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A bordism approach to string topology
"... Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a comp ..."
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Cited by 18 (1 self)
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Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a compact manifold. 1.
Poincaré Duality at the Chain Level, and a BV Structure on the Homology of the Free Loops Space of a Simply Connected Poincaré Duality Space
, 2003
"... We show that the simplicial chains, C•X, on a compact, triangulated, and oriented Poincaré duality space, X, of dimension d, can be endowed with an A ∞ Poincaré duality structure. Using this, we show that the shifted Hochschild cohomology, HH • (C • X, C•X)[d], of the cochain algebra, C • X, with ..."
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Cited by 3 (1 self)
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We show that the simplicial chains, C•X, on a compact, triangulated, and oriented Poincaré duality space, X, of dimension d, can be endowed with an A ∞ Poincaré duality structure. Using this, we show that the shifted Hochschild cohomology, HH • (C • X, C•X)[d], of the cochain algebra, C • X, with values in the chains, C•X, has a BV structure. This is achieved by using the A ∞ Poincaré duality structure to obtain a particular vector space isomorphism between HH • (C • X, C • X), which carries a multiplication, ∪ , and HH • (C • X, C•X), which carries a ∆ operator. It is argued in [T2] that due to the particular properties of this isomorphism, the transport of the multiplication ∪ from the domain onto the range yields a BV structure on the shifted Hochschild cohomology HH • (C • X, C•X)[d]. For a simply connected space X, the Hochschild cohomology, HH • (C • X, C•X), of the cochain algebra with values in the chains, is identified [J] with the homology, H•(LX), of the free loop space. Thus, for a simply connected Poincaré duality space, X, the shifted homology H•(LX)[d] admits a BV structure. For a manifold M, Chas and
Duality in Gerstenhaber algebras
 the Journal of Pure and Appl. Algebra
, 2005
"... Let C be a differential graded coalgebra, ΩC the Adams cobar construction and C ∨ the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH ∗ (C ∨ , C ∨ ) and HH ∗ (ΩC; ΩC). This result permits to ..."
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Cited by 2 (1 self)
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Let C be a differential graded coalgebra, ΩC the Adams cobar construction and C ∨ the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH ∗ (C ∨ , C ∨ ) and HH ∗ (ΩC; ΩC). This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of ChasSullivan, when the field of coefficients is of characteristic zero.
The BV algebra on . . .
, 2002
"... Given a unital A∞algebra with a reasonably nice∞innerproduct, as introduced in [16]. In this paper it is shown, that its Hochschildcohomology has the structure of a BValgebra, whose induced Gerstenhaber structure is ..."
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Given a unital A∞algebra with a reasonably nice∞innerproduct, as introduced in [16]. In this paper it is shown, that its Hochschildcohomology has the structure of a BValgebra, whose induced Gerstenhaber structure is