In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X.
The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map.
The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f* = g*.
In contrast, a pushforward for de Rham cohomology for example is given by integration-along-fibers.
For example, if C is the singular chain complex associated to a topological space X, then this is the definition of the singular cohomology of X with coefficients in G. Now, let f: C → C' be a map of chain complexes (for example, it may be induced by a continuous map between topological spaces, see Pushforward (homology)).