Awesome

AT

A Haskell rewrite of Kenzo, a collection of algorithms for 'effective algebraic topology'. The algorithms and implementations in Kenzo were created by Francis Sergeraert, Julio Rubio Garcia, Xavier Dousson, Ana Romero and many collaborators.

Writing it from scratch myself is the only chance I have of understanding it!

Examples

See the examples/ folder.

...
> let x = totalSpace s3 (Wbar kz1) fibration
> putStr "π₄ S³ is: "
> print (homology x !! 4)
π₄ S³ is: ℤ/2

> homology (Wbar (WbarDiscrete (Zmod 3)))
[ℤ,0,ℤ/3,0,ℤ/3,0,ℤ/(3^2),ℤ/3,ℤ/3,ℤ/3,ℤ/3 ⊕ ℤ/3,^C

Central Concepts of Kenzo

A simplicial set X is described by a type a, containing whatever data is required to specify X, and a type GeomSimplex a, whose elements correspond to non-degenerate simplices (in Kenzo called 'geometric simplices'). Like Kenzo we also allow a predicate on GeomSimplex a specifying when an element actually describes a geometric simplex and when it is 'spurious'.

An actual simplex of X is a geometric simplex together with a 'formal degeneracy operator', which is a list of degeneracy operators in a normal form. Face maps are implemented as functions from geometric simplices to (possibly degenerate) simplices, and the extension of these face maps to all simplices is forced by the simplicial identities.

A simplicial set is of finite type if there is a finite number of geometric simplices for each dimension, and there is a function giving a list of these simplices for any dimension n. It is not required that there are finitely many geometric simplices overall.

The normalised chain complex N(X) of X has each N(X)_n given by the free abelian group on the set of nondegenerate n-simplices of X, with the boundary of a simplex calculated similar to usual (the alternating sum of face maps), but ignoring any degenerate faces.

If C(X) is the ordinary chain complex of simplicial chains of X, the quotient map C(X) -> N(X) is a quasi-isomorphism, and so if X is of finite type, then the homology of X can be computed via N(X).

But many unavoidable simplicial sets (like K(ℤ,n) and loop spaces ΩX) are not of finite type, and so we need some other way to calculate their homology. This is where 'effective homology' comes in.

A reduction between chain complexes C and D is a strong deformation retract of chain complexes. The data of a reduction unwinds to a triple (f : C -> D, g : D -> C, h : C -> C) where f and g are degree 0, the homotopy operator h is degree 1, and certain equations involving these hold. A (strong chain) equivalence between two chain complexes C and D is a span of reductions l : E -> C and r : E -> D.

An effective homology structure on C is an equivalence between C and a chain complex F of finite type.

A simplicial set with effective homology is a simplicial set X equipped with an effective homology structure on N(X).

The point of Kenzo is that although constructions on simplicial sets sometimes do not preserve levelwise finiteness, they do extend to effective homology structures. And so if we begin with a finite simplicial complex and perform some constructions using it, then we can often compute the homology of the result even if the actual simplicial sets are now far too complicated to get a handle on.

Plan

Homological Algebra

• Definitions
  • Chain Complex
  • Bicomplex
    • Tot
  • Reduction
    • Perturbation
  • Strong Equivalence
    • Composition via span
• Constructions
  • Tensor (of chain complex)
    • Functoriality
  • Hom (of chain complex)
  • 'Bicone' (specialised pushout for surjections)
  • Bar
    • Commutative algebra structure
    • Functoriality
  • Cobar
    • Of 1-reduced
    • Of 0-reduced
    • Functoriality

Simplicial Sets

• Definitions
  • Simplicial Set
  • Simplicial Morphism
  • Simplicial Group
  • Kan Structure
  • SSet With Effective Homology
  • Principal Fibrations
  • Discrete Vector Fields
  • Coalgebra Structure on Chains
  • Algebra Structure on Chains of Groups
  • Kan Structure on Chains of Groups
• Finite Examples
  • Spheres
    • Treat S¹ separately (not 1-reduced)
  • Moore Spaces
  • Projective Spaces
  • Lens Spaces
• Eilenberg-MacLane Spaces
  • K(ℤ,1)
  • K(ℤ/2,1) (Can be made particularly efficient)
  • K(ℤ/p,1)
• Constructions
  • Products
    • Group structure
  • Total Space of Principal Fibration
  • Loop Space
    • Of 1-reduced
    • Of 0-reduced
    • Group structure
    • Canonical twisting function X -> GX
  • Classifying Space
    • For 0-reduced group
    • For non-reduced group
    • Special case for discrete groups
    • Group Structure
    • Canonical twisting function WG -> G
  • Suspension
    • 0-reduced
    • General Kan suspension
  • Pushouts (of 1-reduced sSets)
  • Other Finite Homotopy Colimits
  • 'Nerve' taking a ChainComplex back to a sAb?

Effective Homology

• Classifying Spaces
  • Direct Reduction of K(ℤ,1) to S¹
  • Of 0-reduced Abelian sGrps
  • Of General sGrps
• Products
  • Eilenberg-Zilber reduction
  • Use specialised contraction maps for efficiency
• Fibrations
  • Total Space from Base and Fibre ('Serre' problem)
    • 1-reduced Fibre
    • 0-reduced Fibre
  • Fibre from Base and Total Space ('Eilenberg-Moore' problem)
• Loop Space
  • 1-reduced
  • 0-reduced
• Colimits
  • Suspension
  • Pushouts (of 1-reduced sSets)
  • Finite Homotopy Colimits
• Discrete Vector Fields
  • Induced Reduction
  • Products
  • Fibrations
  • K(ℤ,1)
    • Naive but easy
    • Polynomial time but complicated
  • K(ℤ/n,1)?
  • Classifying Spaces for 0-reduced sAb
• Homotopy Groups
  • Whitehead Tower (for 1-reduced sSet)
  • Postnikov Tower?
• Cohomology Operations
  • Over fields using "minimal models" and

Misc TODOs

• Fix space leaks, jeez
• Pretty-printing for everything (unicode sub/superscripts in output?)
• Docs for everything
• Move this list to Github issues
• Consolidate some files? Eg. Sum, Shift into ChainComplex
• Use bit operations eg from bits-extra for degeneracy operators.
• Short-circuits: e.g. composing with zero/id for morphism/reduction/equivalence
• Make sure things are being aggressively inlined
• Make homology calculation do less work: should just need SNF of one matrix and the rank of another.
• Improve Smith normal form code, would be better to call out to some existing library instead of rolling our own. The options appear to be LinBox or FLINT. The former appears to support sparse matrices better
• Check homology of K(G,n) calculations against known results <!-- eg [Clement's thesis](#ref-clement%3Athesis) -->
• Add methods to produce representatives of homology classes
• Rewrite Bar to be a perturbed TensorCoalgebra?
• Rename basepoint to geomBasepoint say

Notes

• I have switched a little terminology: I believe Kenzo uses 'effective' for finite-type things and 'locally effective' for what I am calling effective things, but I find this a bit confusing.
• In Kenzo, every sSet is conflated with its chain complex of normalised chains, here I have kept the two separate.
• Avoid over-engineering the Haskell as much as possible.
• That being said, the use of Constrained.Category is a bit of a mess.
• There may be a way to unify some of the algorithms via bicomplexes and the 'generalised Eilenberg-Zilber theorem' relating the diagonal and total complexes. But the EZ-theorem only gives a strong deformation retract in special cases, in general it is just a chain homotopy equivalence.
• The classifying space functor Wbar factors through the 'total bisimplicial set' functor. But it would be difficult to describe the total functor on bisimplicial spaces algorithmically, because its definition involves the equaliser of certain face maps. So it only makes sense to implement bicomplexes and not bisimplicial sets.
• Auto-formatting the code: fourmolu -o -XTypeApplications -i $(find . -name '*.hs')
• Running Kenzo with SBCL:

  > rlwrap sbcl
  (require :asdf)
  (load "kenzo.asd")
  (asdf:load-system "kenzo")
  (in-package :kenzo)
  
  (finite-ss-table '(a b 1 c (b a)))
  

  etc.
• classes.lisp in Kenzo contains the meaning of some of the 4 letter abbreviations
  • ABSM = ABstract SiMplex
  • GMSM = GeoMetric SiMplex
  • CMBN = CoMBinatioN
  • CFFC = CoeFFiCient
  • GNRT = GeNeRaTor
  • CMPR = CoMPaRison
  • CMPRF = CoMPaRison Function
  • ICMBN = Internal-CoMBiNation
  • STRT = STRaTegy
  • bsgn = BaSe GeNerator
  • dffr = DiFFeRential
  • grmd = GRound MoDule
  • efhm = EFfective HoMology
  • idnm = IDentification NuMber
  • orgn = ORiGiN
  • vctr = VeCToR
  • intr-mrph = INTeRnal-MoRPHism (the class of actual functions implementing a morphism of simplicial sets or chain complexes)
  • sbtr = SuBTRact
  • crpr = CarRtesian PRoduct
  • BRGN = BaR GeNerator
  • TNPR = TeNsor PRoduct
• Unguessable CL functions
  • (ash x n) = bit shift x left by n
  • (add x y) and (sbtr x y) can sometimes actually be a use of the perturbation lemma(!!), depending on the types of the arguments.

References

Code:

• Kenzo homepage
• Kenzo documentation, likely out of date with the code in places
• 'Official' Kenzo mirror on GitHub, there are three different versions of the code here, I am not clear on what is gained/lost between them. There are online Jypter notebooks that let you play with Kenzo (I believe Kenzo-9) without having to figure out how to install and operate a Common Lisp environment
• Fork by Ana Romero + collaborators, has some added features over the mirror above, worth looking at resolutions.lisp and homotopy.lisp, but is missing all the discrete vector field code
• Modules written by Ana Romero, mostly to do with spectral sequences

Papers:

Everything even remotely relevant to effective algebraic topology that I can find (not all of which is relevant for implementation). Some of the documents have multiple versions; I have tried to link to the most recent in each case. Some material is repeated in different references.

<span class="csl-left-margin">[1] </span><span class="csl-right-inline">Medina-Mardones, A.M. 2022. An axiomatic characterization of Steenrod’s cup-i products.</span>

<span class="csl-left-margin">[2] </span><span class="csl-right-inline">Medina-Mardones, A.M. 2022. New formulas for cup-i products and fast computation of Steenrod squares.</span>

<span class="csl-left-margin">[3] </span><span class="csl-right-inline">Franz, M. 2021. Szczarba’s twisting cochain and the Eilenberg-Zilber maps. Collectanea Mathematica. 72, 3 (2021), 569–586.</span>

<span class="csl-left-margin">[4] </span><span class="csl-right-inline">Guidolin, A. and Romero, A. 2021. Computing higher Leray-Serre spectral sequences of towers of fibrations. Foundations of Computational Mathematics. 21, 4 (2021), 1023–1074.</span>

<span class="csl-left-margin">[5] </span><span class="csl-right-inline">Kaufmann, R.M. and Medina-Mardones, A.M. 2021. Cochain level May-Steenrod operations. Forum Math. 33, 6 (2021), 1507–1526.</span>

<span class="csl-left-margin">[6] </span><span class="csl-right-inline">Chen, R. 2020. E_∞-Rings and Modules in Kan Spectral Sheaves. University of Michigan.</span>

<span class="csl-left-margin">[7] </span><span class="csl-right-inline">Romero, A., Rubio, J., Sergeraert, F. and Szymik, M. 2020. A new Kenzo module for computing the Eilenberg-Moore spectral sequence. ACM Communications in Computer Algebra. 54, 2 (2020), 57–60.</span>

<span class="csl-left-margin">[8] </span><span class="csl-right-inline">Romero, A., Rubio, J. and Sergeraert, F. 2019. An implementation of effective homotopy of fibrations. Journal of Symbolic Computation. 94, (2019), 149–172.</span>

<span class="csl-left-margin">[9] </span><span class="csl-right-inline">Romero, A. and Sergeraert, F. 2019. The Eilenberg-Zilber theorem via discrete vector fields.</span>

<span class="csl-left-margin">[10] </span><span class="csl-right-inline">Zhechev, S. 2019. Algorithmic aspects of homotopy theory and embeddability. Institute of Science; Technology Austria.</span>

<span class="csl-left-margin">[11] </span><span class="csl-right-inline">Sergeraert, F. 2018. The homological hexagonal lemma. Georgian Mathematical Journal. 25, 4 (2018), 603–622.</span>

<span class="csl-left-margin">[12] </span><span class="csl-right-inline">Vejdemo-Johansson, M. 2018. Algorithms in A^∞-algebras. Georgian Mathematical Journal. 25, 4 (2018), 629–635.</span>

<span class="csl-left-margin">[13] </span><span class="csl-right-inline">Chen, R., Kriz, I. and Pultr, A. 2017. Kan’s combinatorial spectra and their sheaves revisited. Theory and Applications of Categories. 32, (2017), No. 39, 1363–1396.</span>

<span class="csl-left-margin">[14] </span><span class="csl-right-inline">Romero, A. and Sergeraert, F. 2017. A Bousfield-Kan algorithm for computing the effective homotopy of a space. Foundations of Computational Mathematics. 17, 5 (2017), 1335–1366.</span>

<span class="csl-left-margin">[15] </span><span class="csl-right-inline">Hess, K. 2016. The Hochschild complex of a twisting cochain. Journal of Algebra. 451, (2016), 302–356.</span>

<span class="csl-left-margin">[16] </span><span class="csl-right-inline">Filakovský, M. 2015. Algorithmic construction of the postnikov tower for diagrams of simplicial sets. Masaryk University.</span>

<span class="csl-left-margin">[17] </span><span class="csl-right-inline">Romero, A. and Sergeraert, F. 2015. A combinatorial tool for computing the effective homotopy of iterated loop spaces. Discrete & Computational Geometry. 53, 1 (2015), 1–15.</span>

<span class="csl-left-margin">[18] </span><span class="csl-right-inline">Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L. and Wagner, U. 2014. Computing all maps into a sphere. Journal of the ACM. 61, 3 (2014), Art. 17, 44.</span>

<span class="csl-left-margin">[19] </span><span class="csl-right-inline">Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L. and Wagner, U. 2014. Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM Journal on Computing. 43, 5 (2014), 1728–1780.</span>

<span class="csl-left-margin">[20] </span><span class="csl-right-inline">Cohen, C. and Mörtberg, A. 2014. A coq formalization of finitely presented modules. Interactive theorem proving (2014), 193–208.</span>

<span class="csl-left-margin">[21] </span><span class="csl-right-inline">Filakovský, M. 2014. Effective homology for homotopy colimit and cofibrant replacement. Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum. 50, 5 (2014), 273–286.</span>

<span class="csl-left-margin">[22] </span><span class="csl-right-inline">Sergeraert, F. 2013. Discrete vector fields and fundamental algebraic topology.</span>

<span class="csl-left-margin">[23] </span><span class="csl-right-inline">Krčál, M., Matoušek, J. and Sergeraert, F. 2013. Polynomial-time homology for simplicial Eilenberg-MacLane spaces. Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics. 13, 6 (2013), 935–963.</span>

<span class="csl-left-margin">[24] </span><span class="csl-right-inline">Filakovský, M. 2012. Effective chain complexes for twisted products. Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum. 48, 5 (2012), 313–322.</span>

<span class="csl-left-margin">[25] </span><span class="csl-right-inline">Romero, A. and Rubio, J. 2012. Computing the homology of groups: The geometric way. Journal of Symbolic Computation. 47, 7 (2012), 752–770.</span>

<span class="csl-left-margin">[26] </span><span class="csl-right-inline">Romero, A. and Sergeraert, F. 2012. Discrete vector fields and fundamental algebraic topology.</span>

<span class="csl-left-margin">[27] </span><span class="csl-right-inline">Romero, A. and Sergeraert, F. 2012. Effective homotopy of fibrations. Applicable Algebra in Engineering, Communication and Computing. 23, 1-2 (2012), 85–100.</span>

<span class="csl-left-margin">[28] </span><span class="csl-right-inline">Rubio, J. and Sergeraert, F. 2012. Constructive homological algebra and applications.</span>

<span class="csl-left-margin">[29] </span><span class="csl-right-inline">Stevenson, D. 2012. Décalage and Kan’s simplicial loop group functor. Theory and Applications of Categories. 26, (2012), No. 28, 768–787.</span>

<span class="csl-left-margin">[30] </span><span class="csl-right-inline">Spiwack, A. 2011. <span class="nocase">Verified Computing in Homological Algebra</span>. Ecole Polytechnique X.</span>

<span class="csl-left-margin">[31] </span><span class="csl-right-inline">Heras, J. 2010. Pushout construction for the Kenzo systems.</span>

<span class="csl-left-margin">[32] </span><span class="csl-right-inline">Álvarez, V., Armario, J.A., Frau, M.D. and Real, P. 2010. Cartan’s constructions and the twisted Eilenberg-Zilber theorem. Georgian Mathematical Journal. 17, 1 (2010), 13–23.</span>

<span class="csl-left-margin">[33] </span><span class="csl-right-inline">Berciano Alcaraz, A., Rubio, J. and Sergeraert, F. 2010. A case study of A_∞-structure. Georgian Mathematical Journal. 17, 1 (2010), 57–77.</span>

<span class="csl-left-margin">[34] </span><span class="csl-right-inline">Heras, J. 2010. Effective homology of the pushout of simplicial sets. Proceedings of the XII encuentros de álgebra computacional y aplicaciones (2010), 152–156.</span>

<span class="csl-left-margin">[35] </span><span class="csl-right-inline">Heras, J., Pascual, V., Romero, A. and Rubio, J. 2010. Integrating multiple sources to answer questions in algebraic topology. Proceedings of the 10th ASIC and 9th MKM international conference, and 17th calculemus conference on intelligent computer mathematics (Paris, France, 2010), 331–335.</span>

<span class="csl-left-margin">[36] </span><span class="csl-right-inline">Hess, K. and Tonks, A. 2010. The loop group and the cobar construction. Proceedings of the American Mathematical Society. 138, 5 (2010), 1861–1876.</span>

<span class="csl-left-margin">[37] </span><span class="csl-right-inline">Romero, A. 2010. Computing the first stages of the Bousfield-Kan spectral sequence. Applicable Algebra in Engineering, Communication and Computing. 21, 3 (2010), 227–248.</span>

<span class="csl-left-margin">[38] </span><span class="csl-right-inline">Álvarez, V., Armario, J.A., Frau, M.D. and Real, P. 2009. Algebra structures on the comparison of the reduced bar construction and the reduced W-construction. Communications in Algebra. 37, 10 (2009), 3643–3665.</span>

<span class="csl-left-margin">[39] </span><span class="csl-right-inline">Romero, A., Ellis, G. and Rubio, J. 2009. Interoperating between computer algebra systems: Computing homology of groups with Kenzo and GAP. ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation (2009), 303–310.</span>

<span class="csl-left-margin">[40] </span><span class="csl-right-inline">Sergeraert, F. 2009. Triangulations of complex projective spaces.</span>

<span class="csl-left-margin">[41] </span><span class="csl-right-inline">Barakat, M. and Robertz, D. 2008. homalg: A meta-package for homological algebra. J. Algebra Appl. 7, 3 (2008), 299–317.</span>

<span class="csl-left-margin">[42] </span><span class="csl-right-inline">Thomas, S. 2008. The functors W̄ and Diag ∘ Nerve are simplicially homotopy equivalent. Journal of Homotopy and Related Structures. 3, 1 (2008), 359–378.</span>

<span class="csl-left-margin">[43] </span><span class="csl-right-inline">Berciano, A., Jiménez, M.J. and Real, P. 2006. Reducing computational costs in the basic perturbation lemma. Computer algebra in scientific computing. V.G. Ganzha, E.W. Mayr, and E.V. Vorozhtsov, eds. Springer, Berlin. 33–48.</span>

<span class="csl-left-margin">[44] </span><span class="csl-right-inline">Romero, A., Rubio, J. and Sergeraert, F. 2006. Computing spectral sequences. Journal of Symbolic Computation. 41, 10 (2006), 1059–1079.</span>

<span class="csl-left-margin">[45] </span><span class="csl-right-inline">González-Díaz, R. and Real, P. 2003. Computation of cohomology operations of finite simplicial complexes. Homology Homotopy Appl. 83–93.</span>

<span class="csl-left-margin">[46] </span><span class="csl-right-inline">Clément, A. 2002. Integral cohomology of finite Postnikov towers. Université de Lausanne.</span>

<span class="csl-left-margin">[47] </span><span class="csl-right-inline">Díaz, R.G. 2000. Cohomology operations: A combinatorial approach. University of Seville.</span>

<span class="csl-left-margin">[48] </span><span class="csl-right-inline">Real, P. 2000. Homological perturbation theory and associativity. Homology, Homotopy and Applications. 2, (2000), 51–88.</span>

<span class="csl-left-margin">[49] </span><span class="csl-right-inline">Dousson, X. 1999. Homologie effective des classifiants et calculs de groupes d’homotopie. l’Université Joseph Fourier.</span>

<span class="csl-left-margin">[50] </span><span class="csl-right-inline">Goerss, P.G. and Jardine, J.F. 1999. Simplicial homotopy theory. Birkhäuser Verlag, Basel.</span>

<span class="csl-left-margin">[51] </span><span class="csl-right-inline">González-Díaz, R. and Real, P. 1999. A combinatorial method for computing Steenrod squares. J. Pure Appl. Algebra. 89–108.</span>

<span class="csl-left-margin">[52] </span><span class="csl-right-inline">Hurado, P.R., Álvarez, V., Armario, J.A. and González-Díaz, R. 1999. Algorithms in algebraic topology and homological algebra: The problem of the complexity. Zapiski Nauchnykh Seminarov POMI. 258, (1999), 161–184, 358.</span>

<span class="csl-left-margin">[53] </span><span class="csl-right-inline">Rubio Garcia, J., Sergeraert, F. and Siret, Y. 1999. Kenzo: A symbolic software for effective homology computation. Institut Fourier.</span>

<span class="csl-left-margin">[54] </span><span class="csl-right-inline">Forman, R. 1998. Morse theory for cell complexes. Advances in Mathematics. 134, 1 (1998), 90–145.</span>

<span class="csl-left-margin">[55] </span><span class="csl-right-inline">Kadeishvili, T. and Saneblidze, S. 1998. Iterating the bar construction. Georgian Mathematical Journal. 5, 5 (1998), 441–452.</span>

<span class="csl-left-margin">[56] </span><span class="csl-right-inline">Real, P. 1996. An algorithm computing homotopy groups. Math. Comput. Simulation. 42, 4-6 (1996), 461–465.</span>

<span class="csl-left-margin">[57] </span><span class="csl-right-inline">Real, P. 1996. On the computability of the Steenrod squares. Ann. Univ. Ferrara Sez. VII (N.S.). 42, (1996), 57–63 (1998).</span>

<span class="csl-left-margin">[58] </span><span class="csl-right-inline">Morace, F. 1994. Cochaînes de brown et transformation d’Eilenberg-Mac Lane: Réécriture en dimension deux et homologie. Paris 7.</span>

<span class="csl-left-margin">[59] </span><span class="csl-right-inline">Morace, F. and Prouté, A. 1994. Brown’s natural twisting cochain and the Eilenberg-Mac Lane transformation. J. Pure Appl. Algebra. 97, 1 (1994), 81–89.</span>

<span class="csl-left-margin">[60] </span><span class="csl-right-inline">Real Jurado, P. 1993. Algoritmos de cálculo de homología efectiva de los espacios clasificantes. Universidad de Sevilla, Departamento de Geometría y Topología.</span>

<span class="csl-left-margin">[61] </span><span class="csl-right-inline">Rubio, J. and Sergeraert, F. 1993. Locally effective objects and algebraic topology. Computational algebraic geometry (Nice, 1992). F. Eyssette and A. Galligo, eds. Birkhäuser Boston, Boston, MA. 235–251.</span>

<span class="csl-left-margin">[62] </span><span class="csl-right-inline">Gugenheim, V.K.A.M., Lambe, L.A. and Stasheff, J.D. 1991. Perturbation theory in differential homological algebra. II. Illinois J. Math. 35, 3 (1991), 357–373.</span>

<span class="csl-left-margin">[63] </span><span class="csl-right-inline">Huebschmann, J. and Kadeishvili, T. 1991. Small models for chain algebras. Math. Z. 207, 2 (1991), 245–280.</span>

<span class="csl-left-margin">[64] </span><span class="csl-right-inline">Gugenheim, V.K.A.M. and Lambe, L.A. 1989. Perturbation theory in differential homological algebra. I. Illinois J. Math. 33, 4 (1989), 566–582.</span>

<span class="csl-left-margin">[65] </span><span class="csl-right-inline">Lambe, L. and Stasheff, J. 1987. Applications of perturbation theory to iterated fibrations. Manuscripta Mathematica. 58, 3 (1987), 363–376.</span>

<span class="csl-left-margin">[66] </span><span class="csl-right-inline">Sergeraert, F. 1987. Homologie effective. I. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics. 304, 11 (1987), 279–282.</span>

<span class="csl-left-margin">[67] </span><span class="csl-right-inline">Sergeraert, F. 1987. Homologie effective. II. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics. 304, 12 (1987), 319–321.</span>

<span class="csl-left-margin">[68] </span><span class="csl-right-inline">Eilenberg, S. and Mac Lane, S. 1954. On the groups H(Π,n). II. Methods of computation. Ann. of Math. (2). 60, (1954), 49–139.</span>

<span class="csl-left-margin">[69] </span><span class="csl-right-inline">Eilenberg, S. and Mac Lane, S. 1953. On the groups H(Π,n). I. Ann. of Math. (2). 58, (1953), 55–106.</span>