TY - JOUR

T1 - Four-manifold systoles and surjectivity of period map

AU - Katz, Mikhail

PY - 2003

Y1 - 2003

N2 - P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in Χ(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of X, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima λi of a lattice, as deforming a λ2-bound into λ 2-bound.

AB - P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in Χ(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of X, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima λi of a lattice, as deforming a λ2-bound into λ 2-bound.

KW - Conformal invariants

KW - Conway-Thompson lattices

KW - Period map

KW - Systole

UR - http://www.scopus.com/inward/record.url?scp=0242319579&partnerID=8YFLogxK

U2 - 10.1007/s00014-003-0774-9

DO - 10.1007/s00014-003-0774-9

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AN - SCOPUS:0242319579

SN - 0010-2571

VL - 78

SP - 772

EP - 786

JO - Commentarii Mathematici Helvetici

JF - Commentarii Mathematici Helvetici

IS - 4

ER -