Chromatic number: Topology I know the significance in graph theory, but how does it transfer to topology (relatively rigorously)? To elaborate, in an expository article, Martin Gardner gives 'chromatic number' as an invariant of a number of surfaces (see chapter 18,the colossal book of mathematics), namely a torus, klein bottle, sphere, square, 'tube', and projective plane. It is easy to see how betti numbers fit into this, how do chromatic numbers fit in? (Again, I am looking for some degree of rigour!)
 A: There are two ways to define the chromatic number $\gamma(S)$ of a surface $S$. One is to say that $\gamma(S)$ is the supremum of chromatic numbers of finite simple graphs which are embeddable in $S$. (Here the chromatic number of a finite graph is the minimal number of colors in which vertices of the graph can be colored so that no two adjacent vertices share the same color.) An equivalent definition is to say that $\gamma(S)$ is the minimal number of colors needed to color any "map" of $S$, where a "map" is a finite cover of $S$ by closed topological disks so that any two disks intersect in a topological arc, or a point, or are disjoint. The value of $S$ (as the function of the Euler characteristic $\chi$ of $S$ (if the latter is connected, closed and orientable) is given by the Heawood Conjecture:
$$
\gamma(S)= \lfloor \frac{1}{2}\left( 7 + \sqrt{49- 24\chi}\right)\rfloor.
$$
This conjecture holds for oriented surfaces, but for nonoriented surfaces, a modification is needed in the case of the Klein bottle, where 
the correct value is 6 rather than 7, as predicted by the conjecture. 
In case you are interested in proofs, take a look at the references in the link, as well as this Wikipedia article.  
