On the torsion in the cohomology of certain mapping spaces (Q1922751)

The authors prove a counterintuitive result which states that if \(X\) has \(p\)-torsionfree integral cohomology, then so does \(\text{Map}(BV;X)\). Here \(BV\) is the classifying space of an elementary Abelian \(p\)-group \(V\). The conditions under which this fact holds are those under which the natural map \(\theta_x: T_VH^*(X;\psi/p)\to H^*(\text{Map}(BV,X);\psi/p)\) is an isomorphism. Here \(T_V\) is the Lannes \(T\) functor and \(\theta_x\) is a natural map which is an isomorphism under appropriate finiteness conditions. Actually, they prove a generalization of this result concerning the image of \(\theta_x\). As an example, they can prove a theorem of Borel concerning when \(B_G\) has \(\text{mod }p\) homology when \(G\) is a compact Lie group.