nLab
rational 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

Sullivan models for mapping spaces

Spectral sequence for rational homotopy of mapping spaces

A spectral sequence computing the rational homotopy of mapping spaces:

  • Samuel B. Smith, A based Federer spectral sequence and the rational homotopy of function spaces, Manuscripta Math (1997) 93: 59 (doi:10.1007/BF02677458)

based on

  • Herbert Federer, A Study of Function Spaces by Spectral Sequences, Transactions of the American Mathematical Society Vol. 82, No. 2 (Jul., 1956), pp. 340-361 (jstor:1993052)

Last revised on September 12, 2019 at 07:17:53. See the history of this page for a list of all contributions to it.