Decomposition of polyhedra: The point on the third Hilbert’s problem. (Décomposition des polyèdres: Le point sur le troisième problème de Hilbert.) (French) Zbl 0589.51032

Cartier, Pierre

Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. No. 646, Astérisque 133/134, 261-288 (1986).

This is a very nice expository article on the third Hilbert’s problem. The paper contains four sections.
In the first section the author presents some interesting historical remarks on this problem: Euclid and the plane areas, The theory of plane areas in the XIX-th century, The third Hilbert’s problem and the Dehn’s invariant.
The second section contains a proof of the theorem of J. P. Sydler [Comment. Math. Helvet. 40, 43–80 (1965; Zbl 0135.20906)] and some open questions.
The next section presents some homological comments. Here is presented, in terms of the Hadwiger’s invariants, the results due to B. Jessen and A. Thorup [Math. Scad. 43, 211–240 (1978; Zbl 0398.51009)] which asserts that two polytopes \(P\) and \(P'\) are equivalent by decomposition and translation if and only if they have the same invariants of Hadwiger. In this section are given also some applications of the above mentioned result.
In the last section the author gives some remarks on the conjecture of Friedlander [E. M. Friedlander and G. Mislin, Comment. Math. Helvet. 59, 347–361 (1984; Zbl 0548.55016)] and J. Milnor [Comment. Math. Helv. 58, 72–85 (1983; Zbl 0528.20033)].
The paper is written in a very clear style and ends with a complete bibliography.
[For the entire collection see Zbl 0577.00004.]

Reviewer: Dorin Andrica

MSC:

51M20

Polyhedra and polytopes; regular figures, division of spaces

Keywords:

decomposition of polyhedra; third Hilbert’s problem; bibliography

Citations:

Zbl 0577.00004; Zbl 0135.20906; Zbl 0398.51009; Zbl 0548.55016; Zbl 0528.20033