Faulkner's "living map" refers to the area where the story takes place, the ficitonal Yoknapatawpha County, Mississipi. It is close to the action of the Civil War for the protagonists Bayard Satoris (white) and his best friend, the slave Ringo.
Here is a passage that refers to the "living map" of the history the friends are witnessing:
Behind the smokehouse that summer, Ringo and I had a living map. Although Vicksburg was just a handful of chips from the woodpile and the River a trench scraped into the packed earth with the point of a hoe, it (river, city, and terrain) lived, possessing even in miniature that ponderable though passive recalcitrance of topography which outweighs artillery, against which the most brilliant of victories and the most tragic of defeats are but the loud noises of a moment. To Ringo and me it lived, if only because of the fact that the sunimpacted ground drank water faster than we could fetch it from the well, the very setting of the stage for conflict a prolonged and wellnigh hopeless ordeal in which we ran, panting and interminable, with the leaking bucket between wellhouse and battlefield, the two of us needing first to join forces and spend ourselves against a common enemy, time, before we could engender between us and hold intact the pattern of recapitulant mimic furious victory like a cloth, a shield between ourselves and reality, between us and fact and doom (3-4).