## A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT (905)

### BibTeX

@MISC{Ginot905achen,

author = {Grégory Ginot and Thomas Tradler and Mahmoud Zeinalian},

title = {A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT},

year = {905}

}

### OpenURL

### Abstract

Abstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes which are complexes whose differentials are modeled by an arbitrary simplicial set much in the same way that the ordinary Hochschild differential is modeled by the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold, which is an analogue of the loop product in string topology. As an application we show that this product is homotopy invariant. We prove Hochschild-Kostant-Rosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Contents

### Citations

404 | An Introduction to Homological Algebra, Cambridge University Press - Weibel - 1994 |

261 | Simplicial objects in algebraic topology - May - 1992 |

224 | Rational homotopy theory - Quillen - 1969 |

115 |
Iterated path integrals
- Chen
- 1997
(Show Context)
Citation Context ...nition 1.2.2], a p-form ω ∈ Ω p (M Y ) on M Y is given by a p-form ωφ ∈ Ω p (U) for each plot φ : U → M Y , which is invariant with respect to smooth transformations of the domain. Let us recall from =-=[C]-=- that a smooth transformation between two plots φ : U → M Y and ψ : V → M Y is a smooth map θ : U → V such that the following diagram φ♯ U × Y � M ��� θ ��� ��� ψ♯ ��� V × Y commutes. The invariance o... |

114 | A homotopy theoretic realization of string topology - Cohen, Jones |

114 |
Cyclic Homology. Grundlehren der Mathematischen Wissenschaften
- Loday
- 1992
(Show Context)
Citation Context ...the fundamental connection between the Hochschild chain complex and the circle, which, for instance gives rise to the cyclic structure of the Hochschild chain complex and thus to cyclic homology, see =-=[L]-=-. This connection is also at the heart of the relationship between the Hochschild complex of the differential forms Ω • M on a manifold M, and the differential forms Ω • (LM) on the free loop space LM... |

95 |
The cyclotomic trace and algebraic K-theory of spaces
- Bökstedt, Madsen
- 1993
(Show Context)
Citation Context ...nctor sd2 : ∆ → ∆ is defined by sd2([n − 1]) = [2n − 1], and, for any map, f : [n − 1] → [m − 1], by sd2(f) : [2n − 1] → [2m − 1], sd2(f) : i + nj ↦→ f(i) + mj where 0 � i � n − 1 and j ∈ {0, 1}, see =-=[BHM]-=-. The edgewise subdivision sd2(X•) of a simplicial set X• is the composition X• ◦ sd2. There is a natural homeomorphism D : |sd2(X•)| ∼ → |X•|(see [BHM, Lemma 1.1]) induced by the maps ∆n−1 × X2n−1 → ... |

49 |
Iterated integrals of differential forms and loop space homology
- Chen
- 1973
(Show Context)
Citation Context ...Chen(M Y ), which, by definition, is given by the image of the iterated integral map Chen(M Y ) = Im(ItY• : CH Y• • (Ω, Ω) → Ω• (MY )) ⊂ Ω• (MY ). Chen showed in the case of the circle Y• = S1 • (cf. =-=[C2]-=-), that Chen(M S1) is in fact quasi-isomorphic to Ω• (MY ) by showing that its kernel Ker(ItS1 •) is acyclic. In the case of a general simplicial set Y•, this task turns out to become quite more elabo... |

44 | Infinitesimal computations in topology, Inst. Hautes Études - Sullivan - 1977 |

38 | Differential forms on loop spaces and the cyclic bar complex,” Topology 30
- Getzler, Jones, et al.
- 1991
(Show Context)
Citation Context ...mplex of the differential forms Ω • M on a manifold M, and the differential forms Ω • (LM) on the free loop space LM of M, which is the space of smooth maps from the circle S 1 to the manifold M; see =-=[GJP]-=-. At the core of this connection is the fact that the Hochschild complex is the underlying complex of a simplicial module whose simplicial structure is modelled on a particular simplicial model S1 • o... |

35 |
Relative algebraic K-theory and cyclic homology
- Goodwillie
- 1986
(Show Context)
Citation Context ... algebra, then CH X• (R•) is a bisimplicial commutative differential graded algebra. Note that for the standard Hochschild chain complex over S 1 •, this definition was first introduced by Goodwillie =-=[Go]-=-. By the (generalized) Eilenberg-Zilber theorem [MacL, GJ], there is a natural quasi-isomorphism EZ : CH X• (R•) → diag(CH X• (R•))•, where diag(CH X• (R•))• is the diagonal simplicial set associated ... |

12 | Cyclic homology of commutative algebras - Burghelea, Vigué-Poirrier - 1986 |

11 | Rational BV-algebra in string topology
- Félix, Thomas
(Show Context)
Citation Context ...nnected compact manifold M, the (dualized) iterated integral ItΣ• : ( ⊕ g≥0 H−•(Map(Σg , M)), ⊎) → ( ⊕ g≥0 HH−• Σ g (Ω, Ω), ∪) is an isomorphism of algebras using rational homotopy • techniques as in =-=[FT]-=-. As a corollary of this, it follows that the surface product ⊎ is homotopy invariant meaning, that, if M and N are 2-connected compact manifolds with equal dimensions, and i : M → N is a homotopy equ... |

9 | Higher string topology on general spaces - Hu |

9 | The cohomology ring of free loop spaces - Menichi |

9 | Hodge decomposition for higher order Hochschild homology
- Pirashvili
(Show Context)
Citation Context ...ct new complexes whose module structure and differential are combinatorially governed by a given simplicial set X•, much in the same way that the ordinary Hochschild complex is governed by S1 • ; see =-=[P]-=-. However carrying the construction to higher dimensional simplicial sets turns out to require (associative and) commutative algebras. The result of these constructions define for any (differential gr... |

6 | Notes on string topology, String topology and cyclic homology - Cohen, Voronov - 2006 |

6 | On operations for Hochschild homology - McCarthy - 1993 |

4 |
Higher order Hochschild Cohomology
- Ginot
- 2005
(Show Context)
Citation Context ...required to include the genus zero case of the 2-sphere in this framework. To this end, we first recall a cup product for genus zero defined in by a simplicial model Pinchg,h : Σ g+h • → Σ g • ∨Σ h • =-=[G]-=-, and then define a left and right action, ˜∪, of CH Σ0 • • (A, B) on CH Σg • • (A, B). Taking advantage of the fact that one can choose different simplicial models for a given space, we then show tha... |

3 | Grégory Ginot, Behrang Noohi, and Ping Xu, String topology for stacks, preprint: math.AT/0712.3857v1 - Behrend - 2007 |

3 |
Cochain algebras of mapping spaces and finite group actions. To appear in Topology and its Applications (2003) jean-claude.thomas@univ-angers.fr CRM Barcelona Département de mathématique Institut d’Estudis Catalans Faculté des Sciences Apartat 50E 2, Boul
- Patras, Thomas
(Show Context)
Citation Context ...f algeCorollary 2.4.7. ItY• bras. Remark 2.4.8. The proof of Proposition 2.4.6 is essentially the same as the proof given by Patras and Thomas in [PT, Proposition 2], and could have been deduced from =-=[PT]-=-. We will use the relationship with [PT] in the next subsection. 2.5. Chen’s iterated integrals as a quasi-isomorphism. In this subsection, we show that the iterated integral map ItY• : CH Y• • (Ω, Ω)... |

2 |
Σ-models and String Topology” in Graphs and Patterns in Math. and Theoretical Phys
- Sullivan
(Show Context)
Citation Context ...ld (co)homology over compact surfaces Σg of genus g. The collection of compact surfaces of any genus is naturally equipped with a product similar to the loop product of string topology [CS], also see =-=[S2]-=-. The idea behind this product, that we call the surface product, is shown in the following picture. (1.1) wedge pinch In Section 3.1, we describe an explicit simplicial model for the string topology ... |

1 | Micheline Vigué, The Hochschild cohomology of a closed manifold - Félix, Thomas |

1 | Micheline Vigué, Rational string topology - Félix, Thomas |

1 |
Poincaré duality algebras and commutative differential graded algebras
- Lambrechts, Stanley
(Show Context)
Citation Context ...M)) is the dual of the map δ g,h .44 G. GINOT, T. TRADLER, AND M. ZEINALIAN We now want to dualize the Hochschild cup product for surfaces. Since M is a Poincaré duality space, by the main result of =-=[LS]-=-, there exists a differential graded commutative algebra (A, d), weakly equivalent to (Ω•M, ddR), which is simply connected, finite dimensional and is equipped with a trace Adim(M) ǫ → R such that: • ... |