I recently started learning about CW complexes. Although my understanding of them is somewhat nascent, I see that one can deduce a number of useful properties of a space if one can show it is a CW complex. For example, if $X$ is a CW complex, then we can immediately conclude that $X$ is paracompact. While I see the utility of CW complexes, I still do not quite see where the definition of a CW complex comes from. In other words, I don't yet see what motivated the development of CW complexes.

My question is the following:

Are there any sources out there which discuss the historical development of CW complexes and the motivation behind their development?

Note: I am aware that History of Topology by I.M. James contains a chapter entitled “Development of the Concept of a Complex” but I away from my university for the next month and thus do not have access to the book.

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    $\begingroup$ As far as I know, the concept of a CW complex was developed more or less in its entirety by J.H.C. Whitehead in two seminal articles in 1949. They were supposed to be a generalization of simplicial complexes, at once more general, easier to construct, not tied to an embedding into Euclidean space and very amenable to algebraic topology. $\endgroup$ – Miha Habič Aug 18 '12 at 13:44
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    $\begingroup$ I had a look at my copy of Burde's and Zieschang's contribution to History of topology. It contains a one-page section on CW-complexes which essentially is an expansion on what Ronnie says in his answer. $\endgroup$ – t.b. Aug 18 '12 at 16:33
  • $\begingroup$ I just had a look at that article: curiously, they do not list the later 1949 papers. I suspect this is just a mistake, since the notion of CW (closure finite with the weak topology) is not, I think, in the earlier papers. Ioan James once said in a lecture that JHCW took a year to prove his product theorem for CW-complexes! ($K \times L$ is CW if one of them is locally finite.) In JHCW's undergraduate lectures on homology, he was concerned with giving "block" structures on simplicial complexes which gave easier calculations. $\endgroup$ – Ronnie Brown Aug 19 '12 at 11:21

To give more details than Miha, the two papers are

(CHI) Whitehead, J. H.C., Combinatorial homotopy. I. Bull. Amer. Math. Soc. 55 (1949) 213–245.

(CHII) Combinatorial homotopy. II. Bull. Amer. Math. Soc. 55 (1949) 453–496.

It was the first paper that developed the notion of CW-complex, and proved their most used properties.

However the roots of these papers go back to very original earlier papers of his developing what is now called “Simple homotopy type”, particularly

Whitehead, J. H.C. On incidence matrices, nuclei and homotopy types. Ann. of Math. 2 42 (1941) 1197–1239.

which introduced the notion of a “membrane complex”, which is basically the notion of a space obtained by attaching cells. But the development of the notion of adjunction space needs a separate account! This last paper was rewritten by Whitehead, using (CHI), (CHII), to become

(SHT) Simple homotopy types. Amer. J. Math. 72 (1950) 1–57.

which became a foundation paper for algebraic K-theory.

The key aspect of a CW-complex $X$ is that you can develop properties by induction on the skeleta $X^n$; for example the topology on $X$ is arranged so that a map $f: X \to Y$ is continuous if and only if the restrictions $f|X^n$ are continuous for all $n$.

It should be useful for students to study adjunction spaces and the case of finite cell complexes from the book Topology and Groupoids.

The paper (CHII) is no less original, but has not been so extensively used; it is I believe important for the future. It contains a theorem determining $\pi_2(X \cup _\lambda\{e^2_\lambda\},X,x)$ as a free crossed $\pi_1(X,x)$-module; this theorem is sometimes stated but rarely proved in texts on algebraic topology. Some main ideas of the book Nonabelian Algebraic Topology (EMS,2011) derive from and considerably generalise this result and put many other results of CHII, e.g. on relations with chain complexes with operators, in a wider context.

Later: Relevant to this paper is the following:

Ellis, G.J. "Homotopy classification the J.H.C. Whitehead way". Exposition. Math. 6(2) (1988) 97--110,

which shows how the work of (CHII) includes work published later by P. Olum and others.

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    $\begingroup$ @t.b. Thanks t.b. for editing in the links. $\endgroup$ – Ronnie Brown Aug 18 '12 at 22:23

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