Algebras realized by \(n\) rational homotopy types. (English) Zbl 0735.55008
Using rational perturbation theory as developed by Halperin, Stasheff and Schlessinger, the author proves the following theorem.
Let \(n\) be a natural number and let \(X=S^ 5\vee S^{4n+4}\vee S^{4n+13}\vee S^{4n+17}\vee\dots\vee S^{8n+9}\). Then there are exactly \(n+1\) rational homotopy types \(Y\) such that \(H^*(X;{\mathbb{Q}})\approx H^*(Y;{\mathbb{Q}})\) (as algebras).
A similar example shows that for any \(n\) there is a product of Eilenberg- MacLane spaces \(X\) such that there are exactly \(n+1\) rational homotopy types \(Y\) such that \(\pi_ *(\Omega X)\otimes {\mathbb{Q}}\approx\pi_ *(\Omega Y)\otimes{\mathbb{Q}}\) (as Lie algebras).
Let \(n\) be a natural number and let \(X=S^ 5\vee S^{4n+4}\vee S^{4n+13}\vee S^{4n+17}\vee\dots\vee S^{8n+9}\). Then there are exactly \(n+1\) rational homotopy types \(Y\) such that \(H^*(X;{\mathbb{Q}})\approx H^*(Y;{\mathbb{Q}})\) (as algebras).
A similar example shows that for any \(n\) there is a product of Eilenberg- MacLane spaces \(X\) such that there are exactly \(n+1\) rational homotopy types \(Y\) such that \(\pi_ *(\Omega X)\otimes {\mathbb{Q}}\approx\pi_ *(\Omega Y)\otimes{\mathbb{Q}}\) (as Lie algebras).
Reviewer: H.M.Unsöld (Berlin)
