The strong ring of simplicial complexes

The strong ring R is a commutative ring generated by finite abstract simplicial complexes. To every G ∈ R belongs a Hodge Laplacian H = H(G) = D = (d + d∗)2 determining the cohomology and a unimodular connection operator L = L(G). The sum of the matrix entries of g = L−1 is the Euler characteristic χ(G). For any A,B ∈ R the spectra of H satisfy σ(H(A × B)) = σ(H(A)) + σ(H(B)) and the spectra of L satisfy σ(L(A×B)) = σ(L(A)) · σ(L(B)) as L(A×B) = L(A)⊗ L(B) is the matrix tensor product. The inductive dimension of A×B is the sum of the inductive dimension of A and B. The dimensions of the kernels of the form Laplacians Hk(G) in H(G) are the Betti numbers bk(G) but as the additive disjoint union monoid is extended to a group, they are now signed with bk(−G) = −bk(G). The maps assigning to G its Poincaré polynomial pG(t) = ∑ k=0 bk(G)t k or Euler polynomials eG(t) = ∑ k=0 vk(G)t k are ring homomorphisms from R to Z[t]. Also G → χ(G) = p(−1) = e(−1) ∈ Z is a ring homomorphism. Kuenneth for cohomology groups H(G) is explicit via Hodge: a basis for H(A × B) is obtained from a basis of the factors. The product in R produces the strong product for the connection graphs. These relations generalize to Wu characteristic. R is a subring of the full Stanley-Reisner ring S, a subring of a quotient ring of the polynomial ring Z[x1, x2, . . . ]. An object G ∈ R can be visualized by ts Barycentric refinement G1 and its connection graph G′. Theorems like Gauss-Bonnet, PoincaréHopf or Brouwer-Lefschetz for Euler and Wu characteristic extend to the strong ring. The isomorphism G → G′ to a subring of the strong Sabidussi ring shows that the multiplicative primes in R are the simplicial complexes and that connected elements in R have a unique prime factorization. The Sabidussi ring is dual to the Zykov ring, in which the Zykov join is the addition, which is a spherepreserving operation. The Barycentric limit theorem implies that the connection Laplacian of the lattice Z remains invertible in the infinite volume limit: there is a mass gap containing [−1/5, 1/5] for any dimension d.