Configuration Spaces Woes
I’ve been neglecting this blog a lot. Juggling research, teaching, organizing a seminar, and a personal life leaves little time for writing articles! (Wait, isn’t that the same complaint as last time?)
Most prominently I’ve been spending a lot of time working on my paper about the Lambrechts–Stanley model for configuration spaces (see my previous post). The good news is, I’m done (or as done as one can be with a paper). I’ve just uploaded the third version of the paper on the arXiv, and I’ve submitted it. I’ve finally managed to remove this bothersome hypothesis about the Euler characteristic of the manifold, and I’ve fixed an issue about my use of the propagator (PA forms are hard). The new abstract is:
We prove the validity over ℝ of a CDGA model of configuration spaces for simply connected manifolds of dimension at least 4, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on a Poincaré duality model of the manifold. We moreover prove that our model is compatible with the action of the Fulton–MacPherson operad when the manifold is framed, by relying on Kontsevich’s proof of the formality of the little disks operads. We use this more precise result to get a complex computing factorization homology of framed manifolds.
I’ve also had the opportunity to give three talks about the paper: at the annual meeting of the GDR Topologie Algébrique in Amiens, at the conference in the honor of Said Zarati in Tunis, and at the seminar of the LAGA at Paris 13. I’d like to thank the organizers of these events, and especially Sadok Kallel and Bruno Vallette for inviting me at the latter two.
Hopefully I will now have more time to fill this blog! (Or more realistically, do more research.)