Van den Bergh isomorphisms in string topology

The purpose of this paper is to give applications of the Eilenberg-Moore type spectral sequence converging to the relative loop homology algebra of a Gorenstein space, which is introduced in the previous paper due to the authors. Moreover, it is proved that the spectral sequence is functorial on the category of simply-connected Poincaré duality spaces over a space.2010 Mathematics Subject Classification: 55P35, 55T20

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“…Then it is readily seen that η is an isomorphism of graded vector spaces. See [20,Section 9] for such an isomorphism in more general setting.…”

Section: The Emss Calculations Of the Loop Homologymentioning

confidence: 99%

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

Kuribayashi

Menichi

Naito

2014

Topology and its Applications

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“…Special cases of such isomorphisms can be traced back to the origins of homological algebra, but systematically they were studied only in 1998 by M. Van den Bergh [39]. For more recent results involving the Van den Bergh isomorphisms, see [5,7,8,10,12,23,26], to cite a few. Below we will use a Fréchet algebra version of the Van den Bergh isomorphisms, which was introduced and applied in [32] to some problems of Topological Homology (see also [35]).…”

Section: Van Den Bergh Isomorphisms For O(x)mentioning

confidence: 99%

Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

Pirkovskii

2014

Journal of Functional Analysis

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Abstract. Let X be a Stein manifold, and let Y ⊂ X be a closed complex submanifold. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y ) as a Fréchet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y ) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y ) over O(X) equals the dimension of X.

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Derived string topology and the Eilenberg-Moore spectral sequence

Kuribayashi

Menichi²,

Naito

2015

Isr. J. Math.

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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 ∨ ) of a commutative Gorenstein algebra A and show that over Q, HH * (A P L (M ), A P L (M ) ∨ ) 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 * (A P L (M ), A P L (M )) of Félix and Thomas.

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Van den Bergh isomorphisms in string topology

Cited by 4 publications

References 26 publications

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

Derived string topology and the Eilenberg-Moore spectral sequence

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