that preserve the standard lattice (this corresponds to integer coefficients) and thus descend to the quotient. At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra: Since the torus is an Eilenberg–MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism. Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of gives a splitting, via the linear maps, as above): The mapping class group of higher genus surfaces is much more complicated, and an area of active research. Coloring a torus The torus's chromatic number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven. (Since the complete graph can be embedded on the torus, and , the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the four color theorem for the plane.) de Bruijn torus In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where n is 1 (one dimension). Cutting a torus A solid torus of revolution can be cut by n (> 0) planes into maximally parts. The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows: 1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... . See also Notes Nociones de Geometría Analítica y Álgebra Lineal, , Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. .