Timeline for Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 3, 2021 at 17:26 | vote | accept | qqqqqqw | ||
| Feb 1, 2021 at 13:06 | comment | added | Connor Malin | @Aleksandar_Milivojevic Yes, that is right. Just an alternative POV. Perhaps the analysis will end up being similar to Cohen’s proof of his result. | |
| Feb 1, 2021 at 7:11 | comment | added | Aleksandar Milivojević | @ConnorMalin can you elaborate on this statement that a model for the wedge is given by a product? You’ll have to introduce new generators in the tensor product of the two cdgas you wrote in order for the result to have the correct cohomology (I’m assuming you want a minimal model, since from a minimal model you easily get one for the based loop space). For example, xz must vanish in cohomology so we need a generator in degree d+1. Then this forces new generators to be introduced, etc; it seems pretty cumbersome to me to get a full description without invoking Gregory Arone’s last 2 paragraphs | |
| Jan 31, 2021 at 15:27 | comment | added | Connor Malin | The space you are looping has a very simple rational model (and is also formal). $\mathbb{C}P^n$ has a Sullivan model $\Lambda \{x,y\}$ with $|x|=2$ and $|y|=2n+1$ and $\partial(y)=x^{n+1}$ and $S^d$ has a Sullivan model $\Lambda \{z\}$ if $d$ is odd or $\Lambda \{z,w\}$ with $\partial w =z^2$ if $d$ is even. A model for the wedge is given by the product of the chain complexes in positive degrees. From here an explicit model of the based loop space can be written down (so do this twice). | |
| Jan 30, 2021 at 9:46 | answer | added | Gregory Arone | timeline score: 19 | |
| Jan 30, 2021 at 4:44 | history | asked | qqqqqqw | CC BY-SA 4.0 |