Title: The strong ring of simplicial complexes,
Abstract: We define a ring R of geometric objects G generated by finite abstract
simplicial complexes. To every G belongs Hodge Laplacian H as the square of the
Dirac operator determining its cohomology and a unimodular connection matrix
L). The sum of the matrix entries of the inverse of L is the Euler
characteristic. The spectra of H as well as inductive dimension add under
multiplication while the spectra of L multiply. The nullity of the Hodge of H
are the Betti numbers which can now be signed. The map assigning to G its
Poincare polynomial is a ring homomorphism from R the polynomials. Especially
the Euler characteristic is a ring homomorphism. Also Wu characteristic
produces a ring homomorphism. The Kuenneth correspondence between cohomology
groups is explicit as a basis for the product can be obtained from a basis of
the factors. The product in R produces the strong product for the connection
graphs and leads to tensor products of connection Laplacians. The strong ring R
is also a subring of the full Stanley-Reisner ring S Every element G can be
visualized by its Barycentric refinement graph G1 and its connection graph G'.
Gauss-Bonnet, Poincare-Hopf or the Brouwer-Lefschetz extend to the strong ring.
The isomorphism of R with a subring of the strong Sabidussi ring shows that the
multiplicative primes in R are the simplicial complexes and that every
connected element in the strong ring has a unique prime factorization. The
Sabidussi ring is dual to the Zykov ring, in which the Zykov join is the
addition. The connection Laplacian of the d-dimensional lattice remains
invertible in the infinite volume limit: there is a mass gap in any dimension.