Minimal models of loop spaces and suspensions. (English) Zbl 0808.55009
This paper gives the solution to the 2 following problems:
(1) Let \(X\) be a space. Determine the Sullivan minimal model for the loop multiplication \(\mu_ x : \Omega X \times \Omega X \to \Omega X\) and the Quillen model for the suspension comultiplication \(\nabla X : \Sigma X \to \Sigma X \vee \Sigma X\).
(2) Let \(\psi : A \to A \otimes A\) be an associative comultiplication of a graded commutative algebra and let \(\varphi : L \to L \sqcup L\) be an associative comultiplication of a 1-connected graded Lie algebra \(L\); describe spaces \(X\) and \(Y\) such that \(\psi\) is \((\mu_ x)_ *\) and \(\varphi\) is \((\nabla Y)_ *\).
(1) Let \(X\) be a space. Determine the Sullivan minimal model for the loop multiplication \(\mu_ x : \Omega X \times \Omega X \to \Omega X\) and the Quillen model for the suspension comultiplication \(\nabla X : \Sigma X \to \Sigma X \vee \Sigma X\).
(2) Let \(\psi : A \to A \otimes A\) be an associative comultiplication of a graded commutative algebra and let \(\varphi : L \to L \sqcup L\) be an associative comultiplication of a 1-connected graded Lie algebra \(L\); describe spaces \(X\) and \(Y\) such that \(\psi\) is \((\mu_ x)_ *\) and \(\varphi\) is \((\nabla Y)_ *\).
Reviewer: Y.Felix (Louvain-La-Neuve)
MSC:
| 55P62 | Rational homotopy theory |
| 55P45 | \(H\)-spaces and duals |
| 55P40 | Suspensions |
| 55P35 | Loop spaces |
Keywords:
Sullivan minimal model for the loop multiplication; Quillen model for the suspension comultiplication; comultiplication; graded Lie algebraReferences:
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