On String Topology of Three Manifolds

Details On String Topology of Three Manifolds On String Topology of Three Manifolds

2003-10-08

arxiv.org/abs/math/0310112v2

Topology 44 (2005), no. 5, 1059--1091

Mathematics

Geometric Topology

38 pages, 13 figures

Scientific paper

Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $\L M$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbolicity'' of the underlying manifold. This is an application of the existing powerful tool and results in three dimensional topology such as the prime decomposition, torus decomposition, Seifert theorem, torus theorem.

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