Development of the concept of a complex. (English) Zbl 0944.57001

James, I. M. (ed.), History of topology. Amsterdam: Elsevier. 103-110 (1999).
Introduction: In algebraic and geometric topology and elsewhere in mathematics the concept of a complex played and plays till now a basic role: in simplicial and cellular homology, in homotopy theory, in obstruction theory, in the study of PL-manifolds, in the homological theory of groups, etc. Spaces which admit a decomposition into cells are often easier to handle than general topological manifolds. Fortunately, the majority of ‘interesting’ topological spaces fall into this category. These advantages counterbalance the cumbersome definitions and proofs of the main properties of complexes, in particular, invariance properties. In the article we will describe the thorny way to the concepts used nowadays. It took quite a long time since the problems considered belonged to other disciplines of mathematics, mainly to analysis and geometry and the topological side became clear only later. For this treatment of the history of complexes we obtained much information from survey articles and books with historical remarks or flavour such as [M. Dehn and P. Heegaard, Enzyklopädie Math. Wissensch., Vol. III AB/3 (1907); J. Dieudonné, Abrégé d’histoire des mathématiques 1700-1900 (1986; Zbl 0656.01001), A history of algebraic and differential topology 1900-1960 (1989; Zbl 0673.55002); J. Stillwell, Classical topology and combinatorial group theory, Grad. Texts Math. 72 (1980; Zbl 0453.57001), Mathematics and its history (1989; Zbl 0685.01002); P. Alexandroff and H. Hopf, Topologie. Bd. I, Grundlehren Math. Wiss. 45 (1935; Zbl 0013.07904); H. Seifert and W. Threlfall, Lehrbuch der Topologie (1934; Zbl 0009.08601)].
For the entire collection see [Zbl 0922.54003].


57-03 History of manifolds and cell complexes
01A60 History of mathematics in the 20th century
57Q05 General topology of complexes