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# Topology of Mapping Spaces

We use the space Map( X,Y ) of continuous maps and its based version Map * ( X,Y ) quite often in algebraic topology.

One of the reasons is Map( X, - ) and Map * ( X, - ) are right adjoint to functors ( - ) × X and ( - ) X . From the viewpoint of homotopical algebra , it is desirable that our category of spaces is closed under various constructions. The mapping space construction is one of the fundamental constructions.

When we study inﬁnite dimensional spaces such as mapping spaces, we need to approximate them by well-known spaces. There are several such models for mapping spaces. These are especially useful for studying iterated loop spaces.

It is useful to regard the mapping space Map( X,F ) as the space of cross sections on the trivial bundle . For example, it is used in the construction of twisted cohomology theories .

There are spectral sequences for computing the homotopy groups of mapping spaces and spaces of sections.

• Federer spectral sequence [ Fed56 ]
• Schultz’ spectral sequence [ ? ] for computing the homotopy groups of the space of sections