2 3 A ug 2 00 7 SINGULAR SURFACES , MOD 2 HOMOLOGY , AND HYPERBOLIC VOLUME , II
نویسنده
چکیده
The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the dimension of H1(M ;Z2) is at most 7, and that it is at most 6 unless M is “strange.” To say that a closed, orientable 3-manifold M , for which H1(M ;Z2) has dimension 7, is strange means that the Z2-vector space H1(M ;Z2) has a 2-dimensional subspace X such that for every homomorphism ψ : H1(M ;Z2) → Z2 with X ⊂ kerψ, the two-sheeted covering space M̃ of M associated to ψ has the property that H1(M̃ ;Z4) is a free Z4-module of rank 6. We state a group-theoretical conjecture which implies that strange 3-manifolds do not exist. One consequence of the main theorem is that if M is a closed, orientable, hyperbolic 3manifold such that rk2M ≥ 4 and the cup product H(M ;Z2)×H1(M ;Z2) → H(M ;Z2) is trivial, then VolM > 1.54. The methods of this paper refine those of [3], in which we obtained an upper bound of 10 for the dimension of H1(M ;Z2) under the same hypothesis.
منابع مشابه
m at h . G T ] 2 3 M ay 2 00 8 SINGULAR SURFACES , MOD 2 HOMOLOGY , AND HYPERBOLIC VOLUME , II
If g is an integer ≥ 2, and M is a closed simple 3-manifold such that π1(M) has a subgroup isomorphic to a genus-g surface group and dimZ2 H1(M ;Z2) ≥ max(3g−1, 6), we show that M contains a closed, incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold such that VolM ≤ 3.08, then dimZ2 H1(M ;Z2) ≤ 5.
متن کاملJu n 20 05 Singular surfaces , mod 2 homology , and hyperbolic volume , I
The main theorem of this paper states that if M is a closed orientable hyperbolic 3-manifold such that the rank of H1(M ;Z/2Z) is at least 11, then VolM > 3.08. The theorem depends on a purely topological result which can be viewed as an analogue of Dehn’s lemma for π1-injective singular surfaces of genus 2. The proof of the main theorem combines this topological result with several deep geomet...
متن کاملSingular Surfaces, Mod 2 Homology, and Hyperbolic Volume, I
The main theorem of this paper states that if M is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank of H1(M ;Z/2Z) is at most 10. The theorem depends on a purely topological result which can be viewed as an analogue of Dehn’s lemma for π1-injective singular surfaces of genus 2. The proof of the main theorem combines this topological result with several deep geomet...
متن کامل20 07 Singular Surfaces , Mod 2 Homology , and Hyperbolic Volume , I
The main theorem of this paper states that ifM is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank ofH1(M ;Z/2Z) is at most 10. The theorem depends on a purely topological result which can be viewed as an analogue of Dehn’s lemma for π1-injective singular surfaces of genus 2. The proof of the main theorem combines this topological result with several deep geometri...
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If M is a simple, closed, orientable 3-manifold such that π1(M) contains a genus-g surface group, and if H1(M ;Z2) has rank at least 4g−1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank of H1(M ;Z2) is at most 6.
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