Download Elliptic structures on 3-manifolds by Charles Benedict Thomas PDF

By Charles Benedict Thomas

This quantity will provide a scientific exposition of recognized effects at no cost activities through finite teams on S. The textual content starts off with initial fabric on Seifert manifolds and staff type. this is often by means of sections facing similar issues together with loose bZe/2 and bZe/3 activities on lens/prism manifolds, the relief theorem -and tangential constitution.

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Example text

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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