# Combinatorial topology

A branch of topology in which the topological properties of geometrical figures are studied by means of their divisions (cf. Division) into more elementary figures (for example, the triangulation of polyhedra into simplexes) or by means of coverings (cf. Covering) by systems of sets. These methods are applied under the most general hypotheses concerning the figures being studied.

#### Comments

Around 1930, combinatorial topology was the name given to a fairly coherent area covering parts of general, algebraic and piecewise-linear (PL) topology. Typical subjects are: simplicial complexes, their fundamental groups and homology groups, surfaces, and topology of 3- or higher-dimensional manifolds (cf. Fundamental group; Homology group; Manifold; Simplicial complex; Surface). Most of these topics have nowadays developed to specialisms in most diverse branches of mathematics.

One of the classical textbooks (in German) has recently been translated to English; cf. [a1].

#### References

[1] | P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) |

[2] | L.S. Pontryagin, "Grundzüge der kombinatorischen Topologie" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |

[a1] | H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) |

**How to Cite This Entry:**

Combinatorial topology.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Combinatorial_topology&oldid=53816