nLab
Sullivan model of mapping space

Contents

Contents

Idea

Given two topological spaces XX, YY one may ask for the rational homotopy type of their mapping space Maps(X,Y)Maps(X,Y). Under good conditions this is a nilpotent space of finite type and hence admits a Sullivan model from which its rationalized homotopy groups and its rational cohomology groups may be read off.

Examples

Free loop spaces

See at Sullivan model of free loop space.

Mapping spaces between spheres

Proposition

Let nn \in \mathbb{N} be a natural number and f:S nS nf\colon S^n \to S^n a continuous function from the n-sphere to itself. Then the connected component Maps f(S n,S n)Maps_f\big( S^n, S^n\big) of the mapping space which contains this map has the following rational homotopy type:

(1)Maps f(S n,S n) {S n×S n1 | neven,deg(f)=0 S 2n1 | neven,deg(f)0 S n | nodd Maps_f\big( S^n, S^n\big) \;\simeq_{\mathbb{Q}}\; \left\{ \array{ S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0 \\ S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0 \\ S^n &\vert& n\, \text{odd} } \right.

where deg(f)deg(f) is the degree of ff.

Moreover, under the canonical morphism expressing the canonical action of the special orthogonal group SO(n+1)SO(n+1) on S n=S( n+1)S^n = S\big( \mathbb{R}^{n+1}\big) (regarded as the unit sphere in (n+1)(n+1)-dimensional Cartesian space) we have that on ordinary homology

H (SO(n+1)) H (Maps f=id(S n,S n)) \array{ H_\bullet\Big( SO\big( n+ 1 \big) \Big) &\longrightarrow& H_\bullet\Big( Maps_{f = id}\big( S^n, S^n \big) \Big) }

the generator in {H 2n+1(SO(n+1),) | neven H n(SO(n+1),) | nodd\left\{ \array{ H_{2n+1}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{even} \\ H_{n}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{odd} } \right. maps to the fundamental class of the respective spheres in (1), all other generators mapping to zero.

(Møller-Raussen 85, Example 2.5, Cohen-Voronov 05, Lemma 5.3.5)

See at Sullivan model of a spherical fibration for more on this.

References

Last revised on March 5, 2019 at 02:25:25. See the history of this page for a list of all contributions to it.