Abstract
We study the Birman exact sequence for compact $3$--manifolds, obtaining a complete picture of the relationship between the mapping class group of the manifold and the mapping class group of the submanifold obtained by deleting an interior point. This covers both orientable manifolds and non-orientable ones.
Highlights
The mapping class group Mod(M ) of a manifold M is the group of isotopy equivalence classes of self-homeomorphisms of M
The first map is given by sending a loop ρ in M based at p to the isotopy class of homeomorphisms that results from pushing the point p once around ρ
In Definition 2.1, we chose to restrict ourselves to a class of homeomorphisms that all have a common fixed point other than p it would immediately follow that ker(ΦM ) ∼= 1
Summary
The mapping class group Mod(M ) of a manifold M is the group of isotopy equivalence classes of self-homeomorphisms of M. If the use of the Poincare Conjecture and the Orthogonalisation Theorem are removed, the proof remains valid for all compact, connected 3–manifolds whose prime connected summands with finite fundamental group are all either S3, P2 × I or Seifert fibred. For the 3–manifold M = S × S1 the group K from Theorem 6.4 will contain KS × Z There are four such surfaces: an annulus, a Mobius band, a torus and a Klein bottle. Applying this construction yields four of the exceptional cases in the statement of Theorem 6.4. The development of this paper has been aided by comments from and discussion with Steven Boyer, Martin Bridson, Allen Hatcher, Andy Putman, Saul Schleimer and a referee of this paper
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