bibkey: looijenga1997prym
authors: Eduard Looijenga
year: 1997
title: Prym representations of mapping class groups
doi: 10.1023/A:1004909416648
url: https://doi.org/10.1023/A:1004909416648
claim: Finite abelian cover homology gives independent arithmetic Prym representations on suitable finite-index mapping-class subgroups; this does not identify a prescribed quotient of the whole mapping-class group.
strata_touched: []
license: citation-only
triage: anchor

Prym blocks and global factorization

Verified locator

Eduard Looijenga, Prym representations of mapping class groups, Geometriae Dedicata 64 (1997), 69-83. DOI: 10.1023/A:1004909416648. https://doi.org/10.1023/A:1004909416648

Source scope

Finite abelian cover homology supplies arithmetic Prym representations with independence and almost-surjectivity properties on suitable finite-index mapping-class subgroups. The cyclic-cover image result is also stated in Bader, Prym representations of the handlebody group, Theorem 1.5, DOI 10.1007/s10711-024-00911-5. Bader’s handlebody theorem does not substitute for a theorem about the whole mapping-class group.

Correspondence to the theory

Sections 33-39 of SURFACE_CONGRUENCE_OBSERVERS fix N2=ker(pi_g -> H1(Sigma_g,F2)). For odd l, the homology kernel of pi_g/[N2,N2]N2^l splits into a 2g-dimensional trivial block and 4^g-1 sign blocks of dimension 2g-2. Projectivization removes lift ambiguity.

For a separating curve of type (h,g-h), the canonical lift has square-zero homology increment of rank (4^h-1)(4^(g-h)-1) and exact outer order l. In genus at least three, separating twists distinguish the sign-block kernels on the Johnson kernel. Their full joint image and the symplectic permutation action yield the specified congruence quotient PSp(2g-2,F_l)^(4^g-1) semidirect Sp(2g,F2).

The volume derives concrete subgroups satisfying the curve-orbit inclusion for those specified inputs. For a fixed nonabelian finite simple quotient of the whole mapping-class group, factorization through this action is equivalent to factorization through ordinary homology modulo 2l. Normality and transitive block permutation prevent selection of one Prym factor as a global quotient.

Boundaries

The rank formula concerns the canonical lift, not every deck-modified lift. The full-product and global simple-quotient results require g>=3 and odd l. Their PSp simplicity input is classical; rank-one and even-field exceptions are outside this domain.

Specified positive congruence examples do not handle arbitrary prescribed finite-index subgroups. Prime-power refinements over the same mod-two cover retain the stated obstruction for a fixed nonhomological simple target; they do not exclude other finite characteristic quotients or other covers. Cross-characteristic semisimple block formulas require l not dividing the base-group order and cannot be replaced by defining-characteristic formulas.