Sphere geometry and invariants

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Green function g(x, x) = (1+A′)−1 xx , where A ′ is the adjacency matrix of the connection graph G′ of the complex G. By unimodularity ψ(G) = det(1 +A′) = ∏ x(−1) = φ(G), the Fredholm matrix 1 + A′ is in GL(n,Z), where n is the number of simplices in G. We show that the set of possible unit sphere topologies in G1 are combinatorial invariants of the complex G, and establish so that also the Green function range of G is a combinatorial invariant. The unit sphere character formula g(x, x) = i(x) applies especially for the prime graph G(n) and prime connection graph H(n) on square free integers in {2, . . . , n} playing the role of simplices. In G(n), integers a, b are connected if a|b or b|a and where in H(n) two a, b are connected if gcd(a, b) > 1. The Green function g(x, x) in H(n) relate there to the index values i(x) in G(n). To prove the invariance of the unit sphere topology we use that all unit spheres in G1 decompose as S −(x) +S(x), where + is the join and S− is a sphere. The join renders the category X of simplicial complexes into a monoid, where the empty complex is the 0 element and the cone construction adds 1. The augmented Grothendieck group (X,+, 0) contains the graph and sphere monoids (Graphs,+, 0) and (Spheres,+, 0). The Poincaré-Hopf functionals G → i(G) = 1 − χ(G) or G → iG(x) = 1 − χ(SG(x)) as well as the volume are multiplicative functions on (X,+). For the sphere group, both i(G) as well as ψ(G) are characters. The join + can be augmented by a product · so that we have a commutative ring (X,+, 0, ·, 1) in which there are both additive and multiplicative primes and which contains as a subring of signed complete complexes ±Ki isomorphic to the integers (Z,+, 0, ·, 1). Both for addition + and multiplication ·, the question of unique prime factorization appears open. Date: February 12, 2017. 1991 Mathematics Subject Classification. 05C99, 11C20, 05E4.