What is the GLMY path homology of circulant digraphs with finite connection sets?

Why the modulus stops mattering above 2 max S

A circulant digraph has vertices that are residues modulo n, with a directed edge from a to a+s for each step s in a connection set S. Its path complex is the GLMY complex: chains are spanned by allowed paths whose consecutive steps lie in S, and only those chains whose boundary stays allowed are kept. The paper's central move is to replace the finite cyclic group by an infinite cyclic lift, where steps become ordinary integers and no wraparound occurs.

The payoff is a threshold. For finite positive S containing 1, and for any field K, once n > 2 max S the paper proves a split exact sequence relating the path homology of the lift, the path homology of the circulant, and the homology one degree lower in the lift. It also identifies the circulant's homology with the homology of a finite step-word complex C*(S;K) built from the short additive relations among the steps. Because that complex no longer depends on n, homology is stable in all degrees over every field, including fields whose characteristic divides n.

Short additive relations drive the step-word complex

The step-word complex is where the geometry becomes combinatorics. An allowed path is recorded as a word in the steps of S, and deleting an interior vertex merges two consecutive steps into their sum. If that sum is again in S the face is allowed; if it is not, the face is forbidden and its coefficient must cancel against other words producing the same merged step.

The paper's motivating example makes the mechanism concrete. For the circulant on five vertices with steps 1 and 2, a word containing the substring 22 produces a merged step 4, which is not an edge, so that word's coefficient must vanish. Two words differing only by swapping adjacent 12 and 21 produce the same merged step 3, so their coefficients must be opposite. Together these rules leave exactly two independent chains per starting vertex in each positive degree, giving dimension 10 in every degree n ≥ 1, and the homology is one-dimensional in degrees 0 and 1 and zero above.

This example also shows why the finite computation is delicate: the chain spaces are infinite-dimensional, yet the homology collapses. The step-word complex is the tool that makes this collapse systematic rather than case-by-case.

Collision-free steps give exterior homology in high degrees

The first broad structural result concerns step sets with no additive collisions. If (S+S) ∩ S is empty and every unordered pair of steps has a distinct sum, then for n > 2 max S the path homology of the circulant is isomorphic, as a graded vector space, to the exterior algebra on the vector space spanned by S. In other words, the Betti numbers are the binomial coefficients of |S|, and homology is nonzero in degrees up to |S|.

The word collision-free matters here. The condition rules out exactly the merges that would let a forbidden face be cancelled by a legal one, so the short-relation complex reduces to its degree-one generators with no further relations. The paper later refines this into a ring statement: under the hypotheses of its short-relation proposition, the path cohomology ring is isomorphic to the cohomology of a lattice, which is exterior on degree-one generators, and the degree-one generators square to zero even in characteristic two.

Three positive steps split into seven additive patterns

For connection sets of the form S = {1, a, b} with 1 < a < b, the paper gives a complete classification under an exact Freiman admissibility condition on n. The answer is stated through a single integer r(a,b) that depends only on the additive shape of the three steps, not on n: the path homology in degree m is the m-th exterior power of K^{r(a,b)}.

The value of r(a,b) is 1 when (a,b) is (2,3) or (2,4); it is 3 when a ≥ 3 and b is not one of a+1, 2a−1, 2a; and it is 2 in all remaining cases. The seven additive patterns behind this formula are tabulated in the paper. As concrete checks, the Betti sequences for {1,2,4}, {1,2,5}, and {1,4,6} are respectively (1,1), (1,2,1), and (1,3,3,1), followed by zeros.

The practical reading is that for three steps the entire homology profile is decided by which of seven additive configurations the steps realize. Two step sets with the same pattern have the same Betti numbers, and a change of pattern is what changes the answer.

Rectangular types and arithmetic progressions

For arbitrarily many steps, the paper computes two structured families. A d-dimensional rectangular additive type, satisfying the hypotheses of the relevant theorem, gives path homology isomorphic to the m-th exterior power of K^{d+1} for the circulant and of K^{d} for the lift, over every field and every admissible modulus. The extra dimension in the circulant comes from the lift's degree shift in the exact sequence.

Arithmetic progressions are the sharpest instance: for S = {1, 1+q, …, 1+kq}, the stable Betti sequence is (1,2,1,0,…). So regardless of how many steps the progression contains, only degrees 0, 1, and 2 are nonzero, with dimensions 1, 2, and 1. This is a strong contrast with the collision-free case, where the number of nonzero degrees grows with |S|. Adding steps in arithmetic progression does not add homology degrees; it collapses them.

Doubling strings and their boundary ranks

The last family is the opposite extreme. In characteristic zero, suppose unordered pair sums in S are unique and the only legal pair sums are doublings, meaning a sum s+t lies in S only when s = t. Let D be the set of steps s with 2s in S, and let b = |S| − |D|. Then for n > 2 max S the dimension of the degree-m path homology is the binomial coefficient of b choose m.

The paper goes further and records boundary ranks. Writing a_m for the rank of the boundary map from degree m to degree m−1 in the step-word complex, it has a_0 = a_1 = 0 and a generating function for the ranks in terms of b and d = |D|. It also gives a basis of path homology: for each increasing tuple of indices, the alternating sum of step-word chains over all permutations is a cycle, and summing over all initial vertices produces a basis element.

Two worked cases show how the parameters interact. For S = {1,2,4} one has (b,d) = (1,2), chain dimensions 2m+1 for m ≥ 1, and Betti sequence (1,1,0,…). For S = {1,2,5} one has (b,d) = (2,1), chain dimension 3 in degree one and 4 in degrees m ≥ 2, and Betti sequence (1,2,1,0,…). The same Betti sequence can therefore arise from different chain-level data, which is why the paper reports ranks and not only Betti numbers.

Where the stability theorem applies and where it stops

The stability statement is conditional in a specific way: it requires S to be a finite set of positive steps containing 1, and it requires n > 2 max S. That inequality is what removes wraparound, so that a merged step is an ordinary integer sum rather than a residue. The three-step classification adds its own exact Freiman admissibility condition on n, and the doubling-string formula is stated in characteristic zero with unique unordered pair sums and only doubling legal sums.

Direction also matters. The paper compares a directed circulant with its symmetrization and shows the homology can change: for n = 5 and S = {1,2}, the directed circulant has homology K in degrees 0 and 1 and zero above, while the symmetric circulant, which is a complete graph there, has homology K only in degree 0. Adding reverse steps creates new fillings, so results for directed circulants should not be transferred to undirected circulant graphs without checking.

The most useful distinction for a reader is between the two regimes the paper separates. When legal merges are absent or collision-free, homology spreads across many degrees and is read off from the exterior algebra on the steps. When legal merges are abundant, as in arithmetic progressions, homology concentrates in low degrees. The step-word complex, not the size of the connection set, is what decides which regime a given S falls into.

Sources used in this answer