I thought it was homology that is dual to cohomology. But reading Bott & Tu I am now confused. In the intro, page 2 and onwards, it briefly describes a dual point of view and the relation between homotopy and cohomology. It sounds like homotopy and cohomology are dual to each other, although it is not precisely defined.
For example, at the bottom of page 2, it says "the two concepts are dual to each other". The question is, is this saying homotopy and cohomology are dual to each other in general, or is it just that the concepts are "somewhat dual" to each other? Are they dual to each other and if so in which sense and to what extent?
Homotopy and cohomology are dual in a vague sense called "Eckmann-Hilton duality". As the linked page says, this is more a heuristic rather than a precise notion of duality. Certainly in the formal mathematical sense of vector space duality, then homology/cohomology are trying to be dual to each other (and this is achieved when working with field coefficients and finite CW complexes, for example). Homotopy/cohomology are dual in the sense that homotopy groups are determined by homotopy classes of maps out of spheres, and spheres have just a single nonvanishing reduced cohomology group; "dually," cohomology groups are determined by homotopy classes of maps into certain spaces called Eilenberg-Mac Lane spaces, which have just a single nonvanishing homotopy group. If we write $S^n$ for a sphere, as usual, and $K_n$ for an Eilenberg-Mac Lane space, we have this informal notion of duality with $$ [S^n, -] \quad \longleftrightarrow \quad [-, K_n]. $$