On Primes, Graphs and Cohomology
Abstract
The counting function on the natural numbers defines a discrete MorseSmale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the PoincaréHopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 DOI:
 10.48550/arXiv.1608.06877
 arXiv:
 arXiv:1608.06877
 Bibcode:
 2016arXiv160806877K
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Number Theory;
 53A55;
 05C99;
 52C99;
 57M15;
 68R99;
 53C65
 EPrint:
 26 pages, 5 figures