Rational homotopy groups and Koszul algebras

Papadima, Stefan; Suciu, Alexander I.

Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 53-58.

Soient X et Y deux CW-espaces de type fini (X connexe, Y simplement connexe), tels que l'anneau de cohomologie $H^{*}(\mathrm{Y},\mathbb{Q})$ soit un k-recalibrage de $H^{*}(\mathrm{X},\mathbb{Q})$. Si $H^{*}(\mathrm{X},\mathbb{Q})$ est une algèbre de Koszul, alors l'algèbre de Lie graduée ${\pi }_{*}\left(\Omega \mathrm{Y}\right)\otimes \mathbb{Q}$ est le k-recalibrage de ${\mathrm{gr}}_{*}\left({\pi }_1\mathrm{X}\right)\otimes \mathbb{Q}$. Si Y est un espace formel, alors l'implication réciproque est vraie aussi, et l'espace Y est coformel. De plus, si X est formel, avec algèbre de cohomologie de Koszul, on trouve des isomorphismes de groupes filtrés entre le complété de Malcev de π₁X, le complété de $[\Omega {\mathrm{S}}^{2\mathrm{k}+1},\Omega \mathrm{Y}]$, et le groupe de Milnor–Moore d'applications de cogèbres entre $H_{*}(\Omega {\mathrm{S}}^{2\mathrm{k}+1},\mathbb{Q})$ et $H_{*}(\Omega \mathrm{Y},\mathbb{Q})$.

Let X and Y be finite-type CW-spaces (X connected, Y simply connected), such that the ring $H^{*}(\mathrm{Y},\mathbb{Q})$ is a k-rescaling of $H^{*}(\mathrm{X},\mathbb{Q})$. If $H^{*}(\mathrm{X},\mathbb{Q})$ is a Koszul algebra, then the graded Lie algebra ${\pi }_{*}\left(\Omega \mathrm{Y}\right)\otimes \mathbb{Q}$ is the k-rescaling of ${\mathrm{gr}}_{*}\left({\pi }_1\mathrm{X}\right)\otimes \mathbb{Q}$. If Y is a formal space, then the converse holds, and Y is coformal. Furthermore, if X is formal, with Koszul cohomology algebra, there exist filtered group isomorphisms between the Malcev completion of π₁X, the completion of $[\Omega {\mathrm{S}}^{2\mathrm{k}+1},\Omega \mathrm{Y}]$, and the Milnor–Moore group of coalgebra maps from $H_{*}(\Omega {\mathrm{S}}^{2\mathrm{k}+1},\mathbb{Q})$ to $H_{*}(\Omega \mathrm{Y},\mathbb{Q})$.

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