Analytical geometry by Pogorelov, A.V.

By Pogorelov, A.V.

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What corresponds to the classifying space construction in geometry is now the bar tilde construction. Inside the bar tilde construction, Massey products show up which determine the differentials in the resulting bar construction spectral sequence. Stasheff referred to these operations as Yessam operations. History relates that once, at the end of a talk of Jim’s, S. Mac Lane asked the question: Who was Yessam? Let me recall a warning, one of Jim’s favorite warnings in this context: When the differential of an A∞ -algebra is zero, the conditions force the algebra to be strictly associative but there may still be nontrivial higher operations encapsulating additional information, as the example of the Borromean rings already shows where the nontriviality of the Massey product reflects the triple linking.

Dokl. : Deformation theory and rational homotopy type. Pub. Math. Sci. IHES. : Homologie des espaces fibr´es. Pub. Math. Sci. : Homotopy associativity of H-spaces. I, II. Trans. Am. Math. Soc. : The de Rham bar construction as a setting for the Zumino, Fadde’ev, etc. descent equations. , White, A. ) Proceedings of the Symposium on Anomalies, Geometry, and Topology, Argonne, Chicago. : Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. In: Quantum groups (Leningrad 1990).

In that counterexample to Riemann’s conjecture about the infinitely great, one deforms flat Minkowskian spacetime by introducing a nonzero pseudo-Riemannian curvature. For instance, one can consider a nonzero constant curvature, de Sitter space (with positive curvature and SO(4, 1) symmetry) or anti de Sitter space (AdS, with negative curvature and SO(3, 2) symmetry). The latter two symmetry groups are further deformations of the Poincar´e group (or Lie algebra) in the sense of Gerstenhaber. Being simple and therefore with zero cohomology (Whitehead’s lemma), the Gerstenhaber theory of deformations [Ge64] shows that “the buck stops there” in the category of Lie groups or algebras.

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