## 1 Introduction. (2002)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Mcclure021introduction.,

author = {James E. Mcclure and Jeffrey H. Smith},

title = {1 Introduction.},

year = {2002}

}

### OpenURL

### Abstract

### Citations

198 |
Vogt – Homotopy invariant algebraic structures on topological spaces
- Boardman, M
- 1973
(Show Context)
Citation Context ...of operad hadn’t yet been defined at that time). Boardman and Vogt proved the ⇐= direction for general A∞ operads (except that they used PROP’s instead of operads), but still assuming Y connected, in =-=[5, 6]-=-. May defined the concept of operad in [24] and proved the ⇐= direction for connected Y ; he proved the general version (for group-complete actions) in [25]. The =⇒ direction (for PROP’s, which implie... |

128 |
Intersection homology theory, Topology 19
- Goresky, MacPherson
- 1980
(Show Context)
Citation Context ...d in [13]) is that C∗M is canonically quasi-isomorphic to an E∞ chain algebra. The third goal of this paper is to give a new treatment of the chain-level intersection pairing, based on the account in =-=[5]-=- but with some improvements (in particular, the version given here avoids the need for the rather complicated arguments in [5, Section 7] by making fuller use of the technical felicities of [3]). The ... |

124 | Batalin-Vilkovisky algebras and two-dimensional topological field theories
- Getzler
- 1994
(Show Context)
Citation Context ...[3]). The results of this paper will be applied in [13] to prove two theorems about the ChasSullivan operations. Let S∗ denote the singular chain functor and let F be the framed little 2-disks operad =-=[4]-=-. Theorem A. The Batalin-Vilkovisky structure on the homology of LM is induced by a natural action of an operad quasi-isomorphic to S∗F on a chain complex quasi-isomorphic to S∗(LM). (Theorem A is the... |

121 | Operads in algebra, topology and physics - Markl, Shnider, et al. - 2002 |

49 | Piecewise Linear Topology - Hudson - 1969 |

33 | The Eckmann-Hilton argument and higher operads
- Batanin
(Show Context)
Citation Context ...eded in later sections as a way of interpolating between monoidal and symmetric monoidal structures. (The definition of functor-operad was discovered independently, in a different context, by Batanin =-=[4]-=-). In Section 5 we digress to offer motivation for the definition of ⊠; the definition itself, and the verification that ⊠ is indeed a symmetric monoidal structure, is given in Section 6. The main res... |

28 | Algebraic topology. McGraw-Hill Book Co - Spanier - 1966 |

26 | Combinatorial models for real configuration spaces and En–operads, from: “Operads
- Berger
- 1997
(Show Context)
Citation Context ...product of its 0-th space with ∆• ; this is used in the proof of Theorem 9.1 (the n = 2 case is the “fiberwise prismatic subdivision” used in [18]). In Section 12 we use a technique of Clemens Berger =-=[5]-=- to show that the operad Dn defined in Section 9 is weakly equivalent to the little n-cubes operad Cn; this completes the proof of Theorem 9.1. The basic idea is to generalize the fact that two spaces... |

24 |
algebras, modules and motives, Astérisque 233
- Kriz, May, et al.
- 1995
(Show Context)
Citation Context ...ion 9 gives the formal definition of “partially defined commutative algebra.” I use Leinster’s concept of homotopy algebra [11] for this purpose rather than the Kriz-May definition of partial algebra =-=[8]-=- (but I will use the term “Leinster partial algebra” instead of “homotopy algebra,” since the latter term seems excessively generic). Leinster’s definition seems to have all of the advantages and none... |

12 |
Infinite loop spaces, Annals of Mathematics Studies 90
- Adams
- 1978
(Show Context)
Citation Context ...y any operad weakly equivalent to it; such operads are called En operads. There is a highly developed technology that provides sufficient conditions for a space to have an action by an E∞ operad (see =-=[1]-=-, for example) or an E1 operad ([17], [24]). Much less is known about actions of En operads for 1 < n < ∞. In this paper we consider the important special situation where the space (or spectrum) Y on ... |

10 |
Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers de Top. et Géom
- Batanin
- 1993
(Show Context)
Citation Context ... an introduction to the ideas we begin in Sections 2 and 3 with the n = 1 (that is, the A∞) case. In Section 2 we recall the monoidal structure □ on the category of cosimplicial spaces due to Batanin =-=[2]-=-. In Section 3 we give a very simple proof of the fact (first shown in [3] and [18]) that if X • is a monoid with respect to □ then Tot(X • ) is an A∞ space; the proof is based on an idea due to Beili... |

9 |
Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200
- Dold
- 1972
(Show Context)
Citation Context ...unt in [5] but with some improvements (in particular, the version given here avoids the need for the rather complicated arguments in [5, Section 7] by making fuller use of the technical felicities of =-=[3]-=-). The results of this paper will be applied in [13] to prove two theorems about the ChasSullivan operations. Let S∗ denote the singular chain functor and let F be the framed little 2-disks operad [4]... |

9 |
Intersections and transformations of complexes and manifolds
- Lefschetz
- 1926
(Show Context)
Citation Context ...t of Mathematics, Purdue University 150 N. University Street West Lafayette, IN 47907-2067 Let M be a compact oriented PL manifold. The chain-level intersection pairing was introduced by Lefschetz in =-=[9]-=- as a tool for constructing the intersection pairing on the homology of M. The chain-level pairing is a basic ingredient in Chas and Sullivan’s construction [2] of a Batalin-Vilkovisky structure on th... |

7 |
The homology of iterated loop spaces, Lecture
- Cohen, Lada, et al.
(Show Context)
Citation Context ...he “approximation theorem” [24, Theorem 2.7]. This theorem gives a model for Ω n Σ n Z, built from Z and Cn. Using this model, Fred Cohen has given a complete description of the homology of Ω n Σ n Z =-=[7]-=-. Another basic fact is that the model splits stably as a wedge of pieces of the form Cn(k)+ ∧Σk Z∧k (where + means add a disjoint basepoint and ∧ is the smash product); this is called the Snaith spli... |

2 |
Piecewise linear topology, in Handbook of geometric topology
- Bryant
- 2002
(Show Context)
Citation Context ...d in Section 8. 13 Proof of Proposition 12.2. Throughout this section and the next we fix an integer k ≥ 2. A PL homotopy is a PL map h : X × I → Y , where X and Y are PL spaces and I is the interval =-=[0, 1]-=- with its usual PL structure. It will be convenient to have notation for the standard inclusion maps X → X × I. We write i0 (resp., i1) for the map which takes x to (x, 0) (resp., (x, 1)). We need a s... |

2 |
String topology, Preprint available at http://front.math.ucdavis.edu/math.GT/9911159. i j>i i j<i j i j j>i Pierre Deligne
- Chas, Sullivan
- 1975
(Show Context)
Citation Context ... pairing was introduced by Lefschetz in [9] as a tool for constructing the intersection pairing on the homology of M. The chain-level pairing is a basic ingredient in Chas and Sullivan’s construction =-=[2]-=- of a Batalin-Vilkovisky structure on the homology of the free loop space of M. For a complete understanding of the construction in [2], it seems helpful to have the following theorem. Let C∗M be the ... |

2 |
Homotopy algebras for operads, preprint available at http://front.math.ucdavis.edu/math.QA/0002180
- Leinster
(Show Context)
Citation Context ... the intersection of their Cartesian product with the diagonal in S × S. Section 9 gives the formal definition of “partially defined commutative algebra.” I use Leinster’s concept of homotopy algebra =-=[11]-=- for this purpose rather than the Kriz-May definition of partial algebra [8] (but I will use the term “Leinster partial algebra” instead of “homotopy algebra,” since the latter term seems excessively ... |

2 |
String Topology and the Cobar Construction
- McClure
(Show Context)
Citation Context ...ves C∗M a structure of “partially defined commutative DGA;” this is the second (and main) goal of ∗ Partially supported by NSF. 1this paper (see Theorem 12.1). A consequence of this (to be proved in =-=[13]-=-) is that C∗M is canonically quasi-isomorphic to an E∞ chain algebra. The third goal of this paper is to give a new treatment of the chain-level intersection pairing, based on the account in [5] but w... |

2 | The work and influence of Professor S. Lefschetz in algebraic topology,” in “Algebraic Geometry and Topology. A Symposium - Steenrod - 1957 |

2 |
The combinatorics of iterated loop spaces, preprint math.CT/0301221
- Batanin
(Show Context)
Citation Context ... framed little disks operad. Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. The reader is also referred to the very interesting papers of Batanin (=-=[3, 4]-=-), which treat similar questions from a categorical point of view. Here is an outline of the paper. In Section 2 we motivate the concept of non-symmetric operad. We give the definition of A∞ space and... |

1 |
Über eine Definition von S. Lefschetz in der topologischen
- Keller
- 1969
(Show Context)
Citation Context ... under subdivision. The “looping coefficients” used in Lefschetz’s second definition are tricky to define explicitly (see [10, Subsection 58 on page 216]). The theory has been worked out carefully in =-=[7]-=- (which I have not had an opportunity to consult) and seems to be rather complicated (see the Math Review: 41 # 2663). The chain-level intersection pairing became temporarily obsolete when the cup pro... |