Cancelling the S('1')(,S) when S('1) x M('3) is diffeomorphic to S('1) x N('3).
Haw-yaw Shy · Deep Blue (University of Michigan) · 1993
In this thesis we study the relation between 3-manifolds $M\sp3$ and $N\sp3$ when $S\sp1\times M\sp3$ and $S\sp1\times N\sp3$ are diffeomorphic. When this problem was studied by Conner and Raymond, they showed that $S\sp1\times M$ is diffeomorphic to $S\sp1\times N$ if M and N can be related by a specific relation (their results are very general that M, N could be manifolds of any dimension starting from 3). Manifolds which can be related by such specific relation are called $C-R$ related. $C-R$ related manifolds are $S\sp1$-manifolds and may not be homeomorphic. 3-dimensional $C-R$ related manifolds are special types of Seifert manifolds. Conner and Raymond also provided algorithms for checking whether two Seifert 3-manifolds are $C-R$ related and whether two $C-R$ related Seifert 3-manifolds are of the same homeomorphism type. The purpose of this investigation was to determine if $C-R$ related 3-manifolds are the only source to provide nondiffeomorphic 3-manifolds $M\sp3, N\sp3$ such that $S\sp1\times M\sp3$ and $S\sp1\times N\sp3$ are diffeomorphic. First, we consider the category of all known prime connected closed orientable 3-manifolds. The most difficult part in this category is Seifert manifolds with infinite fundamental groups. Those manifolds, $M\sp3$ and $N\sp3,$ are $S\sp1$-manifolds. Hence $S\sp1\times M\sp3$ and $S\sp1\times N\sp3$ are $T\sp2$-manifolds which can be classified by elements in the cohomology group $H\sp2$($Q; Z\sp2$). After we derived an explicit presentation of $H\sp2$($Q; Z\sp2$) and an algorithm of determining the homeomorphism types of $T\sp2$-manifolds, we were able to explicitly determine the relation between $M\sp3$ and $N\sp3$ when $S\sp1\times M\sp3$ is diffeomorphic to $S\sp1 \times N\sp3.$ Together with the discussion on the other cases in this category, we showed that if $M\sp3$ and $N\sp3$ are two known prime connected closed orientable 3-manifolds with $S\sp1 \times M\sp3$ diffeomorphic to $S\sp1\times N\sp3$ then $M\sp3$ and $N\sp3$ must be diffeomorphic or $C-R$ related. Second, we treated the compact, not closed, non-orientable Seifert manifolds of type NnI. Finally, we treated the connected sums under mild conditions.