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Register a new userTorsion in the homology of finite covers of 3-manifolds
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Let $N$ be a prime 3-manifold that is not a closed graph manifold. Building on a result of Hongbin Sun and using a result of Asaf Hadari we show that for every $kinBbb{N}$ there exists a finite cover $tilde{N}$ of $N$ such that $|operatorname{Tor} H_1(tilde{N};Bbb{Z})|>k$.
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Every closed orientable surface S has the following property: any two connected covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions.
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map ($mathbb{Z}pi$-homology equivalence) to a fixed 3-manifold $N$ and showed that there is a natural surjection from a space of $pi=pi_1N$-decorated graphs to the graded quotient of the filtration over $mathbb{Z}[frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$ the surjection of Garoufalidis--Levine is actually an isomorphism over $mathbb{Q}$. For the proof, we construct a perturbative invariant by applying Fukayas Morse homotopy theoretic construction to a local system of the quotient field of $mathbb{Q}pi$. The first invariant is an extension of the Casson invariant to $mathbb{Z}pi$-homology equivalences to the 3-torus. The results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with $b_1=3$. We also discuss some generalizations of the perturbative invariant for other target spaces $N$.
We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.
Torsion sensitive intersection homology was introduced to unify severa
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