Title: Sphere geometry and invariants,
Abstract: 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 Poincare-Hopf value i(x)=1-X(S(x)), where X is Euler
characteristics and S(x) is the unit sphere of a vertex x in G1, agrees with
the Green function value g(x,x),the diagonal element of the inverse of (1+A'),
where A' is the adjacency matrix of G'. By unimodularity, det(1+A') is the
product of parities (-1)^dim(x) of simplices in 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 is a combinatorial invariant of the
complex G. So, also the Green function range of G is a combinatorial invariant.
To prove the invariance of the unit sphere topology we use that all unit
spheres in G1 decompose as a join of a stable and unstable part. The join
operation + 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 Poincare-Hopf functionals i(G) as well as
the volume are multiplicative functions on (X,+). For the sphere group, both
i(G) as well as Fredholm characteristic are characters. The join + can be
augmented with a product * so that we have a commutative ring (X,+,0,*,1)for
which there are both additive and multiplicative primes and which contains as a
subring of signed complete complexes isomorphic to the integers (Z,+,0,*,1). We
also look at the spectrum of the Laplacian of the join of two graphs. Both for
addition + and multiplication *, one can ask whether unique prime factorization
holds.