Degree-one maps of Seifert manifolds into the Poincaré homology sphere

Solution of the problem of describing minimal Seifert manifolds

“…In 1999 Hayat-Legrand, Matveev, and Zieschang, running a computer experiment (see [3]), noted some periodic dependence between the parameters of the exceptional fibers of Seifert manifolds and the sets of possible degrees of their maps onto the Poincaré sphere. Some periodicity theorems were stated and proven in [4] (in the case of the Seifert manifolds with base "sphere") and [5] (in the case of the Seifert manifolds with base "torus"). These theorems enable us to reduce the solution of Problem 1 to exhaustion of finitely many cases (although their number is very large).…”

“…In this article, using the method for calculation of the degrees of maps which was proposed by Hayat-Legrand, Matveev, and Zieschang and the methods for proving periodicity of the degrees of maps (see [4,5]), we solve Problem 1 in the affirmative by means of finite computer exhaustion in the cases (d1) and (d3) and explicitly in the cases (d2) and (d4); moreover, in (d4) we derive an explicit arithmetic formula for the dependence of the degree of a map on the parameters of the inverse image of the manifold.…”

“…A slight difference between the formulas in Lemma 3 and Theorems 3 and 4 and similar formulas in [3][4][5] is due to some difference in notations: different choice of orientation of edges and 2-cells. In the author's opinion, the notations of this article are most reasonable.…”

“…Recall that the Poincaré homology sphere S 3 /P 120 is the factor space of the sphere by the action of the group of motions of the dodecahedron; moreover, it is also the Seifert manifold M (S 2 ; (2, −1); (3, 1); (5, 1)). Its fundamental group is P 120 ∼ = SL 2 (Z 5 ) ∼ = x, y | x 2 = y 3 = (xy) 5 . We are interested in the geometric presentation P 120 ∼ = a, c | Q 1 , Q 2 , where Q 1 = c 5 a −2 and Q 2 = aca −1 ca −1 c. The authors of [3] use the same presentation; in particular, they give the corresponding Heegaard diagram of S 3 /P 120 .…”