A survey on homological perturbationtheory

J. Huebschmann

USTL, UFR de MathematiquesCNRS-UMR 8524

59655 Villeneuve d’Ascq Cedex, FranceJohannes.Huebschmann@math.univ-lille1.fr

Gottingen, September 14, 2010

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Abstract

Higher homotopies are nowadays playing a prominent role inmathematics as well as in certain branches of theoretical physics.Homological perturbation theory (HPT), in a simple form firstisolated by Eilenberg and Mac Lane in the early 1950’s, is nowa-days a standard tool to handle algebraic incarnations of higherhomotopies. A basic observation is that higher homotopy struc-tures behave much better relative to homotopy than strict struc-tures, and HPT enables one to exploit this observation in variousconcrete situations. In particular, this leads to the effective cal-culation of various invariants which are otherwise intractable.

Higher homotopies and HPT-constructions abound but theyare rarely recognized explicitly and their significance is hardlyunderstood; at times, their appearance might at first glance evencome as a surprise, for example in the Kodaira-Spencer approachto deformations of complex manifolds or in the theory of folia-tions.

The talk will illustrate, with a special emphasis on the compat-ibility of perturbations with algebraic structure, how HPT maybe successfully applied to various mathematical problems aris-ing in group cohomology, algebraic K-theory, and deformationtheory.

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Contents

1 Origins of homotopy and higher homo-topies 4

2 Some 20’th century higher homotopies 5

3 Homological perturbations — HPT 7

4 Perturbation theory for chain equiva-lences 11

5 Iterative perturbations 12

6 Compatibility of HPT with algebraic struc-ture 13

7 Rooted planar trees 22

8 Master equation and HPT 23

9 Higher homotopies, homological pertur-bations, and the working mathemati-cian 25

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1 Origins of homotopy and higher homotopies

Gauß 1833: linking numberGauß 1828: idea of a classifying spaceHilbert: exploration of syzygiesgenerating function for the number ofinvariants of each degree a rational functionhomogeneous ideal I of a polynomial ring S,“number of independent linear conditions” fora form of degree d in S to lie in I” a polynomialfunction of dproblem of counting the number of conditionsalready considered in projective geometry andin invariant theoryCayley 1848: general statement of theproblem, clear understanding of the role ofsyzygies–without the word, introduced bySylvester 1853Cayley somewhat develops what is nowa-days referred to as the Koszul resolutionPoincare 1895: terminology homotopyloop composition

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2 Some 20’th century higher homotopies

Failure of Alexander-Whitney multiplicationof cocycles to be commutativeSteenrod ∪i-products, prompteds(trongly)h(omotopy)c(ommutative) structuresSteenrod operationsA∞-structure system of higher homotopiestogether with suitable coherence conditionsMassey products invariants of certainA∞-structureselementary example Borromean ringsa Massey product of three variables detectsthe simultaneous linking of all three circlesSugawara recognition principle forcharacterizing loop spaces up to homotopy type,associativity problem of loop multiplications

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Stasheff — complete understanding ofhomotopy invariance of associativity— clean recognition principle for loop spaces— classifying space— nested sequence of homotopy associativityconditions— An-spacesany space an A1-space, H-space an A2-spaceevery homotopy associative H-space is A3A∞-space homotopy type of a loop spacealgebraic analogue of An-space incategory of algebrasAn-algebra, the case n =∞ being includedoriginal and motivating example singular chainson the based loop space of a spacevariant L∞-algebraskey observation: A∞-structures behavecorrectly with respect to homotopyat Michigan, Yuri Rainich was one ofStasheff’s mentors as an undergraduatehere Stasheff learned that, in projectivegeometry, associativity was an option

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3 Homological perturbations — HPT

Fundamental means to handlehigher homotopiesEilenberg and MacLane [1953]contraction

(Mπ−→←−∇ N, h)

— chain complexes M and N— chain maps ∇ : M → N and π : N →M— degree 1 morphism h : N → N

requirements

— deformation retraction

π∇ = Id, ∇π − Id = dh + hd

— annihilation properties

h∇ = 0, πh = 0, hh = 0

Eilenberg and Mac Lane [1953]comparison between reduced bar andW-constructions a reductionconjectured that it is a contraction

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Some notions— contraction filtered : N andM filtered chaincomplexes, π, ∇ and h filtration preserving— perturbation of the differential d of a chaincomplex X : (homogeneous) morphism

∂ : X −→ X

of the same degree as d such that— ∂ lowers the filtration— (d + ∂)2 = 0 or, equivalently,

[d, ∂] + ∂∂ = 0

— sum d + ∂, referred to as the perturbeddifferential , again differential— coalgebra perturbation: X a gradedcoalgebra structure such that (X, d) adifferential graded coalgebraperturbed differential d + ∂ compatible withgraded coalgebra structure— similarly algebra perturbationHeller [1954] recursive structure oftriangular complex later identified as aperturbation

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Perturbation lemma.

(Mg−→←−∇ N, h)

filtered contraction∂ perturbation of the differential on N

D =∑n≥0

g∂(h∂)n∇ =∑n≥0

g(∂h)n∂∇

∇∂ =∑n≥0

(h∂)n∇

g∂ =∑n≥0

g(∂h)n

h∂ =∑n≥0

(h∂)nh =∑n≥0

h(∂h)n.

Suppose filtrations on M and N complete.Then: Infinite series converge,D a perturbation of the differential on Mnotation N∂ and MD new chain complexes,

(MDg∂−→←−−∇∂ N∂, h∂)

constitute new filtered contractionnatural in terms of the given data.

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Some historyShi [1962] lurking behind formulasM. Barrat made explicit (unpublished)Brown [1964] in printGugenheim [1972] twisted Eilenberg-Zilber theorem via perturbation lemmaChen [1977] Ω∂[H∗(X)] model for real chainson loop space of a smooth manifoldX by meansof formal power series connection involving deRham complex of XGugenheim [1982] perturbation theory forhomology of loop space, ordinary singularsetting, reworks and extends Chen’s con’nWong [1986] Heidelberg diploma thesis(Diplomarbeit) supervised by H.using perturbation lemma, confirmedEML-conjecture: comparison between reducedbar and W-constructions a contractionMarkl [2006?] operadic perturbation lemmaperturbation lemma under weaker hypothesesBerger and H. [1996]geometric comparison between bar and W-con’s

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4 Perturbation theory for chain equivalences

H. and Kadeishvili [1991]Theorem. Let

(µ,Xα−→←−β

Y, ν)

be a filtered chain equivalence, let ∂ be aperturbation of the differential d on Y , andsuppose that the filtrations on X and Y arecomplete. Then HPT yields formulas for

• a perturbation D : X −→ X of the

differential on X, and for

• new morphisms

α∂ : X −→ Y, β∂ : Y −→ X,

µ∂ : X −→ X, ν∂ : Y −→ Y,

which yield a new filtered chain equivalence

(µ∂, XDα∂−−→←−−β∂

Y∂, ν∂)

which is natural in terms of the given data.Remark about proof: Reduce theconstructions to ordinary perturbation lemmaby means of mapping cyclinder

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5 Iterative perturbations

H. [1986]X a connected simplicial set, simpleπ1, π2, etc. its homotopy groups

kj : Xj−2 −→ K(πj−1, j), j ≥ 3

maps representing the k-invariants of X

Theorem. These maps determine adifferential d on the tensor product

K(π1, 1)⊗ K(π2, 2)⊗ . . .and a chain equivalence between X and theresulting chain complex

(K(π1, 1)⊗ K(π2, 2)⊗ . . . , d)

Main ingredient: Iteration of perturbationlemma, applied to corresponding iteratedfibrationsApplication: Interpretation of the k-invariantsof the algebraic K-theory of a finite fieldConstruction taken up inLambe-Stasheff [1987]models for iterated fibrations

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6 Compatibility of HPT with algebraic structure

H. [1989], [1991]compatibility of HPT- constructionswith algebraic structuresuitable algebraic HPT-constructions to exploitA∞-modules arising in group cohomologyconstruction of suitable free resolutionsexplicit numerical calculations in group cohountil today still not doable by other methodsspectral seq’s show up not collapsing from E2illustrate a typical phenomenon:Whenever a spectral sequence arisesa strong homotopy structure lurking behindspectral sequence invariant thereofhigher homotopy structure finer than spectralsequence itself

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Visit the workshop

Example 1. Group extension

e : 1 −→ N −→ G −→ K −→ 1

free resolutions

M(K) = M](K)⊗d RKM(N) = M](N)⊗d RN

of the ground ring R in the categories of rightRK– and RN–modulesaugmentation maps

ε : M](K) −→ R, ε : M](N) −→ R

M](K) and M](N) connected as gradedmodules, i.e. in degree zero a single copy ofthe ground ring R

explicit HPT-construction of a freeresolution of R in the category of rightRG–modules from the given data:

M(N)⊗RN RG = M](N)⊗d RGa free resolution of RK in the category of rightRG–modules

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write

d0 = M](K)⊗ d (= IdM](K) ⊗ d),

where d the differential on M](N)⊗d RGan RG–linear differential on

M](K)⊗M](N)⊗RGvertical in an obvious senseprojection map

g : M](K)⊗M](N)⊗RG −→ M](K)⊗RKinduces an isomorphism on homology relativeto differential d0 on source, zero differential ontargetobject M](K)⊗RK filtered by degreesdegree filtration of M](K) induces a filtration

Fi = Fi(M](K)⊗M](N)⊗RG)i≥0

for M](K)⊗M](N)⊗RG in the obvious way

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Theorem. There is a differential d on

M](K)⊗M](N)⊗RGhaving the following properties:

1. It can be written as

d = d0 + d1 + d2 + · · ·where, for i ≥ 1, the operator di lowersfiltration by i.

2. The projection map is a chain map

(M](K)⊗M](N))⊗dRG→ M](K)⊗dRKthat is compatible with the differentials,the filtrations, and the right RG–modulestructures.

Moreover, the resulting chain complex

(M](K)⊗M](N))⊗d RGis acyclic and hence a free resolution of theground ring R in the category of right RG–modules. Finally, the higher terms di, i ≥ 1,can be constructed explicitly via HPT.

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Under these circumstances, the series

d1 + d2 + · · ·is a perturbation of d0.Illustration. Metacyclic groupssuch a group G has a presentation of the form

〈x, y; yr = 1, xs = yf , xyx−1 = yt〉where x and y are generators of K = Z/s andN = Z/r respectively, and where

s > 1, r > 1, ts ≡ 1 mod r, tf ≡ f mod r,

in particular, ts−1r and

(t−1)fr integers

Sample results. p a prime whichdivides r and s, and suppose

t 6≡ 1 mod p.

Further, let j0 be the order of t modulo p, i. e.j0 the smallest number j so that

tj ≡ 1 mod p.

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Theorem 1. Suppose that p divides thenumber ts−1

r . Then the mod p cohomologyspectral sequence of the group extensioncollapses from E2, and the multiplicativeextension problem from

E2 = E∞

to H∗(G,Z/p) is trivial. Moreover,

H∗(G,Z/p)

has classes

ω2j0−1, c2j0, |ω2j0−1| = 2j0 − 1, |c2j0| = 2j0,

of filtration zero which restrict to the

classes ωycj0−1y and c

j0y in

H∗(N,Z/p) ∼= Λ[ωy]⊗ P[cy]

respectively, so that, as a gradedcommutative algebra, H∗(G,Z/p) can bewritten as

Λ[ωx]⊗ P[cx]⊗ Λ[ω2j0−1]⊗ P[c2j0].

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Theorem 2. Suppose that p does notdivide the number ts−1

r . Then thecohomology spectral sequence of the groupextension collapses from E3, and themultiplicative extension problem is trivial.Moreover, H∗(G,Z/p) has classes

ω2j0−1, ω4j0−1, . . . , ω2pj0−1, c2pj0,

of filtration zero which restrict to the

classes ωycij0−1y and c

pj0y in

H∗(N,Z/p) ∼= Λ[ωy]⊗ P[cy]

respectively so that, as a gradedcommutative algebra,

H∗(G,Z/p)

is generated by

ωx, cx, ω2j0−1, ω4j0−1, . . . , ω2pj0−1, c2pj0,

subject to the relations

ωaωb = 0,

a, b ∈ 2j0 − 1, 4j0 − 1, . . . , 2pj0 − 1,cxωa = 0,

a ∈2j0 − 1, 4j0 − 1, . . . , 2(p− 1)j0 − 1.19

Example 2.

H. and Kadeishvili [1991]H. [2004]Small models for chain algebrasalgebra and coalgebra perturbation lemmataSample resultTheorem. Let

(µ,Xα−→←−β

Y, ν)

be a chain equivalence, let

(Tµ,T[X ]Tα−−→←−−Tβ

T[Y ],Tν)

be the corresponding filtered chainequivalence of augmented differential gradedalgebras, and let ∂ be a multiplicativeperturbation of the differential on T[Y ] withrespect to the augmentation filtration.Suppose further that X and Y are connected.Then HPT yields formulas for

• a multiplicative perturbation

D : T[X ] −→ T[X ]20

of the differential on T[X ], and for

• new morphisms

T∂α : T[X ] −→ T[Y ]

T∂β : T[Y ] −→ T[X ]

T∂µ : T[X ] −→ T[X ]

T∂ν : T[Y ] −→ T[Y ],

which combine to a new filtered chainequivalence

(T∂µ,TD[X ]T∂α−−−→←−−−T∂β

T∂[Y ],T∂ν)

of augmented differential graded algebraswhere TD[X ] and T∂[Y ] refer to the newchain complexes.

In particular, the new filtered chainequivalence is natural in terms of the givendata.

Application: models for differential gradedalgebras, in particular for chain algebras

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7 Rooted planar trees

Kontsevich-Soibelman [2000]Description of A∞-algebra structures in termsof sums over oriented rooted planar treesendowed with suitable labelsthese sums over oriented rooted planar treesbehind the HPT-constructions that establish(co)algebra perturbation lemma inH. and Kadeishvili [1991]At the time no need to spell out the orientedrooted planar trees explicitlyDetails: H. [2008] (Kadeishvili Festschrift)One application:Minimality theorem forA∞-structures via theperturbation lemmaOriginal minimality theorem:Chen [1977] over the de Rham algebra of smoothmanifoldKadeishvili [1980] for the rational cochainalgebra of a space

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8 Master equation and HPT

Given a coaugmented differential gradedcocommutative coalgebra C and adifferential graded Lie algebra h, a Lie algebratwisting cochain t : C → h is a homogeneousmorphism of degree −1 whose composite withthe coaugmentation map is zero and whichsatisfies

Dt =1

2[t, t].

H. and Stasheff [2002]Data: differential graded Lie algebra g togetherwith contraction

(H(g) ∇−→←−π

g, h)

of chain complexes, the correspondingcoalgebra perturbation of the differential onSc[sg] being written as ∂

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Theorem. The data determine, via HPT,(i) a differential D on Sc[sH(g)] turning thelatter into a coaugmented differential gradedcoalgebra (i. e. D is a coalgebraperturbation of the zero differential) and henceendowing H(g) with an sh-Lie algebrastructure and(ii) a Lie algebra twisting cochain

τ : ScD[sH(g)]→ g

whose adjoint τ , written as

(Sc∇)∂ : ScD[sH(g)]→ C[g],

induces an isomorphism on homology.Furthermore, (Sc∇)∂ admits, via HPT, anextension to a new contraction

(ScD[sH(g)]

(Sc∇)∂−−−−−→←−−−−(Scπ)∂

Sc∂[sg], (Sch)∂)

of filtered chain complexes (notnececessarily of coalgebras).Application: formal solutions of the masterequation

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9 Higher homotopies, homological perturbations, and

the working mathematician

Higher homotopies and homologicalperturbations may be used to solve problemsphrased in language entirely different from thatof higher homotopies and HPT. Higherhomotopies and HPT-constructions occurimplicitly in a number of other situations inordinary mathematics where they are at firstnot even visible.

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List (not exhaustive)

•Kodaira-Nirenberg-Spencer:

Deformations of complex structures

• Frolicher spectral sequence of a

complex manifold

• Toledo-Tong: Parametrix

• Fedosov: Deformation quantization

•Whitney, Gugenheim: Extension of

geometric integration to contraction

•Whitney ground work for Sullivan’stheory of rational differential forms

•Gugenheim integration map in de Rhamtheory sh-multiplicative

•H.: Foliations; requisite higher homotopissdescribed in terms of generalized

Maurer-Cartan algebra

•H.: Equiv. coho and Koszul duality

• Operads

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