Mathematics – Geometric Topology
Scientific paper
2005-03-14
Algebr. Geom. Topol. 5 (2005) 419-442
Mathematics
Geometric Topology
Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-19.abs.html
Scientific paper
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.
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