## Transferring Algebra Structures Up to Homology Equivalence (1998)

Venue: | Math. Scand |

Citations: | 17 - 3 self |

### BibTeX

@ARTICLE{Johansson98transferringalgebra,

author = {Leif Johansson and Larry Lambe},

title = {Transferring Algebra Structures Up to Homology Equivalence},

journal = {Math. Scand},

year = {1998},

volume = {88},

pages = {200--1}

}

### OpenURL

### Abstract

Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a differential graded (co)module over R and maps between modules are graded maps. When we write\Omega we mean\Omega R . The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by jmj. Algebras are assumed to be connected and coalgebras simply connected. (Co)algebras are assumed to have (co)units.(Co)algebras are, unless otherwise stated, assumed to be (co)augmented. The differential in an (co)algebra is a graded (co)derivation. The R-module of maps from M to N (for R-modules M and N) is denoted by hom(M;N) (if the context requires it, we will use a subscript to denote the ground ring). The differential in this module is given by D(f) = df \Gamma (\Gamma1) jf j fd. Note that D is a derivation with respect to the composition operation whenever it is defined. In particular, End(M) = hom(M;M) is an algebra. If A is an algebra and C is a coalgebra, the module hom(C; A) is an algebra with 1 respect to the operation defined by the following diagram C f [ g - A C\Omega C \Delta ? f\Omega g - A\Omega A 6 m (1) This product is called the cup or convolution...

### Citations

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Citation Context ...therefore be identified with the Golod resolution. The key point is that once an SDR has been chosen no more choices are necessary for computing the Golod resolution. The interested reader should see =-=[26]-=- for a correspondence between A1 -structures and the differentials in Eilenberg-Moore type spectral sequences. 7 Co-A1 Structures For the reader's convenience, we present the results dual to those in ... |

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Citation Context ... ; : : : ; v i p ; v j 1 ; : : : ; v jq ) 2=X): Remark 5.2 Note that the representation ofs0 (X) ass0 (\Delta N \Gamma1 )= ker(r 0 ) is related to the Stanley-Reisner ring of the simplicial complex X =-=[25]-=-. 11 Remark 5.3 As is well-known (e.g. [10, pp. 158]), if C and D are free chain complexes (as modules over the ground ring) then any onto map inducing an isomorphism in homology is the projection of ... |

122 |
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Citation Context ..., this problem has a long history. It was first solved in a special case, viz. when the differential in M is zero (so that M = H(A)) and the characteristic of the ground ring is zero by K. T. Chen in =-=[8] and [9] u-=-sing "iterated integrals". The special case of zero differential in M was also done by T. V. Kadeishvili in [19] and independently of this by V. Smirnov in [24]. Again in the special case wh... |

89 |
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Citation Context ...C (X; Q). It is well known (Sullivan--de Rham's theorem) that the transformation (X) - C (X; Q) (24) given by (ff)(s) = Z s ff (25) where s 2 C k (X; Q), ff 2sk (X), yields an isomorphism in homology =-=[4]-=-, [3], [23]. We will from now on assume that X is a finite complex which is connected as a space. This is strictly for convenience. Everything we say generalizes to an arbitrary connected semi-simplic... |

72 | Perturbation theory in differential homological algebra
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Citation Context ...n on the differential of M) was solved in [15] using what is called the tensor trick and [17] using a generalization of the obstruction method. The method of [15], first occurred in the literature in =-=[14]-=- for a special class. It was independently discovered and used in [18] to obtain A1 -structures. Seemingly unrelated at first, Gugenheim and Munkholm [16] gave an inductive formula for lifting cochain... |

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Citation Context ...M constructions). This proposition concludes the proof of theorem 4.1. 5 An Application to de Rham Theory Let X be a simplicial complex. Sullivan has defined a rational version of the de Rham complex =-=[27]-=- denoted bys(X). On the other hand, we have the (normalized) simplicial cochains with rational coefficients C (X; Q). It is well known (Sullivan--de Rham's theorem) that the transformation (X) - C (X;... |

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Citation Context ..." \Delta is defined by the following diagram: C\Omega As" \Delta - C\Omega A C\Omega C\Omega A \Delta\Omega 1 ? 1\Omega \Omega 1 - C\Omega A\Omega A 6 1\Omega m (3) By a classical result of =-=E. Brown, [5]sis a-=- twisting cochain if and only if dC\Omega 1+1\Omega dA +s" \Delta is a differential on C\Omega A, which together with this differential is denoted C\OmegasA and is referred to as the twisted tens... |

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Citation Context ...8 This article is dedicated to Jim Stasheff on his 60th birthday. Abstract Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], =-=[18]-=-, [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show ... |

42 |
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Citation Context ...e unique coextension of f as a coderivation, such thats1 !(f) = sf wheres1 is the projection to X and s as usual is the suspension. This method for constructingsand @ uses the perturbation lemma [6], =-=[11]-=-, [2] in the following way. First one applies the free tensor coalgebra functor T c (\Delta) to (4). T c (r) and T c (f) are the obvious maps and T c (OE) on n tensors is given by T c n (OE) = (OE\Ome... |

37 |
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Citation Context ...be the unique coextension of f as a coderivation, such thats1 !(f) = sf wheres1 is the projection to X and s as usual is the suspension. This method for constructingsand @ uses the perturbation lemma =-=[6]-=-, [11], [2] in the following way. First one applies the free tensor coalgebra functor T c (\Delta) to (4). T c (r) and T c (f) are the obvious maps and T c (OE) on n tensors is given by T c n (OE) = (... |

37 |
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Citation Context ...following diagram: M r - oe f (A; OE): (4) Here OE : A - A is a homotopy: 1 \Gamma rf = dOE \Gamma OEd between rf and the identity on A. Finally fr = 1. Without loss of generality (as demonstrated in =-=[20]-=-) one may assume in addition that the following relations (called the side conditions) hold: OEOE = 0; OEr = 0; f OE = 0: (5) 2 For a given module M we let T (M) denote P\Omega n (s M ), where M denot... |

34 |
On the chain algebra of a loop space
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Citation Context ...mulas in section 4.2 for the construction ofsand @. The differential constructed this way is the one denoted by @ di in theorem 4.1. Finally, we note that these ideas are also relavent to the work in =-=[1]-=-. 2.2 The Tensor Trick We will begin by introducing a notation which will be used throughout this paper. For a map f : T c (X)\Gamma ? X let !(f) : T c (X)\Gamma ? T c (X) be the unique coextension of... |

30 |
On the extended functoriality of Tor and
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(Show Context)
Citation Context ...e classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in =-=[16]-=-. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a di... |

25 | A fixed point approach to homological perturbation theory
- Barnes, Lambe
- 1991
(Show Context)
Citation Context ...ue coextension of f as a coderivation, such thats1 !(f) = sf wheres1 is the projection to X and s as usual is the suspension. This method for constructingsand @ uses the perturbation lemma [6], [11], =-=[2]-=- in the following way. First one applies the free tensor coalgebra functor T c (\Delta) to (4). T c (r) and T c (f) are the obvious maps and T c (OE) on n tensors is given by T c n (OE) = (OE\Omega 1\... |

24 | On the subalgebra generated by the one-dimensional elements in the Yoneda Extalgebra, Algebra, algebraic topology and their interactions (Stockholm - Löfwall - 1986 |

22 |
On the homology theory of fibre spaces
- Kadeishvili
- 1980
(Show Context)
Citation Context ...Lambe September 23, 1998 This article is dedicated to Jim Stasheff on his 60th birthday. Abstract Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],=-=[19]-=-, [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar cons... |

21 |
On a perturbation theory for the homology of the loop space
- Gugenheim
- 1982
(Show Context)
Citation Context ...September 23, 1998 This article is dedicated to Jim Stasheff on his 60th birthday. Abstract Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], =-=[13]-=-, [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar constructi... |

19 |
Extension of C ∞ function algebra by integrals and Malcev completion of π1, Adv
- Chen
- 1977
(Show Context)
Citation Context ...rry Lambe September 23, 1998 This article is dedicated to Jim Stasheff on his 60th birthday. Abstract Given a strong deformation retract M of an algebra A, there are several apparently distinct ways (=-=[9]-=-,[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar... |

6 |
On the multiplicative structure of the de Rham theory
- Gugenheim
- 1976
(Show Context)
Citation Context ...dual cochain complex C (X; Q). 2. The mapsfrom de Rham's theorem is strongly homotopy A1 . 3. The map r from above is strongly homotopy A1 . Remark 5.7 This should be compared with the main result of =-=[12]-=- wheresis shown to be strongly homotopy multiplicative using acyclic models. Proof: Convergence of the obstruction method is given in [17, pp. 244], but we have proven that this is equivalent to the t... |

4 |
Homology of fibre spaces
- Smirnov
- 1980
(Show Context)
Citation Context ...ber 23, 1998 This article is dedicated to Jim Stasheff on his 60th birthday. Abstract Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], =-=[24]-=-,[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of ... |

3 |
On perturbations and
- Gugenheim, Stasheff
- 1986
(Show Context)
Citation Context ...bove results it is clear that ∂∞ is the coderivation induced by the maps � i+j=n sf(ˆπi∪ˆπj) so that we have a proof of this fact independent of [15] and [18]. 4.2 The Obstruction Method Revisited In =-=[17]-=-, Gugenheim and Stasheff constructed a sequence of maps τn and ∂n converging to a twisting cochain and coderivation respectively. We will show that their τn and ∂n converges to the maps given in 3. We... |

2 |
On Perturbations and A1{structures
- Gugenheim, Stashe
- 1986
(Show Context)
Citation Context ...bove results it is clear that @1 is the coderivation induced by the maps P i+j=n sf( i [ j ) so that we have a proof of this fact independent of [15] and [18]. 4.2 The Obstruction Method Revisited In =-=[17]-=-, Gugenheim and Stasheff constructed a sequence of mapssn and @ n converging to a twisting cochain and coderivation respectively. We will show that theirsn and @ n converges to the maps given in 3. We... |

1 |
On a theorem of
- Penna
(Show Context)
Citation Context ...uppose that X has vertices fv 0 ; : : : ; v n g and embed X into the standard simplex \Delta n with corresponding vertices. Let ft 0 ; : : : ; t n g be the barycentric coordinates of \Delta n . Penna =-=[22]-=- has proven the following: Proposition 5.1 0 (\Delta n ) = Q[t 0 ; : : : ; t n ]=(t 0 + : : : + t n \Gamma 1) and if k (\Delta n ) r k - k (X) is the onto map given by restriction, we have ker(r 0 ) =... |