Loop coproducts in string topology and triviality of higher genus TQFT operations - Mathematics > Algebraic TopologyReport as inadecuate




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Abstract: Cohen and Godin constructed positive boundary topological quantum fieldtheory TQFT structure on the homology of free loop spaces of oriented closedsmooth manifolds by associating a certain operations called string operationsto orientable surfaces with parametrized boundaries. We show that all TQFTstring operations associated to surfaces of genus at least one vanishidentically. This is a simple consequence of properties of the loop coproductwhich will be discussed in detail. One interesting property is that the loopcoproduct is nontrivial only on the degree $d$ homology group of the connectedcomponent of $LM$ consisting of contractible loops, where $d=\dim M$, withvalues in the degree 0 homology group of constant loops. Thus the loopcoproduct behaves in a dramatically simpler way than the loop product.



Author: Hirotaka Tamanoi

Source: https://arxiv.org/



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