2012/11/29 by Katsuhiko Kuribayashi, Kuribayashi, Katsuhiko, Luc Menichi +3 · 1 citation
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT
paper · pdf · doi:10.48550/arxiv.1211.6833
40 pages, this version is one of two preprints divided from the first version, an appendix on shriek maps is revised
arxiv created 2013/04/25 · arxiv updated 2013/04/26
Let M be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space H_*(LM). We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively. We also define a generalized cup product on the Hochschild cohomology HH^*(A,A^\vee) of a commutative Gorenstein algebra A and show that over ℚ, HH^*(APL(M),APL(M)^\vee) is isomorphic as algebras to H_*(LM). Thus, when M is a Poincaré duality space, we recover the isomorphism of algebras ℍ_*(LM;ℚ)^≅ HH^*(APL(M),APL(M)) of Félix and Thomas.