A spectral sequence for tangent cohomology of algebras over algebraic operads.

Abstract: We produce a spectral sequence that converges to the operadic cohomology of a fixed algebra over an algebraic operad. Our main tool is that of filtrations arising from towers of cobrations of algebras. These play the role in algebra that cell attaching maps and skeletal filtrations do for topological spaces.

As an application, we consider the rational Adams–Hilton construction on topological spaces, where our spectral sequence is multiplicative and converges to the Chas–Sullivan loop product. We also consider relative Sullivan models of a fibration $p$, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of $p$; and the Quillen model of a space, where our spectral sequence converges to the homotopy groups of the classifying space of the identity component of the self-equivalences of the space.

References:

[1] Moreno-Fernández, J., Tamaroff, P., A spectral sequence for tangent cohomology of algebraic operads, arXiv:2008.00876 (2020).

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)