Download Gems, computers, and attractors for 3-manifolds by Sostenes Lins PDF

By Sostenes Lins

This article presents a advisor to facing 3-manifolds through pcs. Its emphasis is on proposing algorithms that are used for fixing (in perform) the homeomorphism challenge for the smallest of those items. the most important thought is the 3-gem, a different form of edge-colored graph, which encodes the manifold through a ball advanced. Passages among 3-gems and extra regular displays like Heegaard diagrams and surgical procedure descriptions are supplied. a listing of all closed orientable 3-manifolds caused by means of 3-gems as much as 30 vertices is incorporated. to be able to aid the class, quite a few invariants are awarded, together with the recent quantum invariants.

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He uses this to prove that every free action by C3 on a lens space is equivalent to a free linear action. 1. Consider one parameter families of embedded 2-spheres (for S3) or embedded Heegard tori (for a lens space) which sweep out the total space of the group action. A typical family is a map (S2 x [0,17, S2 x {0,1}) (T2 x [0,11, T2 2. x -r (S3,{x1,x2}) or {0,1}) -> (L(p,q), {c1,c2}) Take such families which are in general position with respect to C3 = {1,A,A2}, that is, Et , AEt , A2Et are in general position for all except a finite number of critical values of t.

F-1 ([y J x = u j=1 X. 3 2j-2 , y B2 n , Zj-1 I n A), Y. = f-1(Cy Y = Bo J=1 J , y 2n n JY. 2j-1 , B = 3=0 vB. J I n A) 2j n , D = J=p U B2 . ;. x; ;e A There are deformation retractions of X*(and Y*) LEMMA D onto D* We construct an equivariant retraction of Xj onto Proof. B . 2j-2 . The argument for Y. is similar. Let a = be a typical simplex of A such that a n Xj 0 There are three possibilities:(i) f maps each vertex to y 2j-2 , when there is nothing to prove. Fvo s y Let 2j x e a n X.

Inductively we may suppose that all such have been removed. Consider the one remaining curve C1 in K n AK. be a small tubular neighbourhood of C1 such that N Let N n AN = 0. Write C. R = M - N - (AN) L (A2N)u , so that R - K - AK - A2K = R1 U R2 (disjoint union) when R1 and R2 are A-invariant solid tori. The orbit space is 47 a union of 3 solid tori, which intersect in pairs along boundary annuli. 1 this is enough to establish the existence of a Seifert fibration (with base space S2 and at most 3 exceptional fibres).

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