1 - Combinatorial homotopy

Summary

The first extract is from J. H. C. Whitehead1 s fundamental paper on CW-complexes, which are a most useful class of spaces in which to do homotopy theory. There is always an analogy between what we can do topologically with a space, and what we can do algebraically with its chain groups, etc.; in this class of spaces the analogy reaches its maximum strength. The main prerequisite for reading this extract is a sound knowledge of general topology. On p. 40 Whitehead also uses the homotopy extension property for the pair Eⁿ, Sⁿ⁻¹. Whitehead also makes two references to his earlier papers; the first, on p. 40, is to a geometrical construction which the reader can supply for himself; the second, on p. 42, is to the subdivision argument referred to in §1 above. At the foot of p. 41 Whitehead uses the word ‘cellular’; a map f:X → Y between CW-complexes is said to be cellular if f(Xⁿ) ⊂ Yⁿ for each n, where Xⁿ is as defined on p. 33.

4. Cell complexes.16 By a cell complex, K, or simply a complex, we mean a Hausdorff space, which is the union of disjoint (open) cells, to be denoted by e, eⁿ, eⁿ_i, etc., subject to the following condition. The closure, ēⁿ of each n-cell, eⁿ∈X, shall be the image of a fixed n-simplex, σⁿ, in a map, f:σⁿ→ ēⁿ, such that

(4.1) (a) f|σⁿ−∂σⁿ is a homeomorphism onto eⁿ,

(b)where ∂eⁿ=f∂σⁿ = ēⁿ−eⁿand Kⁿ⁻¹is the (n−1)- section of K, consisting of all the cells whose dimensionalities do not exceed n − 1.