3-Manifolds and Knots

Abstract

Among the usual invariants of algebraic topology the fundamental groups carry most information for 3-manifolds. Consider, e.g., a closed orientable 3-manifold M 3. Then the 0-th and 3-rd homology groups are isomorphic to ℤ, and the first is the abelianized π1(M 3). By Poincare duality, see [Novikov 1986, p. 52], H 1(M) ≅H 2(M) and by the universal coefficient theorem H 1(M) ≅ Hom(H 1(M),ℤ)⊕Ext(H 0(M 3),ℤ) ≅ ℤ P1 where p 1 is the first Betti number. All other homology and cohomology groups are trivial. So homology and cohomology is to a great extent determined by the first homology group (here we do not consider the ring structure of H*(M 3)). Now the following questions arise: Which groups can be fundamental groups of 3-manifolds? To what extent does the fundamental group characterize the manifold? Let us first deal with the first question. Only a few groups are fundamental groups of 2-manifolds. The situatuion is entirely different for dimension 4 because all finitely presentable groups appear as fundamental groups of 4-manifolds (see below). Hence it is not surprising that there is no obvious answer to the question of what happens in dimension 3.