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rational n-sphere

Contents

Contents

Idea

The rational nn-sphere is the rationalization of the n-sphere.

As a topological space this is a rational topological space closest to the nn-sphere, in the sense of rational homotopy theory.

Sullivan models

The minimal Sullivan model of a sphere S 2k+1S^{2k+1} of odd dimension is the dg-algebra with a single generator ω 2k+1\omega_{2k+1} in degre 2k+12k+1 and vanishing differential

dω 2k+1=0. d \omega_{2k+1} = 0 \,.

The minimal Sullivan model of a sphere S 2kS^{2k} of even dimension, for k1k \geq 1. is the dg-algeba with a generator ω 2k\omega_{2k} in degree 2k2k and another generator ω 4k1\omega_{4k-1} in degree 4k+14k+1 with the differential defined by

dω 2k=0 d \omega_{2k}= 0
dω 4k+1=ω 2kω 2k. d \omega_{4k+1} = \omega_{2k}\wedge \omega_{2k} \,.

One may understand this form the central fact of rational homotopy theory (the proposition here):

An nn-sphere has rational cohomology concentrated in degree nn. Hence its minimal Sullivan model needs at least one closed generator in that degree. In the odd dimensional case one such is already sufficient, since the wedge square of that generator vanishes and hence produces no higher degree cohomology classes. But in the even degree case the wedge square ω 2kω 2k\omega_{2k}\wedge \omega_{2k} needs to be canceled in cohomology. That is accomplished by the second generator ω 4k1\omega_{4k-1}.

Again by that proposition, this now implies that the rational homotopy groups of spheres are concentrated, in degree 2k+12k+1 for the odd (2k+1)(2k+1)-dimensional spheres, and in degrees 2k2k and 4k14k-1 in for the even 2k2k-dimensional spheres. (These are the non-torsion homotopy groups of spheres appearing in the Serre finiteness theorem.)

For instance the 4-sphere has rational homotopy in degree 4 and 7. The one in degree 7 being represented by the quaternionic Hopf fibration.

Hence, odd dimensional nn-spheres are rationally homotopy equivalent to Eilenberg-MacLane spaces K(,n)K(\mathbb{Z},n), while even-dimensional spheres are not.

Applications

Hopf invariant

Proposition

By standard results in rational homotopy theory, every continuous function

S 4k1ϕS 2k S^{4k-1} \overset{\phi}{\longrightarrow} S^{2k}

corresponds to a unique dgc-algebra homomorphism

CE(𝔩S 4k1)CE(𝔩ϕ)CE(𝔩S 2k) CE \big( \mathfrak{l}S^{4k-1} \big) \overset{ CE(\mathfrak{l}\phi) }{\longleftarrow} CE \big( \mathfrak{l}S^{2k} \big)

between Sullivan models of n-spheres.

The unique free coefficient of this homomorphism CE(𝔩ϕ)CE(\mathfrak{l}\phi) is the Hopf invariant HI(ϕ)HI(\phi) of ϕ\phi:

Last revised on June 3, 2019 at 11:20:29. See the history of this page for a list of all contributions to it.