nLab combinatorial homotopy

As detailed in the entry algebraic homotopy, in his ICM talk of 1950, Henry Whitehead said:

The ultimate aim of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that ‘analytic’ is equivalent to ‘pure’ projective geometry.

then later

A statement of the aims of ‘algebraic homotopy’ might thus include the following homotopy classification problem (from the same source, J.H.C.Whitehead, ICM 1950).

Classify the homotopy types of polyhedra, $X$, $Y$, … , by algebraic data.

Compute the set of homotopy classes of maps, $[X,Y]$, in terms of the classifying data for $X$, $Y$.

He had already started on this program in two (famous) papers Combinatorial homotopy I and Combinatorial homotopy II. The term combinatorial homotopy was, it seems, in part inspired by the similar usage in combinatorial group theory and his hope would seem to have been that the ideas from the theory of group presentations would allow new insights on combinatorially defined ‘complexes’.

References

• J. H. C. Whitehead, 1950, Algebraic Homotopy Theory , in Proc. Int. Cong. of Mathematics, Harvard, volume 2, 354 – 357. 51

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

• J. H. C. Whitehead, Combinatorial Homotopy II, Bull.

  Amer. Math. Soc., 55 (1949), 453–496.