Synthetic Systolic Geometry

• Synthetic Systolic Geometry is an axiomatic framework that constructs manifolds via explicit cutting, gluing, and scaling to achieve sharp systolic invariants.
• It derives precise inequalities and subadditivity rules through combinatorial manipulations of atomic manifold structures.
• This approach facilitates applications in topology and quantum error correction by linking geometric properties with concrete, constructive methods.

Synthetic systolic geometry is an axiomatic and construction-driven approach to systolic geometry, emphasizing explicit procedures—cutting, gluing, product, and scaling—that generate and bound systolic invariants of Riemannian manifolds. Rather than relying on deep analytic techniques or curvature bounds, synthetic systolic geometry derives sharp inequalities and extremal constructions by working with “atomic” manifolds as building blocks and precise combinatorial manipulations. This framework provides a unified perspective on systolic phenomena across dimensions, with critical applications to geometry, topology, and quantum error correction.

1. Systolic Structures: Definitions and Scale-Invariant Quantities

Given a closed, oriented Riemannian manifold $(M,g)$ of dimension $m \ge 2$, the systole $\sys(M,g)$ is the length of a shortest non-contractible loop. For surfaces $(S,g)$, a variant is the homological systole $\sys_h(S,g)$, the shortest non-separating closed geodesic. The key scale-invariant is the systolic ratio: $SR(M,g) = \frac{\sys(M,g)^m}{\vol(M,g)}$ For surfaces of genus $k$, the optimal systolic and homological systolic ratios are defined as

$SR(k) = \sup_{(S,g),\,\genus(S)=k} SR(S,g)$

$SR_h(k) = \sup_{(S,g),\,\genus(S)=k} SR_h(S,g)$

These ratios provide a quantitative measure of the interplay between the minimal topology-representing cycle and the global volume, forming the backbone of systolic geometry (Akrout et al., 2013, Fetaya, 2011).

2. Synthetic Constructions: Cutting, Pasting, and Product Operations

Central to synthetic systolic geometry is the procedure of constructing new manifolds from old while controlling systolic invariants. The core construction for surfaces is as follows:

1. Begin with two surfaces $(S_1,g_1)$, $(S_2,g_2)$ each normalized such that $\sys=1$, and with systolic ratios near optimal.
2. On each, select a systole $a_i$, partitioned into two equal arcs. Cut each surface along one of these arcs, producing surfaces with geodesic boundary of length 1.
3. Glue the boundary geodesics pairwise between the two open surfaces, yielding a closed surface of genus $k_1+k_2$.

This construction preserves the systole ($\sys=1$), sums the volumes, and induces precise control over the systolic ratio: $$SR(k_1+k_2) \geq \left(\frac{1}{SR(k_1)}+\frac{1}{SR(k_2)}\right)^{-1}$$ The mechanism preventing the creation of new, shorter non-contractible loops is a homotopic lemma: any arc homotopic (with endpoints fixed) to a boundary segment of length $<1$ must have length at least that of the segment, ruling out "short-cuts" that would reduce the systole. An analogous procedure applies for the homological systole using a thin cylinder attachment (Akrout et al., 2013).

The product construction extends the synthetic method to higher dimensions: for two manifolds $(M^m,g)$, $(N^n,h)$ scaled so $\sys(M) = \sys(N) = 1$, their product inherits $\sys(M\times N) = 1$, $\vol(M\times N) = \vol(M)\vol(N)$, and thus

$SR(M \times N) = SR(M) \cdot SR(N)$

This yields multiplicative systolic inequalities and facilitates higher-dimensional extremal constructions.

3. Intersystolic Inequalities and Asymptotic Behavior

Synthetic manipulations yield the fundamental subadditivity and multiplicativity inequalities. For surfaces: $$SR(k_1+k_2) \geq \left(\frac{1}{SR(k_1)}+\frac{1}{SR(k_2)}\right)^{-1}$$ and likewise for $SR_h$. Writing $\sigma(k) = 1/SR(k)$, this is subadditivity: $\sigma(k_1+k_2) \leq \sigma(k_1) + \sigma(k_2)$, valid for all Riemannian metrics without curvature restrictions.

Asymptotic bounds obtained via synthetic pasting and hyperbolic constructions yield, for infinitely many genera,

$SR(k) \gtrsim \frac{4}{9\pi} \frac{(\log k)^2}{k}$

which propagates via repeated pasting to all large $k$ for the homological systolic ratio: $$SR_h(k) \gtrsim \frac{4}{9\pi} \frac{(\log k)^2}{k}$$ The upper bound, adapted from filling radius methods, is

$$SR_h(k) \leq \frac{(\log(195k)+8)^2}{\pi(k-1)} \qquad(k \geq 76)$$

These bounds are achieved through explicit metric constructions and induction on genus, employing capping and separation via systoles (Akrout et al., 2013).

4. Systolic Geometry in Higher Dimensions and Freedom Phenomena

In higher dimensions, synthetic systolic geometry distinguishes between rigid and freedom behaviors. For surfaces, the classical two-dimensional inequality

$(\sys_1(Σ,g))^2 \leq C \cdot \operatorname{Area}(Σ,g)$

establishes universal bounds, with the flat torus as an extremal case. The combinatorial systole $\operatorname{csys}_p(M,S)$ generalizes the notion via minimal numbers of simplices in nontrivial cycles on a triangulation.

In dimension 3 and above, explicit constructions (notably the Freedman–Meyer–Luo mapping torus and surgery example) demonstrate that no universal bound exists for products of systoles. For certain $P_g^3$, as genus $g \to \infty$,

$$\frac{\operatorname{sys}_3(P_g;\mathbb{Z}_2)}{\operatorname{sys}_2(P_g;\mathbb{Z}_2)\cdot \operatorname{sys}_1(P_g;\mathbb{Z}_2)} \to 0$$

This shows “Z₂-systolic freedom”: there is no uniform constant bounding volume by systole products in higher dimensions, in sharp contrast to the 2D case (Fetaya, 2011).

5. Synthetic Systolic Geometry and Applications to Quantum Codes

Synthetic systolic geometry has direct consequences for the theory of homological quantum error-correcting codes. For a triangulated closed surface $(Σ,S)$ with dual cellulation, the distance $d$ of the corresponding CSS quantum code satisfies

$$d = \min\{\operatorname{csys}_1(Σ,S), \operatorname{csys}_1(Σ,S^*)\}$$

Analyzing the combinatorial–Riemannian correspondence in 2D yields the $\sqrt{n}$-bound on code distance relative to code length $n$: any homological code from a surface satisfies $d = O(\sqrt{n})$. Crucially, the breakdown of universal systolic bounds in higher dimensions (systolic freedom) opens the possibility for codes surpassing this limit, a major motivation for the synthetic perspective (Fetaya, 2011).

6. Axiomatic and Category-Theoretic Formulation

Synthetic systolic geometry can be formalized axiomatically using atomic objects (e.g., genus-minimizing extremal surfaces), gluing axioms (cutting and regluing along geodesics, ensuring subadditivity), tensor product axioms (product manifolds, realizing multiplicative scaling), and homogeneity via scaling invariance (normalizing systole to 1 at each step).

This suggests the possibility of an abstract category whose objects are geometric manifolds equipped with invariants ($\sys$, $\vol$), morphisms as gluings and products, and all systolic inequalities and constructions realized combinatorially. Such a formulation synthesizes Gromov’s philosophical framework with concrete geometric strategies, enabling explicit control over systolic invariants and their propagation across dimensions (Akrout et al., 2013, Fetaya, 2011).