Brackets by any other name

<p style='text-indent:20px;'>Brackets by another name - Whitehead or Samelson products - have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I <i>sketch</i> the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.</p> <p style='text-indent:20px;'>In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns; unless particularized, <i>bracket</i> will be the generic term including product and brace. The path leads beyond binary to multi-linear <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-ary operations, either for a single <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> or for whole coherent congeries of such assembled into what is known now as an <inline-formula><tex-math id="M3">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>-algebra, such as in homotopy Gerstenhaber algebras. It also leads to more subtle invariants. Along the way, attention will be called to interaction with 'physics'; indeed, it has been a two-way street.</p>

Log smooth deformation theory via Gerstenhaber algebras

We construct a $$k\left[ \!\left[ Q\right] \!\right] $$ k Q -linear predifferential graded Lie algebra $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ associated to a log smooth and saturated morphism $$f_0: X_0 \rightarrow S_0$$ f 0 : X 0 → S 0 and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, 2019. arXiv:1902.11174) whereof $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields; this method is closely related to recent developments in mirror symmetry.

Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure.We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.

Representation stability for homotopy groups of configuration spaces

We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of [6], and that their derived dual integral homotopy groups are finitely generated as{{\mathsf{FI}}}-modules in the sense of [4]. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-{{\mathsf{FI}}}-space to properties of its dual homotopy groups. We also discuss several other applications, including free Lie and Gerstenhaber algebras.

A direct computation of the cohomology of the braces operad

We give a self-contained and purely combinatorial proof of the well-known fact that the cohomology of the braces operad is the operad ${{\mathsf{Ger}}}$ governing Gerstenhaber algebras.

A spectral sequence for the homology of a finite algebraic delooping

In the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.