Talk:Surface bundle over the circle

Conjecture

1.- By "infinite fundamental group", I assume that $\pi _{1}(M)=\mathbb {Z}$ qualifies as such, right? linas 20:09, 30 September 2005 (UTC)

Euclid proved there are infinitely many primes; so I guess the infinitude of the integers is known. Charles Matthews 22:31, 30 September 2005 (UTC)

So...that's definitely a possible fundamental group of a 3-manifold. But if you consider all the other conditions of the fibering conjecture, it's not allowed. H_1 infinite means there's a non-separating surface; irrreduciblity means it can't be a sphere. Compressing the surface gives a pi_1-injective surface, so the 3-manifold group contains a surface subgroup, which can't be the case if it's the integers (since the sphere is excluded).

Actually, the conclusion that the manifold fibers implies that the group contains a surface subgroup. The surface can't be a sphere (because of irreducibility), so we have a more complicated surface whose fundamental group injects into the fundamental group of the 3-manifold. --C S 02:59, 1 October 2005 (UTC)

Open Question

Which of the Seifert fiber spaces (NnII,3|0) and (NnIII,3|0) correspond to the surface bundle over $S^{1}$ constructed with fiber $N_{3}$ (genus three non-orientable surface: T#P=K#P) and the y-homeomorphism as the monodromy? Juan Marquez 30-Sept-2005

Simpler?

Can this be simplified to be more accessable to the general public? (Actually, it may not be of interest to the general public so it may not matter, but would an expert come here for this?) I added a technical tag so this will be looked at. RJFJR 13:30, 4 August 2006 (UTC)

You're making the mistaken assumption that there are only two groups of people, "the general public", and the "experts". There are a wide range of people ranging from knowing no mathematics to knowing a lot of mathematics. A great many of these people you would probably tag as "expert", but actually have no knowledge of this topic. Some pictures and such would help, so I'll leave the tag for now. --C S (Talk) 02:04, 11 September 2006 (UTC)