Cohomological products and transversality

By making use of geometric cocycles we give a definition of transversality for maps between polyhedra, which extends the mock transversality given by Buoncristiano, Rourke and Sanderson in London Math. Soc. Lect. Note ser. 18 (1975).This definition allows us to introduce a product between PL maps that, in the particular case of the geometric cycles and cocycles, gives a description of the cup and cap products in terms of transversality.