# Stratified String Topology and 3D-Manifolds

# Stratified String Topology and 3D-Manifolds

I] Closed String Homology has various homotopy invariant algebraic structures [power operations, Lie bracket and a direct sum decomposition] which recognize the form of the known structure of three manifolds arising from Thurston's Geometrization. The proof by Chas-Gadgil is based on Chas-Goldman disjunction of curves on surfaces for the toroidal irreducible cases and elementary calculations/constructions for the pure geometry cases and to recognize the connected sum cases. These hark back to the thesis of Hossein Abbaspour. II] Working in covariant open sets of open strings on the diagonal in MxM Somnath Basu used the open string coalgebra to distinguish certain homotopy equivalent but non-diffeomorphic Lens Spaces. III] The Closed String Homology and its structure recognizing the form of geometrization for link complements can be described directly in three space by considering the algebraic topology of closely related covariant open sets of open strings in space with boundary conditions on the link.[joint with M.Sullivan] A main problem is to make these algorithms as finitistic as possible.