# Who first considered constructibility of simplicial complexes?

A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. Who first came up with this definition? I had thought it was M. Hochster [Ann. Math. 96 (1972), 318-337], but a number of sources cite E. Zeeman [Seminar on Combinatorial Topology, Fascicule 1, Expos\'es I \'a V inclus, Institut des Hautes Etudes Scientifiques, 1963]. I've looked at Zeeman's book and don't see where he considers this condition, even under a different name. Can anyone point to the appropriate part of Zeeman? (Disclosure: This is a referee's suggestion for a paper I am revising, and I would like to make the citation pinpoint if possible.)

If you want the first use of the term "constructible" in this context, then your reference to Mel Hochster's work is right-on. But if you want the actual notion, then things get slightly hazy. I think the oldest (implicit) use is Newman's two papers from 1926 called something like The foundations of combinatorial analysis situs (not near my reference books, so the title might be off).

Update Although I remain parted from my books, I should thank Joe O'Rourke for locating the desired references, namely

Newman, Maxwell Herman Alexander. "On the foundations of combinatorial analysis situs." In Proc. Royal Acad. Amsterdam, vol. 29, pp. 610-641. 1926.

And also, the OP might benefit from the introduction (paragraph 2) of this paper

Frank Lutz, Small examples of non-constructible simplicial balls and spheres,

which supports both assertions: that Newman's work is the earliest where the idea of constructibility appears, predating Alexander by $\epsilon$; and also that Hochster's paper is the first place where it gets its present name.

• Newman, Maxwell Herman Alexander. "On the foundations of combinatorial analysis situs." In Proc. Royal Acad. Amsterdam, vol. 29, pp. 610-641. 1926. – Joseph O'Rourke Mar 11 '16 at 20:54
• Alexander's paper is "The combinatorial theory of complexes," Ann. Math. 31, no. 2 (1930), 292-320, which acknowledges a debt to Newman. That said, these sources do not seem to consider constructibility as a property worth studying in its own right, although there are theorems along the lines of "If two d-balls intersect in a (d-1)-ball, then the union is a d-ball." These authors are concerned only with homeomorphism, whereas the reason to consider constructibility is more modern: a constructible space is Cohen-Macaulay "for the right reason" of Reisner's formula plus Mayer-Vietoris. – Jeremy Martin Mar 27 '16 at 16:51