Looks at the role of the fundamental group in determining the topology of a given 3-manifold. This title covers the essential ideas and techniques such as: Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, and free groups. It concludes with a list of problems.
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2-sphere 2-sphere bundle abelian annulus assume boundary components bounds a 2-cell bundle over S1 Choose compact 3-manifold completes the proof conclusion conjugate connected sum contradiction covering map covering translations cube with handles denote double cover double curve element epimorphism fake 3-cell ffj(F ffj(M fiber bundle finite index finite sheeted follows free group free product fundamental group genus Heegaard splitting hence Hj(M homeomorphism homology homotopy 3-sphere homotopy equivalent I-bundle incompressible surface induced infinite cyclic irreducible isomorphism isotopic LEMMA Let F loop theorem manifold map f Math monic nonorientable nonseparating nontrivial normal subgroup Note obtain P2-irreducible pairwise disjoint Poincare conjecture position map prime factorization projective plane properly embedded Q contains regular neighborhood residually finite S1 x S1 satisfying Seifert fibered space simple closed curve simplex solid torus sphere theorem subcomplex subgroup of finite submanifold Suppose surface F torsion free triangulation twisted I-bundle universal cover
Page 190 - JHC WHITEHEAD, A certain open manifold whose group is unity, Quart.