Intersection Multiplicity in Loop Spaces and the String Topology Coproduct (2404.03460v3)

Abstract: The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In particular the coproduct is not homotopy-invariant and it seems much harder to compute. In this article we give an overview over the string topology coproduct and use the notion of intersection multiplicity of homology classes in loop spaces to show that the string topology coproduct and the based string topology coproduct are trivial for certain classes of manifolds. In particular we show that the string topology coproduct vanishes on product manifolds where both factors have vanishing Euler characteristic and we show that the based coproduct is trivial for total spaces of fiber bundles with sections. We also discuss implications of these results.