Arnold's seminar at Moscow University April 18, 2000 S.V.Duzhin Configuration spaces of Euclidean frameworks (review). A framework (a.k.a. linkage) F in R^n is a graph whose vertices are points of R^n. A deformation of F is a continuous movement of the vertices of F that preserves the lengths of all edges. The configuration space C(F,n) of a framework F is the set of all its deformations considered modulo the group of orientation-preserving Euclidean isometries of R^n. Many interesting manifolds arise as configuration spaces of Euclidean frameworks. In particular, I will speak about the following theorem of H.Maehara (see comments below): Theorem. Let P denote the pentagonal framework in R^2 with lengths of consecutive edges (a_1,a_2,a_3,a_4,a_5). If the pentagon P is not foldable into a line, then C(P,2) is a surface whose genus can take the following values: 0, 1, 2, 3, 4. The genus is determined by an explicit system of inequalities involving a_i. I will also mention relevant results by Y.Kamiyama and I.J.Schoenberg as well as some examples invented by W.P.Thurston and J.R.Weeks. Some open problems will be formulated. ------------- (1) A.B.Sossinsky informed me that this theorem was proved in 1992 by D.Zvonkin in his student work (published in 1996, Russian J. of Math. Physics). (2) V.A.Vassiliev informed me that this fact was found by D.Freed (pen on napkin) during a dinner in 1993. (3) S.M.Gussein-Zade informed me that it is probably proved in: Gibson, C. G.; Newstead, P. E. On the geometry of the planar $4$-bar mechanism. Acta Appl. Math. 7 (1986), no. 2, 113--135. (4) V.Trushkov informed me that the theorem about pentagons, as well as its analog for quadrangles was one of the problems at the 6th Summer Tournament of Towns.