# Homological perturbation theory

A theory concerning itself with a collection of techniques for deriving chain complexes which are both smaller and chain homotopy equivalent to a given chain complex (cf. also Complex (in homological algebra)). It is motivated by the desire to find effective algorithms in homological algebra. The cornerstone of the theory is an important algorithm which, when convergent, is commonly called the "perturbation lemma" . To understand the statement of the perturbation lemma, some preliminary notation is needed.

## Contents

### Strong deformation retraction data.

It will be assumed that $R$ is a commutative ring with unit and that all chain complexes are over $R$ and free (cf. also Simplicial complex). A strong deformation retract from $X$ to $Y$ consists of two chain complexes $X$ and $Y$ such that there are chain mappings $\nabla : X \rightarrow Y$, $f : Y \rightarrow X$, and a chain homotopy $\phi : Y \rightarrow Y$ such that $f \nabla = 1 _ { X }$ (the identity mapping on $X$) and $D ( \phi ) = 1 _ { Y } - \nabla f$ (cf. also Complex (in homological algebra)). Here it is assumed that the differentials $d _X$ and $d_Y$ of $X$ and $Y$, respectively, are of degree $- 1$, the degree of $\phi$ is $+ 1$ and $D ( \phi ) = d \gamma \phi + \phi d \gamma$, i.e. $\phi$ is a chain homotopy between the identity and $\nabla f$, while $f \nabla$ is the identity. A standard notation for an strong deformation retract is the following:

\begin{equation} \tag{a1} ( X \leftrightarrows _{f} ^{\nabla } Y , \phi ). \end{equation}

The notion of a strong deformation retract is essentially equivalent to what is called a contraction in [a5].

### Side conditions.

There are three additional conditions for a strong deformation retract which are needed to achieve both theoretical and computational results. They are called the side conditions: $\phi \nabla = 0$, $f \phi = 0$, and $\phi \phi = 0$. Fortunately, these may always be satisfied as follows: if the first two conditions do not hold, replace $\phi$ by $\overline { \phi } = D ( \phi ) \phi D ( \phi )$, then the new data given by $\nabla$, $f$, and $\overline{\phi}$ defines a strong deformation retract in which the first two conditions hold. If the third condition does not hold, and the first two do, replace $\overline{\phi}$ by $\Phi = \overline { \phi } d \overline { \phi }$ and the new data given by $\nabla$, $f$, and $\Phi$ defines a strong deformation retract in which all conditions hold [a17].

## Transference problem.

A transference problem consists of a strong deformation retract (a1) together with another differential $d_Y ^ { \prime }$ on $Y$. The difference $t = d _ { Y } ^ { \prime } - d _ { Y }$ is called the initiator. The problem is to determine changes $d ^ { \prime } _{X}$, $\nabla ^ { \prime }$, $f ^ { \prime }$, and $\phi ^ { \prime }$ such that

is a strong deformation retract. A useful variation of the transference problem, equivalent to it, is stated in terms of splitting homotopies. A splitting homotopy for a complex $( Z , d_\text{Z} )$ is a degree-$+ 1$ mapping $\varphi : Z \rightarrow Z$ such that $\varphi ^ { 2 } = 0$ and $\varphi d_Z \varphi = \varphi$. It is not difficult to see that complexes $( Y , d )$ with a splitting homotopy $\phi$ are in bijective correspondence (up to chain equivalence) with strong deformation retracts (a1). The correspondence is given by noting that if one has a strong deformation retract, $\phi$ is indeed a splitting homotopy. Conversely, given a splitting homotopy $\phi$, if $\pi = 1_ Y - D ( \phi )$, then one has $Y = \operatorname { ker } ( \pi ) \oplus \operatorname { im } ( \pi )$ and setting, $X = \operatorname { im } ( \pi )$, a strong deformation retract results by taking $d _X$ to be the restriction of $d_Y$ to $X$, $\nabla$ to be the inclusion mapping, and $f$ to be the projection. The transference problem in these terms is as follows.

Given a splitting homotopy $\phi : Y \rightarrow Y$, and a new differential $d ^ { \prime } = d + t$ on $Y$, find a new splitting homotopy $\phi ^ { \prime }$ on $Y$ (relative to $d ^ { \prime }$) such that, as $R$-modules, $\operatorname { im } ( \pi )$ is isomorphic to $\operatorname { im } ( \pi ^ { \prime } )$ (where $\pi = 1_ Y - D ( \phi )$ and $\pi ^ { \prime } = 1 _ { Y } - D ( \phi ^ { \prime } )$). See [a1] for full details.

### Solution to the transference problem.

The perturbation lemma gives conditions under which the transference problem can be solved. In terms of splitting homotopies, it can be stated quite simply, as follows.

Suppose that $( Y , d_Y )$ is a chain complex and $\phi : Y \rightarrow Y$ is a splitting homotopy. If $t \phi$ is nilpotent in each homogeneous degree $n \geq 0$, then

\begin{equation} \tag{a2} \phi ^ { \prime } = \phi \sum _ { i = 0 } ^ { \infty } ( - 1 ) ^ { i } ( t \phi ) ^ { i } \end{equation}

(which is well-defined since $t \phi$ is nilpotent in each degree) is a splitting homotopy which solves the transference problem. Furthermore, under mild assumptions, any solution to the transference is conjugate to this solution by a chain homotopy equivalence [a1].

Originally, this was stated in terms of strong deformation retracts [a2], [a6] (although the uniqueness result first appears in [a1]). These early works were influenced by [a19]. For that setting, let $\Sigma _ { \infty } = t - t \phi t + \ldots + ( - t \phi ) ^ { n } t +\dots$. It is easy to see that in terms of strong deformation retracts, if the hypotheses of the perturbation lemma hold, the mappings $f _ { \infty } = f - \Sigma _ { \infty } \phi$, $\nabla _ { \infty } = \nabla - \phi \Sigma _ { \infty } \nabla$, $\partial _ { \infty } = d _ { M } + f \Sigma _ { \infty } \nabla$, and $\phi _ { \infty } = \phi \Sigma _ { \infty } \phi$ solve the transference problem, using the fact that $\phi _ { \infty }$ is exactly $\phi ^ { \prime }$ from above along with the correspondence between strong deformation retracts and splitting homotopies.

The formula (a2) and the uniqueness result have far-reaching consequences in homological algebra and topology. Many seemingly unrelated results may be consolidated by these methods and it can also be used to find new results. The main technique is to set up a transference problem and prove convergence of (a2).

## Applications.

An application is given in [a2] to explain the Hirsch complex, and in [a6] to obtain twisted tensor product complexes in the sense of [a3] for (simplicial) fibrations. The application in [a6] was generalized to iterated fibrations in [a11] and these applications were further generalized to obtain much smaller complexes for iterated fibrations in [a17].

Applications to the derivation of "small" resolutions over group rings of nilpotent groups and certain solvable groups and monoids are given in [a14]. Applications to resolutions over certain filtered algebras are given in [a15], as well as the observation that the perturbation lemma gives rise to an exact formula for all the differentials in a wide class of spectral sequences (involving filtered algebras). Computer algebra has been used to obtain concrete calculations using these results [a16], [a14]. To give a quick and rough idea of how this can be done, think of a given filtered augmented algebra $\epsilon : A \rightarrow R$ such that, as $R$-modules, the associated graded object $E _ { 0 } ( A )$ is isomorphic to $A$ (e.g. $R$ is a field) and one has a resolution of the form $X = E _ { 0 } ( A ) \otimes \overline{X}$ of $R$ over $E _ { 0 } ( A )$ with the property that $X$ is a strong deformation retract of the bar-construction resolution $B ( E _ { 0 } ( A ) )$ [a20] (cf. also Standard construction). Since as $R$-modules, $E _ { 0 } ( A )$ is isomorphic to $A$, one can see two differentials on the underlying $R$-module structure of $B ( A )$: The one coming from $E _ { 0 } ( A )$ and the one coming from $A$. Taking the initiator $t$ to be the difference of the two differentials, one has a transference problem. When the hypothesis of the perturbation lemma is satisfied, this gives a resolution of $R$ over $A$ which is as small as the original one over $E _ { 0 } ( A )$. The requirements for all of this are not at all uncommonly found to hold.

Applications to the derivation of (co-) $A _ { \infty }$-structures were given in [a7], [a8], and in [a12]. These applications proceed by setting up a transference problem involving a strong deformation retract of $T ( H ( A ) )$ into $T ( A )$, where $A$ is a differential graded augmented algebra and $T ( . )$ is the tensor module functor. The point is that the underlying module structure for both ordinary Tor and differential Tor [a4] is given by $T ( A )$, the only difference being the differentials. Taking the difference of these differentials to be the initiator, and showing that (a2) converges in this case, one obtains a differential $\partial _ { \infty }$ on $T ( H ( A ) )$ and a strong deformation retract of this new complex into the differential Tor bar-construction $\overline { B } ( A )$. In this case it was shown in [a8], and independently in [a12], that $\partial _ { \infty }$ is actually a co-derivation (the proof of this fact is non-trivial). Thus, the perturbation lemma gives, in this case, an algorithm for deriving an $A _ { \infty }$-structure on $H ( A )$ which is equivalent to $\overline { B } ( A )$. This application has come to be known as the tensor trick. Applications to the homology of loop spaces can be obtained by these methods [a9], [a12].

## Generalized Gugenheim–Munkholm construction.

As hinted at above, homological perturbation theory also involves the consolidation of sometimes apparently unrelated techniques and results. For example, in [a10] it was shown that if one has a strong deformation retract (a1) where both $X$ and $Y$ are differential graded augmented algebras and the mapping $f$ is an algebra mapping, any twisting cochain [a10] $\tau : C \rightarrow X$ for a differential co-algebra $C$ can be lifted to $\hat { \tau } : C \rightarrow Y$ (with $f \hat { \tau } = \tau$). V.K.A.M. Gugenheim and H.J. Munkholm give an inductive formula:

\begin{equation*} \hat { \tau }_{0} = 0, \end{equation*}

\begin{equation*} \hat { \tau }_1 = \nabla \tau ,\, \hat { \tau } _ { n } = \sum _ { i + j = n } \phi ( \hat { \tau } _ { i } \bigcup \hat { \tau } _ { j } ), \end{equation*}

where $\cup$ is the convolution product for mappings $C \rightarrow A$ ($f \cup g = m ( f \otimes g ) \Delta$, where $m$ is the product in $A$ and $\Delta$ is the co-product in $C$). The mapping $\hat{\tau}$ is $\sum _ { n } \hat { \tau } _ { n }$ (conditions for convergence are given in [a10]).

Staying in the special setting of [a10], and furthermore assuming that $X = H ( Y )$ is the homology of the differential graded augmented algebra $Y$, puts one in the (general) formal setting (see [a18] for the characteristic zero case). In this case, the universal twisting cochain $\pi : \overline { B } ( H ( Y ) ) \rightarrow H ( Y )$ lifts to a twisting cochain $\hat { \pi } : \overline { B } ( H ( Y ) ) \rightarrow Y$. It was shown in [a7] that, in this case, not only does the $A _ { \infty }$-structure on $H ( Y )$ collapse to the bar-construction (as it must by formality), but the induced mapping $T ( \nabla ) _ { \infty } : \overline { B } ( H ( Y ) ) \rightarrow \overline { B } ( Y )$ followed by the universal twisting cochain $\pi _ { Y } : \overline { B } ( Y ) \rightarrow Y$, is exactly the mapping $\hat{\tau}$ above.

The construction given in [a10] can be applied in a purely combinatorial way for any strong deformation retract (a1) for any degree-$- 1$ mapping $C \rightarrow X$. Of course, one cannot even talk about twisting cochains in this context since $X$ might not be an algebra (much less $f$ be an algebra mapping). Nevertheless, if this construction is applied to the case when $X = H ( Y )$ and $Y$ is an algebra (no extra assumptions on $X$ or $f$) and $\pi : T ( H ( Y ) ) \rightarrow H ( Y )$ is the module mapping defined combinatorially in exactly the same way as the universal twisting cochain, then the result of [a7] generalizes to this case, i.e. the composite of the mapping $T ( \nabla ) _ { \infty } : ( T ( H ( Y ) ) , \partial _ { \infty } ) \rightarrow \overline { B } ( Y )$ followed by the universal twisting cochain is exactly the mapping $\hat{\tau}$. But in fact, more is known. By a small alteration of the construction, one can actually obtain the $A _ { \infty }$-structure as well:

Thus, the $A _ { \infty }$-structure of the tensor trick is completely determined by the generalized construction $\widehat{\pi}$. The proof of this, which is not immediate, as well as additional results are given in [a13].

All of the references in the papers cited below should be perused for a more complete picture of the applications, but one should keep in mind that this is presently (1998) an active field and new results are constantly evolving.

#### References

[a1] | D. Barnes, L. Lambe, "Fixed point approach to homological perturbation theory" Proc. Amer. Math. Soc. , 112 (1991) pp. 881–892 |

[a2] | R. Brown, "The twisted Eilenberg–Zilber theorem" , Celebrazioni Archimedee del Secolo XX, Simposio di Topologia (1967) pp. 34–37 |

[a3] | E.H. Brown, "Twisted tensor products" Ann. of Math. , 1 (1959) pp. 223–246 |

[a4] | H. Cartan, "Algèbres d'Eilenberg–MacLane et homotopie" Sém. Henri Cartan (1954/5) |

[a5] | S. Eilenberg, S. MacLane, "On the groups $H ( \pi , n )$ I" Ann. of Math. , 58 (1953) pp. 55–106 |

[a6] | V.K.A.M. Gugenheim, "On the chain complex of a fibration" Illinois J. Math. , 3 (1972) pp. 398–414 |

[a7] | V.K.A.M. Gugenheim, L. Lambe, "Perturbation theory in differential homological algebra I" Illinois J. Math. , 33 (1989) pp. 566–582 |

[a8] | V.K.A.M. Gugenheim, L. Lambe, J. Stasheff, "Perturbation theory in differential homological algebra II" Illinois J. Math. , 35 (1991) pp. 359–373 |

[a9] | V.K.A.M. Gugenheim, L. Lambe, J. Stasheff, "Algebraic aspects of Chen's iterated integrals" Illinois J. Math. , 34 (1990) pp. 485–502 |

[a10] | V.K.A.M. Gugenheim, H.J. Munkholm, "On the extended functoriality of Tor and Cotor" J. Pure Appl. Algebra , 4 (1974) pp. 9–29 |

[a11] | J. Hübschmann, "The homotopy type of $F \Psi ^ { q }$, the complex and symplectic cases" Contemp. Math. , 55 (1986) pp. 487–518 |

[a12] | J. Hübschmann, T. Kadeishvili, "Small models for chain algebras" Math. Z. , 207 : 2 (1991) pp. 245–280 |

[a13] | J. Johansson, L. Lambe, "Transferring algebra structures up to homology equivalence" Math. Scand. , 88 : 2 (2001) |

[a14] | L. Lambe, "Resolutions that split off of the bar construction" J. Pure Appl. Algebra , 84 (1993) pp. 311–329 |

[a15] | L. Lambe, "Homological perturbation theory, Hochschild homology and formal groups" Contemp. Math. , 134 (1992) |

[a16] | L. Lambe, "Resolutions via homological perturbation" J. Symbolic Comp. , 12 (1991) pp. 71–87 |

[a17] | L. Lambe, J. Stasheff, "Applications of perturbation theory to iterated fibrations" Manuscripta Math. , 58 (1987) pp. 363–376 |

[a18] | D. Sullivan, "Infinitesimal computations in topology" Publ. Math. IHES , 47 (1977) pp. 269–331 |

[a19] | W. Shih, "Homology des espaces fibrés" Publ. Math. IHES , 13 (1962) pp. 93–176 |

[a20] | S. MacLane, "Homology" , Grundl. Math. Wissenschaft. , 114 , Springer (1967) |

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Homological perturbation theory.

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