Homological Quantum Field Theory

We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal $d$-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space $S(Y)$, with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories \HL and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev.