bibkey: wilkes2019markedtorusrecovery
authors: Gareth Wilkes
year: 2019
title: Profinite rigidity of graph manifolds, II: knots and mapping classes
doi: 10.1007/s11856-019-1908-0
url: https://arxiv.org/abs/1801.06386v2
claim: JSJ and fibration-sensitive profinite classification applies to the stated graph-manifold class; the marked-cover homology calculation concerns a single separating-twist family.
strata_touched: []
license: citation-only
triage: anchor

Marked cover homology and torus gluing

Verified locator

Gareth Wilkes, Profinite rigidity of graph manifolds, II: knots and mapping classes, Israel Journal of Mathematics (2019). DOI: 10.1007/s11856-019-1908-0. https://doi.org/10.1007/s11856-019-1908-0 https://arxiv.org/abs/1801.06386v2

Source scope

Theorems 1.5, 1.6 and 5.4 give the stated JSJ- and fibration-sensitive profinite results; Section 1 specifies edge-gluing conventions and Question 5.1 retains its hypotheses. General unmarked graph manifolds need not all be profinitely rigid. An unmarked profinite isomorphism does not automatically recover a selected fibration or time section.

Related JSJ classification appears in Wilkes, Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds, https://arxiv.org/abs/1605.08244, Theorem 10.9.

Correspondence to the theory

Sections 46-51 of SURFACE_CONGRUENCE_OBSERVERS consider the mapping torus of T_alpha^k, for an essential separating alpha, g>=2 and k!=0. Pure time-cyclic covers all have H1=Z^(2g+1). A specified mod-p surface-homology cover, with a fixed-point time section, has degree D=p^(2g). Put G=1+D(g-1), a=p^(2 min(h,g-h)), b=p^(2 max(h,g-h)), beta=(a-1)(b-1), and q=|k|. Its covered mapping-torus homology is

Z^(2G+1-beta) + (Z/q)^((a-2)(b-2)) + (Z/qa)^(b-2)

• (Z/qb)^(a-2) + Z/(qab).

The torsion exponent is qD. The free rank determines beta, hence a+b=D+1-beta and the unordered separating genera. These determine the unoriented homeomorphism type within the specified one-twist family. The bounded-exponent result uses one finite-coefficient observation. The cover-lattice input is developed in Sections 40-45; the explicit homology decoder is the volume’s derivation, not a quoted theorem of Wilkes.

Boundaries

The cover and deck projection are marked data; the cover is not asserted characteristic in the whole mapping-torus group. This calculation does not recognize arbitrary unmarked three-manifolds or establish a surface principal outer-kernel inclusion. It gives no optimal or polynomial-time size bound.

Sections 52-58 recover signed k relative to an oriented symplectic marking and fixed covering-surface probes. This differs from the unoriented classification, which records |k|. Such probes need not descend to the base. The manifolds have an epimorphism of their fundamental group onto Z and an incompressible gluing torus; no simply connected sphere classification or Ricci-flow estimate is deduced from this family.