By William P. Thurston

ISBN-10: 0691083045

ISBN-13: 9780691083049

This booklet develops a few of the striking richness, attractiveness, and gear of geometry in and 3 dimensions, and the powerful connection of geometry with topology. Hyperbolic geometry is the megastar. a robust attempt has been made to express not only denatured formal reasoning (definitions, theorems, and proofs), yet a residing feeling for the topic. there are various figures, examples, and routines of various trouble.

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Topological Topological spaces. Fibrations. Fibrations. 18 al cover. e. e. to aa homeomorphic to some sorne neighbourhood neighbourhood of the vertex “cross”. On this tree the free group on two generators acts freely in a natural "cross" . this acts a natural way, with orbit space the bouquet S1 V S’. It follows by this sort of argument with orbit space SI SI. It follows 0 that ni(K) is a free group for every 1-complex l-complex (or graph) K. K. / 1 "- \ \ 1 1 \ 1 '\. ...... " ' - - _ . / / Fig.

Fibrations. Fibrations. 7) Important 1. Let Important cases. cases. 1. Let pp :: X X --) Y Y be be aa covering-space covering-space projection projection (see (see above) with (discrete) fiber Fo. Then since ni(Fo, ~0) = 0 for i > (0 denoting denoting above) with (discrete) fiber Fo. (X,Zo) - w(Y, Yo) -5 0; ~o(Fo,Zo) - 0. e. (Le. in in oneoneto-one correspondence with) the number of components of the fiber; in other ta-one correspondence with) the number of components of the fiber; in other 0 words each coset sheet of words each coset corresponds corresponds to to aa sheet of the the covering.

The region U then has as deformation retract the bouquet of the circle and 2-sphere: bouquet of the circle and 2-sphere: u N s2 v s1 = Y. TI(Sl), where infinite cyclic group, with generator t, say. Consider the universal infinite cyclic group, with generator t, say. Consider the universal U, or, rather, the homotopically equivalent universal cover X of the U, or, rather, the homotopically equivalent universal coyer X of the s2 v s1 = Y: S2 V SI = Y: X+Y=S2VS1. 16). 16). Tl (Y) on X is here given by t(X) = x + 1, t (sj2) = s;+1.

### Three-Dimensional Geometry and Topology by William P. Thurston

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