bibkey: corey2024multitwistgraph
authors: Daniel Corey; Wanlin Li
year: 2024
title: The Ceresa class and tropical curves of hyperelliptic type
doi: 10.1017/fms.2024.36
url: https://doi.org/10.1017/fms.2024.36
claim: The multitwist homology formula gives a square-zero increment and a cycle-pairing block in an adapted symplectic basis.
strata_touched: []
license: citation-only
triage: anchor

Multitwist homology and graph lattices

Verified locator

Daniel Corey and Wanlin Li, Forum of Mathematics, Sigma 12 (2024), e54. DOI: 10.1017/fms.2024.36. https://doi.org/10.1017/fms.2024.36

Source scope

Section 3.1, Propositions 3.1 and 3.2, and Section 3.3, equation (3.4), give the multitwist sum formula, square-zero property and cycle-pairing block in an adapted symplectic basis. These are classical homological monodromy inputs. Graph flow and cut lattices are also treated by Bacher, de la Harpe and Nagnibeda, Bulletin de la Societe Mathematique de France 125 (1997), 167-198, DOI 10.24033/bsmf.2303, https://doi.org/10.24033/bsmf.2303.

Correspondence to the theory

Sections 40-45 of SURFACE_CONGRUENCE_OBSERVERS use actual cover homology and the coset dual graph. A positive exponent-one multitwist has a block Q=Z^t Z; its return cokernel has graph-critical-group torsion. Smith invariants give finite-field ranks and composite-modulus periods. For the mod-p homology cover the graph is K_(a,b), with a=p^(2h) and b=p^(2(g-h)).

Sections 52-58 study fixed probes of A_Q=I+U_Q in one symplectic marking. Passive power traces and characteristic polynomials do not distinguish Q. One-probe traces give -v^t Q v, hence only the diagonal in characteristic two. For z=b_i and w=b_j+a_i the two-probe difference is D_ij=-Q_ii-Q_jj+Q_ij, while D_i=-Q_ii. Consequently Q_ii=-D_i and Q_ij=D_ij-D_i-D_j over every commutative ring. This probe calculation is the volume’s derivation, not a theorem attributed to Corey and Li. Over a characteristic-two field with block rank at least two, maximal probe depth two is necessary and sufficient for this recovery task.

Boundaries

The probes share a marking and reset to the same operator. Simple-curve operations on a covering surface need not descend to the base. Signed twist recovery is relative to that oriented marking, whereas the unmarked, unoriented mapping-torus classification retains the absolute twist exponent.

Recovering a homology block does not prove principal outer-kernel containment for a prescribed simple-curve orbit. Deterministic probes of a fixed quotient action still factor through that action and preserve its kernel obstruction. No claim about all periodic-point counts or integral cokernels follows from the passive spectral equality.