Classification of Irreducible Holonomies of Torsion-Free Affine Connections

1999/07/01 by Sergei Merkulov, Lorenz Schwachhöfer · 101 citations
Mathematics · Medicine · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Mathematics #Torsion (gastropod) #Affine transformation #Pure mathematics #Algebra over a field #Surgery #Medicine

paper · doi:10.2307/121098

published in Annals of Mathematics 150(1), 77 (Princeton University)

openalex publication_date 1999/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators.We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization.We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com ∞ and Ass ∞ , involving, e.g., a mysterious mixture of associahedra with cyclohedra.Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.

Citations

• Operads, homotopy algebra and iterated integrals for double loop spaces
• PROP profile of deformation quantization and graph complexes with loops and wheels
• A theorem on holonomy
• Koszul duality for dioperads
• Koszul duality for operads

Cited by

• Connections with Irreducible Holonomy Representations
• On a rigIDity condition for Berwald spaces
• Holonomy [wikipedia]