Monoide des enlacements et facteurs orthogonaux (Monoids of linking pairings and orthogonal summands)

Abstract

A linking pairing is a symetric bilinear pairing lambda: GxG --> Q/Z on a finite abelian group. The set of isomorphism classes of linking pairings is a non-cancellative monoid E under orthogonal sum, which is infinitely generated and infinitely related. We propose a new presentation of E that enables one to detect whether a linking pairing has a given orthogonal summand. The same method extends to the monoid Q of quadratic forms on finite abelian groups. We obtain a combinatorial classification of Q (that was previously known for groups of period 4). As applications, we describe explicitly 3-manifolds having a degree one map onto prescribed (or proscribed) lens spaces. Most of the results extend to 3-manifolds endowed with a parallelization or a spin structure. In particular, the Reidemeister--Turaev function detects the existence of a spin preserving degree one map between a rational homology 3-sphere and a lens space.