Abstract
In an earlier paper of Čadek, Vokřínek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces.
We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg–MacLane space \(K(\mathbb{Z},1)\), represented as a simplicial group, can be equipped with polynomial-time homology (this is a polynomial-time version of effective homology considered in previous works of the third author and co-workers).
To this end, we construct a suitable discrete vector field, in the sense of Forman’s discrete Morse theory, on \(K(\mathbb{Z},1)\). The construction is purely combinatorial and it can be understood as a certain procedure for reducing finite sequences of integers, without any reference to topology.
The Eilenberg–MacLane spaces are the basic building blocks in a Postnikov system, which is a “layered” representation of a topological space suitable for homotopy-theoretic computations. Employing the result of this paper together with other results on polynomial-time homology, in another paper we obtain, for every fixed k, a polynomial-time algorithm for computing the kth homotopy group π k (X) of a given simply connected space X, as well as the first k stages of a Postnikov system for X, and also a polynomial-time version of the algorithm of Čadek et al. mentioned above.
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Notes
Curiously, \(K(\mathbb{Z},1)\) as a topological space almost cannot be simpler—as we mentioned, it is homotopy equivalent to the circle S 1, and other Eilenberg–MacLane spaces are much more complicated. But we need to work with the Kan simplicial model of \(K(\mathbb{Z},1)\) as introduced above, which has infinitely many simplices in every dimension k≥1. As we will see, for effective (or polynomial-time) homology, it is not sufficient to know, for example, that \(H_{2}(K(\mathbb{Z},1))=0\), but we need to be able to actually compute “witnesses” for it; that is, given a 2-cycle z 2 on \(K(\mathbb{Z},1)\), compute a 3-chain for which z 2 is its boundary. This problem would be trivial for the standard simplicial representation of S 1 with one vertex and one edge, but it is not trivial for the considered Kan model of \(K(\mathbb{Z},1)\).
This actually corresponds to the topological fact that the considered \(K(\mathbb{Z},1)\), as a topological space, is homotopy equivalent to S 1; [ ] represents a vertex, and [1] an edge glued to that vertex by both ends, forming an S 1.
We will not define a Kan simplicial set, but we just mention a key property, which is the reason why these simplicial sets are essential to the considered algorithms. Namely, if X is a simplicial set and Y is a Kan simplicial set, then every continuous map |X|→|Y| is homotopic to a simplicial map X→Y. Thus, continuous maps into Y have a combinatorial representation, describing them up to homotopy.
As another, perhaps more sophisticated example, we can mention the computation of the homotopy group π k (X) for a 1-connected simplicial set X: for this, given X, one first produces another simplicial set X′ from X, by a sequence of operations that “kill” the first k−1 homotopy groups, and then π k (X) is computed as H k (X′) using the Hurewicz isomorphism.
One can also consider other kinds of objects with effective homology, such as chain complexes, but for concreteness, we will stick to simplicial sets.
This feature makes it very natural to implement algorithms from this area using functional programming languages, as was done for the package Kenzo; see, e.g., [14].
These chain complexes are over \(\mathbb{Z}\); more generally, one considers chain complexes over a commutative ring R, where the C k are R-modules. These are needed, among others, for homology with coefficients in R. But for our purposes, homology with integer coefficients suffices; if needed, homology groups with other coefficients can be computed using universal coefficient theorems. Alternatively, all of the theory can be built with coefficients from a fixed ring R, provided that R is equipped with sufficiently strong algorithmic primitives.
They did not require the condition hh=0, but simple transformation converts a reduction without this condition into another one satisfying it.
In [20] and in other papers, effective homology is defined in a more general way, using strong equivalence of chain complexes instead of just a reduction. A strong equivalence of C ∗ and \(\tilde{C}_{*}\) means that there is an auxiliary chain complex A ∗ and reductions of A ∗ to both C ∗ and \(\tilde{C}_{*}\). However, here the simpler notion using a single reduction suffices, and this only makes the result formally stronger, since a reduction is a special case of a strong equivalence.
In [17], vector fields are considered in somewhat greater generality, on algebraic cell complexes. Here it is sufficient to stay in the perhaps more intuitive setting of vector fields on simplicial sets.
In a simplicial set, it may happen that σ is a “multiple” face of τ. i.e., σ=∂ i τ holds for several indices i. In such case, we connect τ to σ with multiple edges in the V∂-graph, one edge for each such index i.
Of course, for the main result of this paper, polynomial-time homology for \(K(\mathbb{Z},1)\), parameterization is not needed, but we need it if we want to have a general tool for obtaining polynomial-time homology from a vector field.
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Acknowledgements
We would like to thank Martin Čadek, Lukáš Vokřínek, and Uli Wagner for useful discussions and ongoing collaboration. Moreover, we thank Uli Wagner and Martin Čadek for insightful comments on a preliminary version of the manuscript. The research by J. M. and M. K. was supported by the Institute for Theoretical Computer Science (ITI), Charles University, Prague (project 1M0545 of the Czech Ministry of Education) and by the ERC Advanced Grant No. 267165. The research by M. K. was also supported by the project GAUK 49209.
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Communicated by Peter Buergisser.
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Krčál, M., Matoušek, J. & Sergeraert, F. Polynomial-Time Homology for Simplicial Eilenberg–MacLane Spaces. Found Comput Math 13, 935–963 (2013). https://doi.org/10.1007/s10208-013-9159-7
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DOI: https://doi.org/10.1007/s10208-013-9159-7