## Structure of the loop homology algebra of a closed manifold

Citations: | 4 - 1 self |

### BibTeX

@TECHREPORT{Félix_structureof,

author = {Yves Félix and Jean-claude Thomas and Micheline Vigué-poirrier},

title = {Structure of the loop homology algebra of a closed manifold},

institution = {},

year = {}

}

### OpenURL

### Abstract

The loop homology of a closed orientable manifold M of dimension d is the ordinary homology of the free loop space MS1 with degrees shifted by d, i.e. H∗(M S1) = H∗+d(M S1). Chas and Sullivan have defined a loop product on H∗(M S1) and an intersection morphism I: H∗(M S1) → H∗(ΩM). The algebra H∗(M S1) is commutative and I is a morphism of algebras. In this paper we produce a model that computes the algebra H∗(M S1) and the morphism I. We show that the kernel of I is nilpotent and that the image is contained in the center of H∗(ΩM), which is in general quite small.

### Citations

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227 |
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(Show Context)
Citation Context ...a, and A e = A ⊗ A op be the enveloping algebra. The Hochschild chain complex C∗(A;M) (resp. the Hochschild cochain complex C∗ (A;M)) of A with coefficients in M is the chain (resp. cochain) complex (=-=[12]-=-,[18],[3]) C∗(A;M) = M ⊗Ae IB(A;A;A) (resp. C ∗ (A;M) = HomAe(IB(A;A;A),M) = Hom(IB(A),M)) . More precisely, and in opposition with the standard convention on upper and lower degrees, we have Cn(A;M) ... |

157 |
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(Show Context)
Citation Context ...Hom (IB(H ∗ (M)),lk) ⊗ H ∗ (M),D). □ Remark. In the case M is a formal space which means that C∗ (M) is quasi-isomorphic to H∗ (M, (for instance a simply connected compact Kähler manifold for lk = Q, =-=[8]-=-), one can choose A = H∗ (M) and thus the algebras HH ∗ (H∗(M),H ∗ (M)) and H∗(M S1) are isomorphic. The commutative case. We suppose here that the algebra C∗ (M) is connected by a sequence of quasi-i... |

114 | A homotopy theoretic realization of string topology
- Cohen, Jones
(Show Context)
Citation Context ... ∼=↑ ↑ ∼ = Hq(A ⊗ ΩA∨ ,D) Hq(ε⊗1) −→ Hq(ΩA ∨ ,D) 2The starting point of our construction is the Jones isomorphism between the free loop space homology and the Hochschild cohomology of C ∗ (M), [17], =-=[5]-=-. Φ : H∗(M S1 ) ∼ = ∗ ∗ ∗ −→ HH (C (M), C (M)), and the fact that this morphism is multiplicative with respect to the loop product and the Gerstenhaber product as defined in [5]. In order to obtain ou... |

113 |
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(Show Context)
Citation Context ...d A e = A ⊗ A op be the enveloping algebra. The Hochschild chain complex C∗(A;M) (resp. the Hochschild cochain complex C∗ (A;M)) of A with coefficients in M is the chain (resp. cochain) complex ([12],=-=[18]-=-,[3]) C∗(A;M) = M ⊗Ae IB(A;A;A) (resp. C ∗ (A;M) = HomAe(IB(A;A;A),M) = Hom(IB(A),M)) . More precisely, and in opposition with the standard convention on upper and lower degrees, we have Cn(A;M) = ⊕p+... |

64 |
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(Show Context)
Citation Context ...Hq(ΩM) ∼=↑ ↑ ∼ = Hq(A ⊗ ΩA∨ ,D) Hq(ε⊗1) −→ Hq(ΩA ∨ ,D) 2The starting point of our construction is the Jones isomorphism between the free loop space homology and the Hochschild cohomology of C ∗ (M), =-=[17]-=-, [5]. Φ : H∗(M S1 ) ∼ = ∗ ∗ ∗ −→ HH (C (M), C (M)), and the fact that this morphism is multiplicative with respect to the loop product and the Gerstenhaber product as defined in [5]. In order to obta... |

31 | On the chain algebra of a loop space - Adams, Hilton |

28 |
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(Show Context)
Citation Context ...ra C∗ (M) is connected by a sequence of quasi-isomorphisms to a commutative differential graded algebra (A,d). This is the case in characteristic zero, and when lk is a field of characteristic p > d (=-=[2]-=-, Proposition 8.7). We can also suppose that A is finite dimensional, A0 = lk, A1 = 0, A >d = 0 and Ad = lkω. Note first that the formulas of Theorem 5 simplify as ⎧ ⎪⎨ D(a ⊗ 1) = d(a) ⊗ 1 ⎪⎩ D(1 ⊗ b)... |

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Citation Context ...(t) a polynomial of degree k ≤ cat M, where catM denotes the Lusternik-Schnirelmann category of M (cf section 6), and ¯ G∗(M) is defined as follows. An element x ∈ πq(M) is called a Gottlieb element (=-=[13]-=-), if the map x∨idM : S q ∨M → M extends to the product S q × M. The subgroup generated by the Gottlieb elements is denoted by G∗(M), and the subgroup of π∗(ΩM) corresponding to G∗(M) by the adjunctio... |

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19 |
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(Show Context)
Citation Context ... AMS Classification : 55P35, 54N45,55N33, 17A65, 81T30, 17B55 Key words : free loop space, loop homology, Hochschild cohomology Let M be a simply connected closed oriented d-dimensional manifold. In (=-=[4]-=-), Chas and Sullivan have constructed a product on the desuspension, Hr(M S1) = Hr+d(M S1), of the ordinary homology of the free loop space on M. The product, called loop product, is defined using the... |

14 | Universal enveloping algebras and loop space homology - Halperin - 1992 |

12 |
Cyclic homology of commutative algebras
- Burghelea, Vigué-Poirrier
- 1986
(Show Context)
Citation Context ... = A ⊗ A op be the enveloping algebra. The Hochschild chain complex C∗(A;M) (resp. the Hochschild cochain complex C∗ (A;M)) of A with coefficients in M is the chain (resp. cochain) complex ([12],[18],=-=[3]-=-) C∗(A;M) = M ⊗Ae IB(A;A;A) (resp. C ∗ (A;M) = HomAe(IB(A;A;A),M) = Hom(IB(A),M)) . More precisely, and in opposition with the standard convention on upper and lower degrees, we have Cn(A;M) = ⊕p+q=nM... |

11 |
The radical of the homotopy Lie algebra
- Felix, Halperin, et al.
- 1988
(Show Context)
Citation Context ... dimensional commutative differential graded algebra (A,d) satisfying (A + ) n = 0 for n > cat M0. Thus Theorem 5 (2) implies (1). Denote by L the rational homotopy Lie algebra of the manifold M. In (=-=[10]-=-), the authors prove that R(L) is finite dimensional and that dimR(L)even ≤ catM0. The Poincaré-Birkoff-Witt theorem implies then that the center of UL is a graded subvector space of the tensor produc... |

9 |
Suites inertes dans les algèbres de Lie graduées (“Autopsie d’un meurtre
- Halperin, Lemaire
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(Show Context)
Citation Context ...e algebra generated by u, v and w considered in degree 2. The inclusion i : M → N induces a surjective map π∗(ΩM) ⊗ Q → π∗(ΩN) ⊗ Q, This means that the attachment of the cell is inert in the sense of =-=[16]-=-. Therefore, ([16]), π∗(ΩM) ⊗ Q ∼ = Ab(a,b,c) ∐ Ab(e,f,g) ∐ L(x) 15with |x| = 7. In particular R(L) is zero, and by Theorems 8 and 9, the morphism I is trivial. Example 3. Product of two manifolds. I... |

7 |
Notions of category in differential algebra, In: Algebraic topology - rational homotopy
- Halperin, Lemaire
- 1988
(Show Context)
Citation Context ... −→ H∗(M S1,E) ∼= ←− H∗(ΩM), with θ ′ = H(ρ) −1 ◦ H( ¯ λ) ◦ θM,Y . □ 4 A model for the intersection map I Denote by f : (T(V ),d) → C ∗ (M) a free minimal model for the singular cochain algebra on M (=-=[15]-=-), i.e. (T(V ),d) is a differential graded algebra, f is a quasi-isomorphism of differential graded algebras, and d(V ) ⊂ T ≥2 (V ). The differential graded algebra (T(V ),d) is uniquely defined by th... |

5 | Gottlieb groups, group actions, fixed points and rational homotopy, Lecture - Oprea |

4 |
The loop homology of spheres and projective spaces, preprint
- Cohen, Jones, et al.
- 2002
(Show Context)
Citation Context ... → Hq(ΩM). The morphism I is multiplicative with respect to the loop product on MS1 and the Pontryagin product on ΩM. In ([4]) Chas and Sullivan proves that I is surjective for a Lie group G, and in (=-=[6]-=-), Cohen, Jones and Yan compute the algebra H∗(M S1) and the morphism I when M is a sphere or a projective space. In this paper we produce a very simple model for computing H∗(M S1) and we establish g... |