonzero lattice element.