A method of constructing 3-manifolds and its application to the computation of the μ-invariant

Abstract

If F and G are disjoint compact surfaces with boundary in S3=∂D4, let F′ and G′ be the result of pushing F and G into the interior of D4, keeping ∂F and ∂G fixed. The authors give an explicit cut and paste description of an irregular 3-fold branched cover W4(F,G) of D4 branched along F∪G. If M3=∂W4(F,G), they say that (F,G) "represents M3 by bands''. Their main result is that any closed oriented 3-manifold can be so represented. In particular, any such 3-manifold bounds a simply connected W4 which is an irregular 3-fold branched cover of D4. Moreover, F and G can always be chosen in a rather special form which leads to a formula for the μ-invariant of M3 when M3 is a (Z/2)-homology sphere.