Homology of function spaces (Q1089957)

We compute the homology of the space of pointed maps from X to Y under certain conditions on X and Y. One such set of conditions is that X be the suspension of a connected finite CW complex and that Y be the m-fold suspension of a connected CW complex, with \(m\geq\) (dimension of \(X+\) the connectivity of \(X+2).\) Under these conditions, the space Map\((X,Y)\) has the mod p homology of a product of spaces \(\Omega ^ iY\), the i-fold loops of Y. If we choose a basis for the mod p homology of X, there is one copy of \(\Omega ^ iY\) for each basis element of \(H_ i(X;F_ p)\). Under the above conditions, the homology of \(\Omega ^ iY\) is known, so this really does give a calculation of \(H_ *(\text{Map}(X,Y);F)\) for any field F.