homomorphism for every integer j, where f*(E) denotes the inverse image sheaf or pullback sheaf. If f is the inclusion of a subspace X of Y, f*(E) is the restriction of E to X, often just called E again, and the pullback of a section s from Y to X is called the restriction s|X. Pullback homomorphisms are used in the Mayer–Vietoris sequence, an important computational result. Namely, let X be a topological space which is a union of two open subsets U and V, and let E be a sheaf on X. Then there is a long exact sequence of abelian groups: Sheaf cohomology with constant coefficients For a topological space X and an abelian group A, the constant sheaf AX means the sheaf of locally constant functions with values in A. The sheaf cohomology groups Hj(X,AX) with constant coefficients are often written simply as Hj(X,A), unless this could cause confusion with another version of cohomology such as singular cohomology. For a continuous map f: X → Y and an abelian group A, the pullback sheaf f*(AY) is isomorphic to AX. As a result, the pullback homomorphism makes sheaf cohomology with constant coefficients into a contravariant functor from topological spaces to abelian groups. For any spaces X and Y and any abelian group A, two homotopic maps f and g from X to Y induce the same homomorphism on sheaf cohomology: It follows that two homotopy equivalent spaces have isomorphic sheaf cohomology with constant coefficients. Let X be a paracompact Hausdorff space which is locally contractible, even in the weak sense that every open neighborhood U of a point x contains an open neighborhood V of x such that the inclusion V → U is homotopic to a constant map. Then the singular cohomology groups of X with coefficients in an abelian group A are isomorphic to sheaf cohomology with constant coefficients, H*(X,AX). For example, this holds for X a topological manifold or a CW complex. As a result, many of the basic calculations of sheaf cohomology with constant coefficients are the same as calculations of singular cohomology. See the article on cohomology for the cohomology of spheres, projective spaces, tori, and surfaces. For arbitrary topological spaces, singular cohomology and sheaf cohomology (with constant coefficients) can be different. This happens even for H0. The singular cohomology H0(X,Z) is the group of all functions from the set of path components