An algorithm computing the first two levels of the
Bousfield-Kan spectral sequence
Ana Romero∗
Abstract
In this paper, an algorithm computing the terms E 1 and E 2 of the Bousfield-Kan
spectral sequence of a 1-reduced simplicial set X is presented. The main ingredient is
the computation of the effective homology of RX, RX being the free simplicial Abelian
group generated by X.
Introduction
The Bousfield-Kan spectral sequence [2] was designed to present the Adams spectral sequence [1] in a different way, in the framework of combinatorial topology, to make easier
the study of its algebraic properties. The Adams spectral sequence and its satellite spectral
sequences are the main tools to compute homotopy groups, one of the most challenging
problems in Algebraic Topology. In particular, these spectral sequences have made it possible to determine many stable and unstable sphere homotopy groups, but no constructive
version of this spectral sequence is yet available; in other words no routine translation
work allows a programmer to implement this spectral sequence on a theoretical or concrete
machine to produce an algorithm computing homotopy groups.
The effective homology method [7] provides on the contrary real algorithms for the
computation of homology and homotopy groups of complicated spaces, and it has been concretely implemented in the Kenzo system [3], a Common Lisp program devoted to Symbolic
Computation in Algebraic Topology. In some cases, this technique replaces some common
spectral sequences, as those of Serre or Eilenberg-Moore: when the usual inputs of these
spectral sequences are organized as objects with effective homology, general algorithms are
produced computing for example the homology groups of the total space of a fibration, of an
arbitrarily iterated loop space, of a classifying space, etc. Furthermore, as we have shown
in a previous paper [6], the effective homology technique can also be used to obtain as a byproduct the different components of spectral sequences associated with filtered complexes
(which include those of Serre and Eilenberg-Moore), obtaining in this way real algorithms.
In this work we try to use the effective homology method to construct algorithms computing in this case the Bousfield-Kan spectral sequence associated with a (1-reduced) simplicial set X, which is not defined by means of filtered complexes. As a first step, we have
∗
Partially supported by Ministerio de Educación y Ciencia, project MTM2006-06513
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developed an effective homology version of the constructor R associating to every simplicial
set X the free simplicial Abelian group RX. This result can then be used to obtain the
first two levels of the Bousfield-Kan spectral sequence, the “pages” E 1 and E 2 . The general
algorithm computing the higher levels is not finished yet.
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Effective homology of free simplicial Abelian groups
One of the main ingredients in the definition of the Bousfield-Kan spectral sequence of a simplicial set X is the constructor R producing the free simplicial Abelian group RX, which satisfies πn (RX) ∼
= Hn (X) for every n > 0. Given X a pointed simplicial set, RX is defined as
the free simplicial Abelian group generated by the simplices of X, where the base point and
its degeneracies are put equal to zero. Let us denote Rn X = R(R(· · · RX)) (n times); then
the cosimplicial simplicial Abelian group RX is defined by (RX)n = Rn+1 X, n ≥ −1 (the
definition of the cofaces ∂ i and codegeneracies η i can be found in [2]). The Bousfield-Kan
spectral sequence of X is then the spectral sequence of the cosimplicial simplicial group RX,
1 = π (Rp+1 X) ∩ Ker η 0 ∩ · · · ∩ Ker η p−1 ∼ H (Rp X) ∩ Ker η 0 ∩ · · · ∩
and its first term is Ep,q
= q
q
p−1
Ker η , q > p ≥ 0. Hence, for the computation of this spectral sequence it is necessary to
determine πq (Rp+1 X) ∼
= Hq (Rp X). Furthermore, an ordinary description of these groups
is not sufficient: the generators are necessary in order to determine the kernels Ker η j and
1 .
then the differential maps defined on the groups Ep,q
To compute πq (Rp+1 X) ∼
= Hq (Rp X) with the corresponding generators, we try to use
the effective homology method, which allows one to determine the homology groups of
an initial (complicated) space Y by building a homotopy equivalence between the associated chain complex C∗ (Y ) and an effective (finitely generated) chain complex E∗ , denoted
C∗ (Y ) ⇐⇒ E∗ . The homology groups H∗ (E∗ ) are easily computable and the equivalence provides an isomorphism H∗ (Y ) ∼
= H∗ (E∗ ) which makes it possible to determine the looked-for
homology groups of Y . See [7] for concrete definitions and details about this technique.
In our case, given X a simplicial set with effective homology, we must build an effective
homology version of RX (that is, an equivalence between C∗ (RX) and an effective chain
complex E∗ ), and then an iterative application would compute a version with effective
homology of Rp X for every p. A particular situation is obtained when the simplicial set X
is contractible; in this case a specific algorithm can be developed allowing one to compute
the effective homology of RX, as it was explained in [4]. For the general case, the effective
homology of RX is more complicated, and it involves different constructions of Algebraic
Topology such as the Dold-Kan correspondence or Eilenberg-MacLane spaces. A complete
description of our construction of the effective homology of RX can be found in [5]. The
following algorithm is obtained there.
Algorithm 1.1. .
Input: a 1-reduced pointed simplicial set X with effective homology.
Output: the effective homology of RX, that is, an equivalence C∗ (RX) ⇐⇒ E∗ , where E∗ is
an effective chain complex.
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An iterative application of this algorithm allows in particular to compute the groups
πq (Rp+1 X) ∼
= Hq (Rp X) with the corresponding generators, which makes it possible to
determine the first two stages of the Bousfield-Kan spectral sequence, as explained in the
following section.
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Computation of E 1 and E 2
The level E 1 of the Bousfield-Kan spectral sequence is given by the groups
1
Ep,q
= πq (Rp+1 X) ∩ Ker η 0 ∩ · · · ∩ Ker η p−1 ∼
= Hq (Rp X) ∩ Ker η 0 ∩ · · · ∩ Ker η p−1
If X is a 1-reduced simplicial set with effective homology, then our Algorithm 1.1 provides us the effective homology of the simplicial Abelian group RX. Iterating the process
(taking into account that RX is also 1-reduced), it is possible to obtain the effective homology of Rp X for every p ≥ 1. In this way, if X is an object with effective homology, the
groups Hq (Rp X) ∼
= πq (Rp+1 X) (with the corresponding generators) are computable.
The codegeneracy maps η j are well-defined on these homotopy groups:
η j : πq (Rp+1 X) ∼
= Hq (Rp X) −→ πq (Rp X) ∼
= Hq (Rp−1 X),
0≤j ≤p−1
and as far as Hq (Rp X) and Hq (Rp−1 X) are groups of finite type, these maps can be expressed as finite integer matrices. Therefore, the kernels Ker η j can be computed by means
of elementary operations for each 0 ≤ j ≤ p − 1, and in this way one can determine the
1 (with the corresponding generators).
groups Ep,q
P
p+1
1 → E1
j j
The differential map d1p,q : Ep,q
= p+1
p+1,q is induced by δq
j=0 (−1) ∂ . Then, for
1 (given by means of the coefficients with respect to the generators of the
a class [x] ∈ Ep,q
1
.
group) it is possible to compute the image d1p,q ([x]) = [δqp+1 (x)] in Ep+1,q
This implies that, if X is an object with effective homology, the first level of the
Bousfield-Kan spectral sequence is computable.
Algorithm 2.1. .
Input: a 1-reduced pointed simplicial set X with effective homology.
Output:
1 for each p, q ∈ Z,
• the groups Ep,q
• the differential maps d1p,q for all p, q ∈ Z.
1 → E1
Let us remark now that the differential maps d1p,q : Ep,q
p+1,q can also be expressed
as finite integer matrices. Therefore it is possible to determine their kernels and their
images, and then an elementary algorithm computes the quotient groups
2
Ep,q
=
Ker d1p,q
Im d1p−1,q
We obtain therefore the following algorithm which determines the groups of the second
level of the Bousfield-Kan spectral sequence. However, in this case the differential maps
cannot be computed.
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Algorithm 2.2. .
Input: a 1-reduced pointed simplicial set X with effective homology.
2 }
Output: the groups {Ep,q
p,q≥0 of the Bousfield-Kan spectral sequence of X.
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Conclusions and further work
The algorithms presented in this paper, based on the computation of the effective homology of the simplicial Abelian group RX, allow one to determine the first two stages of the
Bousfield-Kan spectral sequence associated with a (1-reduced) simplicial set X. The implementation is being written in Common Lisp, as a new module for the Kenzo system, as
did before for spectral sequences of filtered complexes [6]. We have already written several
Lisp functions implementing some parts of our algorithms, but the work is not finished yet.
On the other hand, we plan to go on with this work developing new algorithms computing
the higher levels of the spectral sequence, differential maps included. However, this task
seems to be more difficult since there some complicated ingredients (such as cosimplicial
spaces or tower of fibrations) appear.
References
[1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathematics 72 (1960) 20-104.
[2] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture
Notes in Mathematics 304 (Springer-Verlag, 1972).
[3] X. Dousson, J. Rubio, F. Sergeraert, and Y. Siret, The Kenzo program (Institut Fourier,
Grenoble, 1999). http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/
[4] A. Romero, Effective homology of free simplicial Abelian groups: the acyclic case,
Proceedings of X Encuentro de Álgebra Computacional y Aplicaciones (Universidad de
Sevilla, 2006) 143-146.
[5] A. Romero, Effective Homology and Spectral Sequences, PhD. thesis (Universidad de
La Rioja, 2007).
[6] A. Romero, J. Rubio and F. Sergeraert, Computing Spectral Sequences, Journal of
Symbolic Computation 41 No. 10 (2006) 1059-1079.
[7] J. Rubio and F. Sergeraert, Constructive Algebraic Topology, Bulletin des Sciences
Mathématiques 126 (2002) 389-412.
Ana Romero
Departamento de Matemáticas y Computación, Universidad de La Rioja
Edificio Vives. Calle Luis de Ulloa s/n. 26004 Logroño (La Rioja, Spain).
E-mail: [email protected]
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