Computing simplicial representatives of homotopy group elements
CComputing simplicial representatives ofhomotopy group elements ∗ Marek Filakovsk´y Peter Franek Uli WagnerStephan ZhechevNovember 10, 2018
Abstract
A central problem of algebraic topology is to understand the homotopygroups π d ( X ) of a topological space X . For the computational version ofthe problem, it is well known that there is no algorithm to decide whetherthe fundamental group π ( X ) of a given finite simplicial complex X istrivial. On the other hand, there are several algorithms that, given afinite simplicial complex X that is simply connected (i.e., with π ( X )trivial), compute the higher homotopy group π d ( X ) for any given d ≥ π d ( X ), d ≥ abstract finitely generated abeliangroup given by generators and relations, but they work with very implicitrepresentations of the elements of π d ( X ). Converting elements of thisabstract group into explicit geometric maps from the d -dimensional sphere S d to X has been one of the main unsolved problems in the emerging fieldof computational homotopy theory.Here we present an algorithm that, given a simply connected space X , computes π d ( X ) and represents its elements as simplicial maps from asuitable triangulation of the d -sphere S d to X . For fixed d , the algorithmruns in time exponential in size( X ), the number of simplices of X . More-over, we prove that this is optimal: For every fixed d ≥
2, we constructa family of simply connected spaces X such that for any simplicial maprepresenting a generator of π d ( X ), the size of the triangulation of S d onwhich the map is defined, is exponential in size( X ). ∗ The research leading to these results has received funding from Austrian Science Fund(FWF): M 1980. a r X i v : . [ c s . C G ] A ug Introduction
One of the central concepts in topology are the homotopy groups π d ( X ) ofa topological space X . Similar to the homology groups H d ( X ), the homo-topy groups π d ( X ) provide a mathematically precise way of measuring the“ d -dimensional holes” in X , but the latter are significantly more subtle andcomputationally much less tractable than the former. Understanding homotopygroups has been one of the main challenges propelling research in algebraictopology, with only partial results so far despite an enormous effort (see, e.g.,[40, 29]); the amazing complexity of the problem is illustrated by the fact thateven for the 2-dimensional sphere S , the higher homotopy groups π d ( S ) arenontrivial for infinitely many d and known only for a few dozen values of d .For computational purposes, we consider spaces that have a combinatorialdescription as simplicial sets (or, alternatively, finite simplicial complexes) andmaps between them as simplicial maps .A fundamental computational result about homotopy groups is negative:There is no algorithm to decide whether the fundamental group π ( X ) of afinite simplicial complex X is trivial, i.e., whether every continuous map fromthe circle S to X can be continuously contracted to a point; this holds even if X is restricted to be 2-dimensional. On the other hand, given a space X that is simply connected (i.e., pathconnected and with π ( X ) trivial) there are algorithms that compute the higherhomotopy group π d ( X ), for every given d ≥
2. The first such algorithm wasgiven by Brown [5], and newer ones have been obtained as a part of generalcomputational frameworks in algebraic topology; in particular, an algorithmbased on the methods of Sergeraert et al. [49, 45] was described by Real [41].More recently, ˇCadek et al. [9] proved that, for any fixed d , the homotopygroup π d ( X ) of a given 1-connected finite simplicial set can be computed inpolynomial time. On the negative side, computing π d ( X ) is d ispart of the input [2, 8] (and, moreover, W[1]-hard with respect to the parameter d [33]), even if X is restricted to be 4-dimensional. These results form partof a general effort to understand the computational complexity of topologicalquestions concerning the classification of maps up to homotopy [7, 6, 8, 15] andrelated questions, such as the embeddability problem for simplicial complexes (ahigher-dimensional analogue of graph planarity) [32, 34, 10]. By definition, elements of π d ( X ) are equivalence classes of continuous mapsfrom the d -dimensional sphere S d to X , with maps being considered equivalent(or lying in the same homotopy class ) if they are homotopic , i.e., if they can becontinuously deformed into one another (see Section 3 for more details). This follows via a standard reduction from a result of Adjan[1] and Rabin [39] on thealgorithmic unsolvability of the triviality problem of a group given in terms of generators andrelations; we refer to the survey [51] for further background. π d ( X ) as an abstractabelian group, in terms of generators and relations. However, they work withvery implicit representations of the elements of π d ( X ).The main result of this paper is an algorithm that, given an element α of π d ( X ), computes a suitable triangulation Σ d of the sphere S d and an explicitsimplicial map Σ d → X representing the given homotopy class α .Apart from the intrinsic importance of homotopy groups, we see this asa first step towards the more general goal of computing explicit maps withspecific topological properties; instances of this goal include computing explicitrepresentatives of homotopy classes of maps between more general spaces X and Y (a problem raised in [7]) as well as computing an explicit embedding of a givensimplicial complex into R d (as opposed to deciding embeddability ). Moreover,these questions are also closely related to quantitative questions in homotopytheory [21] and in the theory of embeddings [17]. See Section 1.2 for a moredetailed discussion of these questions.Throughout this paper, we assume that the input X is simply connected , i.e.,that it is connected and has trivial fundamental group π ( X ). For the purposeof the exposition, we will assume that X is given as a 1-reduced simplicial set,encoded as a list of its nondegenerate simplices and boundary operators givenvia finite tables. We remark that the class of 1-reduced simplicial sets containsstandard models of 1-connected topological spaces, such as spheres or complexprojective spaces. A more general version of the theorem that also includessimply connected simplicial complexes is discussed in Section 4. Theorem A.
There exists an algorithm that, given d ≥ and a finite -reducedsimplicial set X , computes the generators g , . . . , g k of π d ( X ) as simplicial maps Σ dj → X , for suitable triangulations Σ dj of S d , j = 1 , . . . , k .For fixed d , the time complexity is exponential in the size (number of sim-plices) of X ; more precisely, it is O (2 P (size( X )) ) where P = P d is a polynomialdepending only on d . Any element of π d ( X ) can be expressed as a sum of generators, and ex-pressing the sum of two explicit maps from spheres into X as another explicitmap is a simple operation. Hence, the algorithm in Theorem A can convert any element of π d ( X ) into an explicit simplicial map.Theorem A also has the following quantitative consequence: Fix some stan-dard triangulation Σ of the sphere S d , e.g., as the boundary of a d + 1-simplex.By the classical Simplicial Approximation Theorem [23, 2.C], for any continuousmap f : S d → X , there is a subdivision Σ (cid:48) of Σ and a simplicial map f (cid:48) : Σ (cid:48) → X that is homotopic to f . Theorem A implies that if f represents a generator of π d ( X ), then the size of Σ (cid:48) can be bounded by an exponential function of thenumber of simplices of X .Furthermore, we can show that the exponential dependence on the numberof simplices in X is inevitable: That is, they compute integers r, q , . . . , q k such that π d ( X ) is isomorphic to Z r ⊕ Z q ⊕ . . . ⊕ Z q k . heorem B. Let d ≥ be fixed. Then there is an infinite family of d -dimensional -reduced -connected simplicial sets X such that for any simplicialmap Σ → X representing a generator of π d ( X ) , the triangulation Σ of S d onwhich f is defined has size at least Ω(size( X )) . If d ≥ , we may even assumethat X are -reduced.Consequently, any algorithm for computing simplicial representatives of thegenerators of π d ( X ) for -reduced simplicial set X has time complexity at least Ω(size( X )) . In the boundary case of 1-reduced simplicial sets for d = 2, we don’t knowwhether the lower complexity bound is sub-exponential or not. However, wecan show that the algorithm from Theorem A is optimal in that case as well,see a discussion in Section 5, page 24.In Section 4 and 5, we state and prove generalizations of Theorem A and Bdenoted as Theorem A.1 and B.1. They remove the 1-reduceness assumptionand replace it by a more flexible certificate of simply connectedness, allowingthe input space X to be a more flexible simplicial set or simplicial complex. Computational homotopy theory and applications.
This paper falls intothe broader area of computational topology , which has been a rapidly devel-oping area (see, for instance, the textbooks [11, 54, 35]); more specifically, asmentioned above, this work forms part of a general effort to understand thecomputational complexity of problems in homotopy theory , both because of theintrinsic importance of these problems in topology and because of applicationsin other areas, e.g., to algorithmic questions regarding embeddability of sim-plicial complexes [32, 10], to questions in topological combinatorics (see, e.g.,[31]), or to the robust satisfiability of equations [16].A central theme in topology is to understand the set [
X, Y ] of all homotopyclasses of maps from a space X to a space Y . In many cases of interest, thisset carries additional structure, e.g., an abelian group structure, as in the case π d ( X ) = [ S d , X ] of higher homotopy groups that are the focus of the presentpaper.Homotopy-theoretic questions have been at the heart of the development ofalgebraic topology since the 1940’s. In the 1990s, three independent groups ofresearchers proposed general frameworks to make various more advanced meth-ods of algebraic topology (such as spectral sequences) effective (algorithmic):Sch¨on [48], Smith [50], and Sergeraert, Rubio, Dousson, Romero, and coworkers(e.g., [49, 45, 42, 46]; also see [47] for an exposition). These frameworks yieldedgeneral computability results for homotopy-theoretic questions (including newalgorithms for the computation of higher homotopy groups [41]), and in thecase of Sergeraert et al., also a practical implementation in form of the Kenzosoftware package [24].Building on the framework of objects with effective homology by Sergeraert etal., in recent years a variety of new results in computational homotopy theory4ere obtained [7, 30, 9, 8, 52, 15, 10, 43, 44], including, in some cases, thefirst polynomial-time algorithms , by using a refined framework of objects withpolynomial-time homology [30, 9] that allows for a computational complexityanalysis. For an introduction to this area from a theoretical computer scienceperspective and an overview of some of these results, see, e.g., [6] and thereferences therein. Explicit maps.
As mentioned above, the above algorithms often work withrather implicit representations of the homotopy classes in π d ( X ) (or, more gen-erally, in [ X, Y ]) but does not yields explicit maps representing these homotopyclasses.For instance, the algorithm in [41] computes π d ( X ) as the homology group H d ( F ) of an auxiliary space F = F d ( X ) constructed from X in such a way that π d ( X ) and H d ( F ) are isomorphic as groups. More recently, Romero and Sergeraert [44] devised an algorithm that, givena 1-reduced (and hence simply connected) simplicial set X and d ≥
2, computesthe homotopy group π d ( X ) as the homotopy group π d ( K ) of an auxiliary sim-plicial set K (a so-called Kan completion of X ) with π d ( X ) ∼ = π d ( K ). Moreover,given an element of this group, the algorithm can compute an explicit simplicialmap Σ d → K from a suitable triangulation of S d to K representing the givenhomotopy class. In this way, homotopy classes are represented by explicit maps,but as maps to the auxiliary space K , which is homotopy equivalent to but nothomeomorphic to the given space X .By contrast, our general goal is to is represent homotopy classes by maps intothe given space; in the present paper, we treat, as an important first instance,the case π d ( X ) = [ S d , X ]. Open Problems and Future Work.
Our next goal is to extend the resultshere to the setting of [7], i.e., to represent, more generally, homotopy classesin [
X, Y ] by explicit simplicial maps from some suitable subdivision X (cid:48) to Y (under suitable assumptions that allow us to compute [ X, Y ]). In a subsequent step, we hope to generalize this further to the equivariant setting [
X, Y ] G of [10], in which a finite group G of symmetries acts on thespaces X, Y and all maps and homotopies are required to be equivariant , i.e.,to preserve the symmetries.As mentioned above, one motivation is the problem of algorithmically con-structing embeddings of simplicial complexes into R d . Indeed, in a suitablerange of dimensions ( d ≥ k +1)2 ), the existence of an embedding of a finite k -dimensional simplicial complex K into R d is equivalent to the existence of an Z -equivariant map from an auxiliary complex ˜ K (the deleted product) into thesphere S d − , by a classical theorem of Haefliger and Weber [22, 53]. The proofof the Haefliger-Weber Theorem is, in principle, constructive, but in order to Similarly, the algorithm in [9] constructs an auxiliary chain complex C such that π d ( X )is isomorphic to the homology group H d +1 ( C ) and computes the latter. Similarly as before, the algorithm in [7] computes [
X, Y ] as the set [
X, P ] for some auxiliaryspace P (a stage of a Postnikov system for Y ) and represents the elements of [ X, Y ] ∼ = [ X, P ]as maps from X to P , but not as maps to Y . S d − . Quantitative homotopy theory.
Another motivation for representing homo-topy classes by simplicial maps and complexity bounds for such algorithms isthe connection to quantitative questions in homotopy theory [21, 13] and in thetheory of embeddings [17]. Given a suitable measure of complexity for the mapsin question, typical questions are: What is the relation between the complexityof a given null-homotopic map f : X → Y and the minimum complexity ofa nullhomotopy witnessing this? What is the minimum complexity of an em-bedding of a simplicial complex K into R d ? In quantitative homotopy theory,complexity is often quantified by assuming that the spaces are metric spacesand by considering Lipschitz constants (which are closely related to the sizes ofthe simplicial representatives of maps and homotopies [13]). For embeddings,the connection is even more direct: a typical measure is the smallest number ofsimplices in a subdivision K (cid:48) or K such that there exists a simplexwise linear-embedding K (cid:48) (cid:44) → R d . The remainder of the paper is structured as follows: In Section 2, we give a high-level description of the main ingredients of the algorithm from Theorem A.In Section 3, we review a number of necessary technical definitions regardingsimplicial sets and the frameworks of effective and polynomial-time homology,in particular Kan’s simplicial version of loop spaces and polynomial-time loopcontractions for infinite simplicial sets. In Section 4, we formally describe thealgorithm from Theorem A and give a high level proof based on a number oflemmas which are proved in in subsequent chapters. Section 5 contains the proofof Theorem B. The rest of the paper contains several technical parts needed forthe proof of Theorem A: in Section 6, we describe Berger’s effective Hurewiczinverse and analyze its running time (Theorem 4.2), in Section 7, we provethat the stages of the Whitehead tower have polynomial-time contractible loops(Lemma 4.3). Finally, in Section 8, we show how to reduce the case when theinput is a simplicial complex X sc to the case of an associated simplicial set X and convert a map Σ → X into a map from a subdivision Sd (Σ) into X sc (Lemma 4.5). In this section we present a high-level description of the main steps and ingre-dients involved in the algorithm from Theorem A.
The algorithm in a nutshell.
1. In the simplest case when the space X is ( d − π i ( X ) = 0for all i ≤ d − π d ( X ) ∼ = H d ( X ) between the d th homotopy group and the6 th homology group of X . Computing generators of the homology groupis known to be a computationally easy task (it amounts to solving a linearsystem of equations over the integers). The key is then converting thehomology generators into the corresponding homotopy generators, i.e., tocompute an inverse of the Hurewicz isomorphism. This was described inthe work of Berger [3, 4]. We analyze the complexity of Berger’s algo-rithm in detail and show that it runs in exponential time in the size of X (assuming that the dimension d is fixed).2. For the general case, we construct an auxiliary simplicial set F d togetherwith a simplicial map ψ d : F d → X that has the following properties: • F d is a simplicial set that is d − • ψ d : F d → X induces an isomorphism ψ d ∗ : π d ( F d ) → π d ( X ).Our construction of F d is based on computing stages of the Whiteheadtower of X [23, p. 356]; this is similar to Real’s algorithm, which computes π d ( X ) as H d ( F d ) as an abstract abelian group.The overall strategy is to use Berger’s algorithm on the space F d and com-pute generators of π d ( F d ) as simplicial maps. Then we use the simplicialmap ψ d to convert each generator of π d ( F d ) into a map Σ d → X , andthese maps generate π d ( X ). The main technical task for this step is toshow that Berger’s algorithm can be applied to F d . For this, we need toconstruct a polynomial algorithm for explicit contractions of loops in F d (this space is 1-connected but not 1-reduced in general). Our contributions.
The main ingredients of the algorithm outlined aboveare the computability of stages of the Whitehead tower [41] as simplicial setswith polynomial-time homology and Berger’s algorithmization of the inverseHurewicz isomorphism [3, 4].The idea that these two tools can be combined to compute explicit represen-tatives of π d ( X ) is rather natural and is also mentioned, for the special case of1-reduced simplicial sets, in [44, p. 3]; however, there are a number of technicalchallenges to overcome in order to carry out this program (as remarked in [44, p.3]: “Clemens Berger’s algorithm, quite complex, has never been implemented,severely limiting the current scope of this approach, same comment with respectto the theoretical complexity of such an algorithm.”). On a technical level, ourmain contributions are as follows: • We give a complexity analysis of Berger’s algorithm to compute the inverseof the Hurewicz isomorphism (Theorem 4.2). • We show that the homology generators of the Whitehead stage F d can becomputed in polynomial time (Lemma 4.1). • Berger’s algorithm requires an explicit algorithm for loop contraction—acertificate of 1-connectedness of the space F d . While F d is not 1-reduced7n general, we describe an explicit algorithm for contracting its loop andshow that Berger’s algorithm can be applied.We remark that the Whitehead tower stages are simplicial sets with infinitelymany simplices, and we need the machinery of objects with polynomial-timehomology to carry out the last two steps. In this section, we give the necessary technical definitions that will be usedthroughout this paper. In the first part, we recall the standard definitions forsimplicial sets and the toolbox of effective homology.Afterwards, we present Kan’s definiton of a loop space and further formalizeour definition of (polynomial-time) loop contractions.
Simplicial sets and their computer representation.
A simplicial set X isa graded set X indexed by the non-negative integers together with a collectionof mappings d i : X n → X n − and s i : X n → X n +1 , ≤ i ≤ n called the face and degeneracy operators. They satisfy the following identities: d i d j = d j − d i for i < j,d i s i = d i +1 s i = id for 0 ≤ i < n,d i s j = s j d i − for i > j + 1 ,d i s j = s j − d i for i < j,s i s j = s j +1 s i for i ≤ j. More details on simplicial sets and the motivation behind these formulas can befound in [36, 20].Simplicial maps between simplicial sets are maps of graded sets which com-mute with the face and degeneracy operators. The elements of X n are called n - simplices . We say that a simplex x ∈ X n is (non-)degenerate if it can(not) beexpressed as x = s i y for some y ∈ X n − . If a simplicial set X is also a graded(Abelian) group and face and degeneracy operators are group homomorphisms,we say that X is a simplicial (Abelian) group.A simplicial set is called k -reduced for k ≥
0, if it has a single i -simplex foreach i ≤ k .For a simplicial set X , we define the chain complex C ∗ ( X ) to be a freeAbelian group enerated by the elements of X n with differential ∂ ( c ) = n (cid:88) i =0 ( − i d i ( c ) . locally effective , if its simplices have a specified finiteencoding and algorithms are given that compute the face and degeneracy op-erators. A simplicial map f between locally effective simplicial sets X and Y is locally effective , if an algorithm is given that for the encoding of any given x ∈ X computes the encoding of f ( x ) ∈ Y .We define a simplicial set to be finite if it has finitely many non-degeneratesimplices. Such simplicial set can be algorithmically represented in the followingway. The encoding of non-degenerate simplices can be given via a finite list andthe encoding of a degenerate simplex s i k . . . s i y for i < i < . . . < i k and a non-degenerate y can be assumed to be a pair consisting of the sequence ( i , . . . , i k )and the encoding of y . The face operators are fully described by their actionon non-degenerate simplices and can be given via finite tables. In this way, anysimplicial set with finitely many non-degenerate simplices is naturally locallyeffective. Any choice of an implementation of the encoding and face operatorsis called a representation of the simplicial set. The size of a representation is theoverall memory space one needs to store the data which represent the simplicialset. Geometric realization.
To each simplicial set X we assign a topologicalspace | X | called its geometric realization. The construction is similar to thatof simplicial complexes. Let ∆ j be the geometric realization of a standard j -simplex for each j ≥
0. For each k , we define D i : ∆ k − (cid:44) → ∆ k to be the inclusionof a ( k − i ’th face of a k -simplex and S i : ∆ k → ∆ k − be thegeometric realization of a simplicial map that sends the vertices (0 , , . . . , k ) of∆ k to the vertices (0 , , . . . , i, i, i + 1 , . . . , k − | X | is then defined to be a disjoint union of all simplices X factored by the relation ∼ | X | := ( ∞ (cid:71) n =0 X n × ∆ n ) / ∼ where ∼ is the equivalence relation generated by the relations ( x, D i ( p )) ∼ ( d i ( x ) , p ) for x ∈ X n +1 , p ∈ ∆ n and the relations ( x, S i ( p )) ∼ ( s i ( x ) , p ) for x ∈ X n − , p ∈ ∆ n .Similarly, a simplicial map between simplicial complexes naturally inducesa continuous map between their geometric realizations. Simplicial complexes and simplicial sets.
In any simplicial complex X sc ,we can choose an ordering of vertices and define a simplicial sets X ss thatconsists of all non-decrasing sequences of points in X sc : the dimension of( V , . . . , V d ) equals d . The face operator is d i omits the i ’th coordinate andthe degeneracy s j doubles the j ’th coordinate. Moreover, choosing a maximaltree T in the 1-skeleton of X enables us to construct a simplicial set X := X ss /T in which all vertices and edges in the tree, as well as their degeneracies, are con-sidered to be a base-point (or its degeneracies). The geometric realizations of X sc and X are homotopy equivalent and X is 0-reduced, i.e. it has one vertexonly. Homotopy groups.
Let (
X, x ) be a pointed topological space. The k -th9omotopy group π k ( X, x ) of ( X, x ) is defined as the set of pointed homotopy classes of pointed continuous maps ( S k , ∗ ) → ( X, x ), where ∗ ∈ S k is a dis-tinguished point. In particular, the 0-th homotopy group has one element foreach path connected component of X . For k = 1, π ( X, x ) is the fundamentalgroup of X , once we endow it with the group operation that concatenates loopsstarting and ending in x . The group operation on π k ( X, x ) for k > f ] , [ g ] the homotopy class of the composition S k π → S k ∨ S k f ∨ g → X where π factors an equatorial ( k − x into a point. Homotopygroups π k are commutative for k > π k ( X ). For a simplicial set X , we will use thenotation π k ( X ) for the k ’th homotopy group of its geometric realization | X | .An important tool for computing homotopy groups is the Hurewicz theorem .It says that whenever X is ( d − π d ( X ) → H d ( X ). Moreover, if the element of π d ( X ) is represented by a simpli-cial map f : Σ d → X and (cid:80) j k j σ j represents a homology generator of H d (Σ d ),then the Hurewicz isomorphism maps [ f ] to the homology class of the formalsum (cid:80) j k j f ( σ j ) of d -simplices in X . Effective homology.
We call a chain complex C ∗ locally effective if the el-ements c ∈ C ∗ have finite (agreed upon) encoding and there are algorithmscomputing the addition, zero, inverse and differential for the elements of C ∗ .A locally effective chain complex C ∗ is called effective if there is an algorithmthat for given n ∈ N generates a finite basis c α ∈ C n and an algorithm that forevery c ∈ C ∗ outputs the unique decomposition of c into a linear combinationof c α ’s.Let C ∗ and D ∗ be chain complexes. A reduction C ∗ ⇒⇒ D ∗ is a triple ( f, g, h )of maps such that f : C ∗ → D ∗ and g : D ∗ → C ∗ are chain homomorphisms, h : C ∗ → C ∗ has degree 1, f g = id and f g − id = h∂ + ∂h , and further hh = hg = f h = 0.A locally effective chain complex C ∗ has effective homology ( C ∗ is a chaincomplex with effective homology ) if there is a locally effective chain complex ˜ C ∗ ,reductions C ∗ ⇐⇐ ˜ C ∗ ⇒⇒ C ef ∗ where C ef ∗ is an effective chain complex, and allthe reduction maps are computable. Eilenberg-MacLane spaces.
Let d ≥ π be an Abelian group. AnEilenberg-MacLane space K ( π, d ) is a topological space with the properties π d ( K ( π, d )) (cid:39) π and π j ( K ( π, d )) = 0 for 0 < j (cid:54) = d . It can be shown thatsuch space K ( π, d ) exists and, under certain natural restrictions, has a uniquehomotopy type. If π is finitely generated, then K ( π, d ) has a locally effectivesimplicial model [30]. Globally polynomial-time homology and related notions.
In many aux-iliary steps of the algorithm, we will construct various spaces and maps. Toanalyse the overall time complexity, we need to parametrize all these objectsby the very initial input, which is in our case an encoding of a finite 1-reduced A homotopy F : S k × I → X is pointed if F ( ∗ , t ) = x for all t ∈ I . I be a parameter set so that for each I ∈ I an integersize( I ) is defined. We say that F is a parametrized simplicial set (group, chaingroup, . . . ), if for each I ∈ I , a locally effective simplicial set (group, chaingroup, . . . ) F ( I ) is given. The simplicial set F is locally polynomial-time , if thereexists a locally effective model of F ( I ) such that for each k ∈ N and an encodingof a k -simplex x ∈ F ( I ), the encoding of d i ( x ) and s j ( x ) can be computed intime polynomial in size(enc( x ))+size( I ). The polynomial, however, may dependon k . A polynomial-time map between parametrized simplicial sets F and G isan algorithm that for each k ∈ N , I ∈ I and an encoding of an k -simplex x in F ( I ) computes the encoding of f ( x ) in time polynomial in size(enc( x ))+size( I ):again, the polynomial may depend on k .Similarly, a locally polynomial-time (parametrized) chain complex is an as-signment of a computer representation C ∗ ( I ) of a chain complex with a distin-guished basis in each gradation, such that all these basis elements have someagreed-upon encoding. A chain (cid:80) j k j σ j is assumed to be represented as alist of pairs ( k j , enc( σ j )) j and has size (cid:80) j (size( k j ) + size(enc( σ j ))), where weassume that the size of an integer k j is its bit-size. Further, an algorithm isgiven that computes the differential of a chain z ∈ C k ( I ) in time polynomial insize( z ) + size( I ), the polynomial depending on k . The notion of a polynomial-time chain map is straight-forward.A globally polynomial-time chain complex is a locally polynomial-time chaincomplex EC that in addition has all chain groups EC ( I ) k finitely generated andan additional algorithm is given that for each k computes the encoding of thegenerators of EC ( I ) k in time polynomial in size( I ). Finally, we define a simpli-cial set with globally polynomial-time homology to be a locally polynomial-timeparametrized simplicial set F together with reductions C ∗ ( F ) ⇐⇐ ˜ C ⇒⇒ EC where ˜ C, EC are locally polynomial-time chain complexes, EC is a globallypolynomial-time chain complex and the reduction data are all polynomial-timemaps, as usual the polynomials depending on the grading k .The name “polynomial-time homology” is motivated by the following: Lemma 3.1.
Let F be a parametrized simplicial set with polynomial-time ho-mology and k ≥ be fixed. Then all generators of H k ( F ( I )) can be computed intime polynomial in size( I ) .Proof. For the globally polynomial-time chain complex EF and each fixed j ,we can compute the matrix of the differentials d j : EF ( I ) j → EF ( I ) j − withrespect to the distinguished bases in time polynomial in size( I ): we just evaluate d k on each element of the distinguished basis of EF ( I ) k . Then the homologygenerators of H k ( EC ) can be computed using a Smith normal form algorithmapplied to the matrices of d k and d k +1 , as is explained in standard textbooks(such as [37]). Polynomial-time algorithms for the Smith normal form are non-trivial but known [28].Let x , . . . , x m be the cycles generating H k ( EF ( I )). We assume that reduc-11ions C ∗ ( F ) ( f,g,h ) ⇐⇐ ˜ F ( f (cid:48) ,g (cid:48) ,h (cid:48) ) ⇒⇒ EF are given and all the reduction maps are polynomial. Thus we can compute thechains f g (cid:48) ( x ) , f g (cid:48) ( x ) , . . . , f g (cid:48) ( x m )in polynomial time and it is a matter of elementary computation to verify thatthey constitute a set of homology generators for H k ( F ( I )). Principal bundles and loop group complexes.
In the text we will fre-quently deal with principal twisted Cartesian products: these are simplicialanalogues of principal fiber bundles. The definitions in this section come fromKan’s article [27].We first define the Cartesian product X × Y of simplicial sets X, Y : Theset of n -simplices ( X × Y ) n consists of tuples ( x, y ), where x ∈ X n , x ∈ Y n .The face and degeneracy operators on X × Y are given by d i ( x, y ) = ( d i x, d i y ), s i ( x, y ) = ( s i x, s i y ). Definition 3.2 (Principal Twisted Cartesian product) . Let B be a simplicialset with a basepoint b ∈ B and G be a simplicial group. We call a gradedmap (of degree -1) τ : B n +1 → G n , n ≥ a twisting operator if the followingconditions are satisfied: • d n τ ( β ) = τ ( d n +1 b ) − τ ( d n b ) • d i τ ( β ) = τ ( d i b ) for ≤ i < n • s i τ ( b ) = τ ( s i b ) , i < n , and • τ ( s n b ) = 1 n for all b ∈ B n where n is the unit element of G n .Let B , G , τ be as above. We will define a twisted Cartesian product B × τ G tobe a simplicial set E with E n = B n × G n , and the face and degeneracy operatorsare also as in the Cartesian product, i.e. d i ( b, g ) = ( d i b, d i g ) , with the soleexception of d n , which is given by d n ( b, g ) := ( d n b, τ ( b ) d n ( g )) , ( b, g ) ∈ B n × G n . It is not trivial to see why this should be the right way of representing fiberbundles simplicially, but for us, it is only important that it works, and we willhave explicit formulas available for the twisting operator for all the specificapplications.We remark that in the literature one can find multiple definitions of twistedoperator and twisted product [36, 27, 3] and that they, in essence differ fromeach other based on the decision whether the twisting “compresses” the firsttwo or the last two face operators. Here, we follow the same notation as in [3].12 efinition 3.3.
Let X be a -reduced simplicial set. Then we define GX to bea (non-commutative) simplicial group such that • GX n has a generator σ for each ( n + 1) -simplex σ ∈ X and a relation s n y = 1 for each simplex in the image of the last degeneracy s n . • The face operators are given by d i σ := d i σ for i < n and d n σ := ( d n +1 σ ) − d n σ • The degeneracy operators are s i σ := s i σ . We use the multiplicative notation, with 1 being the neutral element. It isshown in [27] that GX is a discrete simplicial analog of the loop space of X .For algorithmic puroposes, we assume that an elements (cid:81) j σ k j j of GX isrepresented as a list of pairs ( σ j , k j ) and has size (cid:80) j size( σ j ) + size( k j ). Definition 3.4.
Let X be a -reduced simplicial set. We say that a map c : GX → GX is a contraction of loops in X , if d c ( x ) = x and d c ( x ) = 1 foreach x ∈ GX .In case where X has finitely many nondegenerate -simplices, we define thesize size( c ) to be the sum (cid:88) γ ∈ X size( c ( γ )) . Loop contraction for simplicial complexes.
Let X sc be a simplicial com-plex. Let T be a spanning tree in the 1-skeleton of X sc and R a chosen vertex.For each oriented edge e = ( v v ) we define a formal inverse to be e − := ( v v )and we also consider degenerate edges ( v, v ). A loop is defined as a sequence e , . . . , e k of oriented edges in X sc such that • The end vertex of e i equals the initial vertex of e i +1 , and • The initial vertex of e and the end vertex of e k equal R .Every edge e that is not contained in T gives rise to a unique loop l e . Fur-ther, every loop in X sc is either a concatenation of such l e ’s, or can be derivedfrom such concatenation by inserting and deleting consecutive pairs ( e, e − ) anddegenerate edges. Before we formally define our combinatorial version of loopcontraction, we need the following definition. Definition 3.5.
Let S be a set, U ⊆ S , F ( S ) and F ( U ) be free groups generatedby S , U , respectively. Let h U : F ( S ) → F ( S ) be a homomorphism that sendseach u ∈ U to and each s ∈ S \ U to itself. We say that an element x of F ( S ) equals y modulo U , if h U ( x ) = y . An example of an element that is trivial modulo U is the word s u s − , where s ∈ S and u ∈ U . Formally, elements of F ( S ) are sequences of symbols s (cid:15) for (cid:15) ∈ { , − } and s ∈ S withthe relation s s − = 1, where 1 represents the empty sequence. The group operation isconcatenation. efinition 3.6. Let S be the set of all oriented edges and oriented degenerateedges in X sc and assume that a spanning tree T is chosen. Let U be the setof all oriented edges in T , including all degenerate edges. A contraction of anedge α is a sequence of vertices A , A , . . . , A s and B , . . . , B s such that • for each i , { A i , A i +1 , B i +1 } is a simplex of X sc , and • the element of F ( S )( A B )( B A )( A B )( B A ) . . . ( B s A s )( A s A s − )( A s − A s − ) . . . ( A A )(1) equals α modulo U .A loop contraction in a simplicial complex is the choice of a contractionof α for each edge α ∈ X sc \ T .The size of the contraction of α is defined to be the number of vertices in (1)and the size size( c ) of the loop contraction on X sc is the sum of the sizes overall α ∈ X sc \ T . A B A B A B A α Figure 1: The loop ranging over the boundary of this geometric shape equals α , after ignoring edges in the maximal tree and canceling pairs ( e, e − ). Theinterior of the triangles gives rise to a contraction.The geometry behind this definition is displayed in Figure 1. The sequenceof A i ’s and B j ’s gives rise to a map from the sequence of (full) triangles into X sc . The big loop around the boundary is combinatorially described by (1).We can continuously contract all of its parts that are in the tree T to a chosenbasepoint, as the tree is contractible. Further, we can continuously contract allpairs of edges ( e, e − ) and what remains is the original edge α : with all thetree contracted to a point, it will be transformed into a loop that geometricallycorresponds to l α . The interior of the full triangles then constitutes its “filler”,hence a certificate of the contractibility of l α .A loop contraction in the sense of Definition 1 exists iff the space X sc issimply connected. One could choose different notions of loop contraction. Forinstance, we could provide, for each α , a simplicial map from a triangulated2-disc into X sc such that the oriented boundary of the disc would be mappedexactly to l α . The description from Definition 3.6 could easily be converted intosuch map. We chose the current definition because of its canonical and algebraic14ature. The connection between Definitions 3.4 and 3.6 is the content of thefollowing lemma. Lemma 3.7.
Let X sc be a -connected simplicial complex with a chosen ori-entation of all simplices, X ss the induced simplicial set, T a maximal tree in X sc , and X := X ss /T the corresponding -reduced simplicial set. Assume thata loop contraction in the simplicial complex X sc is given, such as described inDefinition 3.6. Then we can algorithmically compute c ( α ) ∈ GX such that d c ( α ) = α and d c ( α ) = 1 , for every generator α of GX . Moreover, thecomputation of c ( α ) is linear in the size of X sc and the size of the simplicialcomplex contraction data.Proof. For each i , the triangle { A i , A i +1 , B i +1 } from Def. 3.6 is in the simplicialcomplex X sc . There is a unique oriented 2-simplex in X ss of the form ( V , V , V )(possibly degenerate) such that { V , V , V } = { A i , A i +1 , B i +1 } . Let as denotesuch oriented simplex by σ i , and its image in GX by σ i . We will define anelement g i ∈ GX such that it satisfies d g i (cid:39) ( A i , A i +1 ) and d g i (cid:39) ( A i , B i +1 ) ( B i +1 , A i +1 ) (2)where (cid:39) is an equivalence relation that identifies any element ( U, V ) ∈ GX with( V, U ) − (note that only one of the symbols ( U, V ) and (
V, U ) is well defined in X ss , resp. X .) Explicitly, we can define g i with these properties as follows: • If σ = ( B i +1 , A i , A i +1 ), then g i := σ i , • If σ = ( A i , A i +1 , B i +1 ), then g i := s ( d σ ) σ i s d ( σ i ) − • If σ = ( A i +1 , B i +1 , A i ), then g i = s d σ i − σ i s ( d σ i ) − • If σ = ( B i +1 , A i +1 , A i ), then g i := σ i − • If σ = ( A i +1 , A i , B i +1 ), then g i := s d σ i σ i − s ( d σ i ) − • If σ = ( A i , B i +1 , A i +1 ), then g i := s ( d σ i ) σ i − s d σ i .Let g := g . . . , g s . The assumption (1) together with equation (2) immediatelyimplies that d g ( d g ) − = α . Thus we define c ( α ) := s d ( g ) g − . Algorithmi-cally, to construct g amounts to going over all the triples ( A i , A i +1 , B i +1 ) froma given sequence of A (cid:48) i s and B j ’s, checking the orientation and computing g i forevery i . Polynomial-time loop contraction.
Let F be a parametrized simplicialset such that each F ( I ) is 0-reduced. Using constructions analogous to thosedefined above, GF is a parametrized locally-polynomial simplicial group whereaswe assume a simple encoding of elements of GF i as follows. If x = (cid:81) j σ jk j ∈ GF ( I ) k where σ j are ( k + 1)-simplices in F ( I ), not in the image of s k , then weassume that x is stored in the memory as a list of pairs ( k j , enc( σ j )) and has size (cid:80) j (size( k j ) + size( σ j )) where some σ i may be equal to σ j for i (cid:54) = j . Face and15egeneracy operators are defined in Definition (3.3) and it is easy to see that forany locally polynomial-time simplicial set F , GF is a locally polynomial-timesimplicial group. Definition 3.8.
Let F be a locally polynomial simplicial set. We say that F has polynomially contractible loops , if there exists an algorithm that for a -simplex x ∈ GF ( I ) computes a -simplex c ( x ) ∈ GF ( I ) such that d x = x , d x = 1 ∈ GF ( I ) , and the running-time is polynomial in size( x ) + size( I ) . We will prove a stronger statement of Theorem A formulated as follows.
Theorem A.1.
There exists an algorithm that, given d ≥ and a finite -reduced simplicial set X (alternatively, a finite simplicial complex) with an ex-plicit loop contraction c (such as in Definition 3.4 or 3.6) computes the genera-tors g , . . . , g k of π d ( X ) as simplicial maps Σ dj → X , for suitable triangulations Σ dj of S d , j = 1 , . . . , k .For fixed d , the time complexity is exponential in the size of X and the size ofthe loop contraction c ; more precisely, it is O (2 P (size( X )+size( c )) ) where P = P d is a polynomial depending only on d . This immediately implies Theorem A, as for a 1-reduced simplicial set, thecontraction c is trivial, given by c (1) = 1.The proof of Theorem A.1 is based on a combination of four statementspresented here as Lemma 4.1, Theorem 4.2, Lemma 4.3 and Lemma 4.5. Each ofthem is relatively independent and their proofs are delegated to further sections.First we present an algorithm that, given a 1-connected finite simplicial set X and a positive integer d , outputs a simplicial set F d and a simplicial map ψ d such that • the simplicial set F d is d − • the simplicial map ψ d : F d → X is polynomial-time and induces an iso-morphism ψ d ∗ : π d ( F d ) → π d ( X ). Whitehead tower.
We construct simplicial sets F d as stages of a so-called Whitehead tower for the simplicial set X . It is a sequence of simplicial sets andmaps · · · (cid:47) (cid:47) F d f d (cid:47) (cid:47) F d − f d − (cid:47) (cid:47) · · · f (cid:47) (cid:47) F f (cid:47) (cid:47) (cid:47) (cid:47) F = X. where f i induces an isomorphism π j ( F i +1 ) → π j ( F i ) for j > i and π j ( F i ) = 0for j < i . We define ψ d = f d f d − . . . f . One can see that F d , ψ d satisfy thedesired properties. 16 emma 4.1. Let d ≥ be a fixed integer. Then there exists a polynomial-timealgorithm that, for a given -connected finite simplicial set X , constructs thestages F , . . . , F d of the Whitehead tower of X .The simplicial sets F k ( X ) , parametrized by -connected finite simplicial sets X , have polynomial-time homology and the maps f k are polynomial-time sim-plicial maps.Proof. The proof is by induction. The basic step is trivial as F = X . Wedescribe how to obtain F k +1 , f k +1 assuming that we have computed F k , 2 ≤ k < d .1. We compute simplicial map ϕ k : F k → K ( π k ( X ) , k ) = K ( π k ( F k ) , k ) thatinduces an isomorphism ϕ k ∗ : π k ( F k ) → π k ( K ( π k ( X ) , k )) ∼ = π k ( X ). Thisis done using the algorithm in [9], as K ( π k ( X ) , k ) is the first nontrivialstage of the Postnikov tower for the simplicial set F k .For the simplicial set K ( π k ( X ) , k ) and for such simplicial sets there is aclassical principal bundle (twisted Cartesian product) (see [36]): K ( π k ( X ) , k − (cid:15) (cid:15) E ( π k ( X ) , k −
1) = K ( π k ( X ) , k ) × τ K ( π k ( X ) , k − δ (cid:15) (cid:15) (cid:15) (cid:15) K ( π k ( X ) , k )2. We construct F k +1 and f k +1 as a pullback of the twisted Cartesian prod-uct: K ( π k ( X ) , k − ∼ = (cid:47) (cid:47) (cid:15) (cid:15) K ( π k ( X ) , k − (cid:15) (cid:15) F k +1 := F k × τ (cid:48) K ( π k ( X ) , k − (cid:47) (cid:47) f k +1 (cid:15) (cid:15) (cid:15) (cid:15) K ( π k ( X ) , k ) × τ K ( π k ( X ) , k − δ (cid:15) (cid:15) (cid:15) (cid:15) F k ϕ k (cid:47) (cid:47) K ( π k ( X ) , k ) . It can be shown that the pullback, i.e. simplicial subset of pairs ( x, y ) ∈ F k × E ( π k ( X ) , k −
1) such that δ ( y ) = ϕ k ( x ), can be identified with the twistedproduct as above [36], where the twisting operator τ (cid:48) is defined as τ ϕ k .To show correctness of the algorithm, we assume inductively, that F k haspolynomial-time effective homology. According to [9, Section 3.8], the simpli-cial sets K ( π k ( X ) , k − E ( π k ( X ) , k − K ( π k ( X ) , k ) have polynomial-timeeffective homology and maps ϕ k , δ are polynomial-time. Further, they are allobtained by an algorithm that runs in polynomial time.17s F k +1 is constructed as a twisted product of F k with K ( π k ( X ) , k ), Corol-lary 3.18 of [9] implies that F k +1 has polynomial-time effective homology and f k +1 is a polynomial-time map. The sequence of simplicial sets F k +1 f k +1 (cid:47) (cid:47) F k ϕ k (cid:47) (cid:47) K ( π k ( X ) , k ) inducesthe long exact sequence of homotopy groups · · · (cid:47) (cid:47) π i ( F k +1 ) f k +1 ∗ (cid:47) (cid:47) π i ( F k ) ϕ k ∗ (cid:47) (cid:47) π i ( K ( π k ( X ) , k )) (cid:47) (cid:47) π i − ( F k +1 ) (cid:47) (cid:47) · · · The reason why this is the case follows from a rather technical argument thatidentifies the simplicial set F k +1 with a so called homotopy fiber of the map ϕ k : F k → K ( π k ( X ) , k ). In more detail, the category of simplicial sets is rightproper [20, II.8.67] and map δ is a so-called Kan fibration [36, 23]. Thismakes the pullback F k +1 coincide with so-called homotopy pullback. Further,the simplicial set E ( π k ( X ) , k −
1) is contractible, hence the homotopy pullbackis a homotopy fiber. The induced exact sequence is due to [38, chapter I.3].The inductive assumption, together with the fact that ϕ k induces an isomor-phism ϕ k ∗ : π k ( F k ) → π k ( K ( π k ( X ) , k )) imply that f k induces an isomorphism π j ( F k +1 ) → π j ( F k ) for j > k and π j ( F k +1 ) = 0 for j ≤ k .The lemma implies that the simplicial sets F k have polynomial-time effectivehomology and maps ψ k = f k f k − . . . f are polynomial-time as they are definedas a composition of polynomial-time maps f i .The following theorem is a key ingredient of our algorithm. Theorem 4.2 (Effective Hurewicz Inverse) . Let d > be fixed and F be an ( d − -connected -reduced simplicial set parametrized by a set I , with polynomial-time homology and polynomially contractible loops.Then there exists an algorithm that, for a given d -cycle z ∈ Z d ( F ( I )) , out-puts a simplicial model Σ d of the d -sphere and a simplicial map Σ d → F ( I ) whose homotopy class is the Hurewicz inverse of [ z ] ∈ H d ( F ( I )) .Moreover, the time complexity is bounded by an exponential of a polynomialfunction in size( I ) + size( z ) . The construction of an effective Hurewicz inverse is the main result of [3]and further details are provided in Section 6. It exploits a combinatorial versionof Hurewicz theorem given by Kan in [26] where π d ( F ) is described in termsof π d − ( (cid:103) GF ) where (cid:103) GF is a non-commutative simplicial group that models theloop space of F . Kan showed that the Hurewicz isomorphism can be identifiedwith a map H d − ( (cid:103) GF ) → H d − ( (cid:103) AF ) induced by Abelianization. Berger thendescribes the inverse of the Hurewicz homomorphism as a composition of the We remark that the paper [9] uses a different formalization of twised cartesian productthan the one employed by us. However, the paper [14], on which the Corollary 3.18 of [9]is based, can be reformulated in context of the definition used here. We do not provide fulldetails, only remark that one has to make a choice of
Eilenberg-Zilber reduction data thatcorresponds to the definition of twisted cartesian product. , , π d ( F ) H d ( F ) h − (cid:111) (cid:111) (cid:15) (cid:15) H d − ( (cid:103) GF ) (cid:79) (cid:79) H d − ( (cid:103) AF ) . (cid:111) (cid:111) Arrow 1 is induced by a chain homotopy equivalence and arrow 3 by Berger’sexplicit geometric model of the loop space. To algorithmize arrow 2, we needan algebraic machinery that includes an explicit contraction of k -loops in (cid:103) GF for all k < d −
1. Those are based partially on linear computations in theAbelian group (cid:103) AF and partially on explicit inductive formulas dealing withcommutators. The lowest-dimensional contraction operation, however, cannotbe algorithmized, without some external input. The possibility of providing itis is the content of the following claim: Lemma 4.3.
Let d ≥ be a fixed integer and I be the set of all -connected -reduced finite simplicial sets with an explicit loop contraction c . Then thesimplicial set F d from Lemma 4.1, parametrized by I , has polynomial-time con-tractible loops. The proof is constructive, based on explicit formulas in our model of F d : thedetails are in Section 7.The core of the algorithm we will describe works with simplicial sets andsimplicial maps between them. If our input is a simplicial complex, we needtools to convert them into maps between simplicial complexes. The next twolemmas address this. Lemma 4.4.
Let Y be a finite simplicial set. Then there exists a polynomial-time algorithm that computes a simplicial complex Y sc with a given orientationof each simplex, and a map γ : Y sc → Y (still understood to be a map be-tween simplicial sets) such that the geometric realization of γ is homotopic toa homeomorphism. This construction is given in [8, Appendix B]. Explicitly, the simplicialcomplex Y sc is defined to be Y sc := B ∗ ( Sd ( Y )), where Sd is the barycentricsubdivision functor and B ∗ a functor introduced in [25]: Y sc can be constructedrecursively by adding a vertex v σ for each nondegenerate simplex σ ∈ Sd ( Y ) andreplacing σ by the cone with apex v σ over B ∗ ( ∂σ ). The subdivision Sd ( Y ) isa regular simplicial set and B ∗ ( Sd ( Y )) coincides with the flag simplicial complexof the poset of nondegenerate simplices of Sd ( Y ). It follows that the geometric A version of this lemma is given as [8, Proposition 3.5]. However, we also need the factthat | Y sc | is homeomorphic to | Y | , which is not explicitly mentioned in this reference, butfollows easily from the construction. | Y sc | is homeomorphic to | Y | . Simplices of Y sc are naturally ori-ented and the explicit description of γ is given in [8, p. 61] and the referencestherein.In our main algoritm, Y = Σ d will be a triangulation of the d -sphere and X a simplicial set derived from a simplicial complex X sc by contracting itsspanning tree into a point. The following lemma shows that we can converta map Σ sc → X into a map (Σ sc ) (cid:48) → X sc between simplicial complexes. Lemma 4.5.
Let d > be fixed. Assume that X sc is a given simplicial complexwith a chosen ordering of vertices and a maximal spanning tree T ; we denote theunderlying simplicial set by X ss . Let p : X ss → X := X ss /T be the projection tothe associated -reduced simplicial set. Let Σ be a given d -dimensional simplicialcomplex with a chosen orientation of each simplex, Σ ss the induced simplicialset, and f : Σ ss → X a simplicial map.Then there exists a subdivision Sd(Σ) and a simplicial map f (cid:48) : Sd(Σ) → X sc between simplicial complexes such that | Σ | = | Sd(Σ) | | f (cid:48) | → | X sc | | p | → | X | is homotopic to | Σ ss | | f | → | X | . Moreover, f (cid:48) can be computed in polynomial time,assuming an encoding of the input f, Σ , X sc , X and T . Thus if Σ is a sphere and f corresponds to a homotopy generator, f (cid:48) isthe corresponding homotopy generator represented as a simplicial map betweensimplicial complexes. We remark that the algorithm we describe works even if d is a part of the input, but the time complexity would be exponential in general,as the number of vertices in our subdivision Sd(Σ) would grow exponentiallywith d .The proof of Lemma 4.5 is given in Section 8. Proof of Theorem A.1.
First assume that a finite simplicial complex X sc isgiven together with a loop contraction. Then the algorithm goes as follows.1. We choose an ordering of vertices and convert X sc into a simplicial set.Choosing a spanning tree and contracting it to a point creates a 0-reducedsimplicial set X homotopy equivalent to X sc . By Lemma 3.7, we canconvert the input data into a list c ( α ) for all generators α of GX inpolynomial time.2. We construct the simplicial set F d from Lemma 4.1 as simplicial set withpolynomial-time effective homology. Hence by Lemma 3.1 we can compute The subdivision Sd ( Y ) has geometric realisation homeomorphic to | Y | by [18, Thm 4.6.4].The realisation of Sd ( X ) is a regular CW complex and B ∗ ( Sd ( Y )) coincides with the firstderived subdivision of this regular CW complex, as defined in [19, p. 137]. The geometricrealisation of the resulting simplicial complex is still homeomorphic to | Y | and | Sd ( Y ) | by [19,Prop. 5.3.8]. The constructed map f does not necessarily preserves orientations: it only maps simplicesto simplices. H d ( F d ) in time polynomial in size( X ). Due to Lemma 4.3and Theorem 4.2, we can convert these homology generators to homotopygenerators Σ dj → F d in time exponential in P (size( X ) + size( c )) where P is a polynomial.3. We compose the representatives of π d ( F d ) with ψ d to obtain represen-tatives Σ dj → X of the generators of π d ( X ), another polynomial-timeoperation. This way, we compute explicit homotopy generators as mapsinto the simplicial set X .4. We use Lemma 4.4 to compute simplicial complexes Σ scj and maps Σ scj → Σ d homotopic to homeomorphisms. The compositions Σ scj → Σ dj → X stillrepresent a set of homotopy generators. Finally, by Lemma 4.5, we cancompute, for each j , a subdivision of the sphere Σ scj and a simplicial mapfrom this subdivision into the simplicial complex X sc , in time polynomialin the size of the representatives Σ scj → X .In case when the input is a 0-reduced simplicial set X with a loop contraction c , only steps 2 and 3 are performed. In either case, the overall exponentialcomplexity bound comes from Berger’s Effective Hurewicz inverse theorem. Similarly as in the proof of Theorem A, we prove a slightly stronger version ofTheorem B that also includes finite simplicial complexes.
Theorem B.1.
Let d ≥ be fixed. Then1. there is an infinite family of d -dimensional -connected finite simplicialcomplexes X such that for any simplicial map Σ → X representing agenerator of π d ( X ) , the triangulation Σ of S d on which f is defined hassize at least Ω(size( X )) .2. there is an infinite family of d -dimensional ( d − -connected and ( d − -reduced simplicial sets X such that for any simplicial map Σ → X representing a generator of π d ( X ) , the triangulation Σ of S d on which f is defined has size at least Ω(size( X )) .Consequently, any algorithm for computing simplicial representatives of the gen-erators of π d ( X ) has time complexity at least Ω(size( X )) . The second item immediately implies Theorem B.In the first item, we don’t assume any certificate for 1-connectedness. How-ever, we suspect that any algorithm that computes representatives of π d ( X ) forsimplicial complexes X must necessarily use some explicit certificate of simpleconnectivity, but so far we have not been able to verify this. Lemma 5.1.
Let d ≥ . . There exists a sequence { X k } k ≥ of d -dimensional ( d − -connected sim-plicial complexes, such that H d ( X k ) (cid:39) Z for all k and for any choice ofa cycle z k ∈ Z d ( X k ) generating the homology group, the largest coefficientin z k grows exponentially in size( X k ) .2. There exists a sequence { X k } k ≥ of d -dimensional ( d − -connected and ( d − -reduced simplicial sets, such that H d ( X k ) (cid:39) Z for all k and forany choice of cycles z k ∈ Z d ( X k ) generating the homology, the largestcoefficient in z k grows exponentially in size( X k ) .Proof of Theorem 2 based on Lemma 5.1. Let { X k } k ≥ be the sequence of sim-plicial sets or simplicial complexes from Lemma 5.1. Since they are ( d − π d ( X k ) (cid:39) H d ( X k ) (cid:39) Z . For each k , letΣ k be a simplicial set or simplicial complex with | Σ k | = S d , and f k : Σ k → X k a simplicial map representing a generator of π d ( X k ). The generator of H d (Σ d )contains each non-degenerate d -simplex with a coefficient ± k is a triangulation of the d -sphere and the d -homology of the d -sphere is generated by its fundamental class). The Hurewicz isomorphism π d ( X k ) → H d ( X k ) maps such a representative to the formal sum of simplices f k (cid:55)→ (cid:88) σ is a d − simplex in (Σ k ) ± f k ( σ ) ∈ C d ( X k ) , which represents a generator of H d ( X k ). It follows from Lemma 5.1 that thenumber of d -simplices in Σ k grows exponentially in size( X k ). Moreover, thecomplexity of any algorithm that computes f k : Σ k → X k is at least the size ofΣ k , which completes the proof.It remains to define the sequence from Lemma 5.1: Proof of Lemma 5.1.
1. We begin by constructing for every d ≥
2, a sequence of { X k } k ≥ of ( d − H d ( X k ) (cid:39) Z for all k , and forany choice of a cycle z k ∈ Z d ( X k ) generating the homology group, thelargest coefficient in z k grows exponentially in size( X k ).We start with d = 2. The idea is to glue X k out of k copies of a triangulatedmapping cylinders of a degree 2 map S → S , i.e. k M¨obius bands, andthen fill in the two open ends with one triangle each ( A and B in Figure 2).The case k = 1 is shown in Figure 2. For k ≥
2, we take k copies of thetriangulated M¨obius band and identify the middle circle of each one tothe boundary of the next one.We observe that, up to homotopy equivalence, X k consists of a 2-disc withanother 2-disc which is attached to it via the boundary map S → S ofdegree 2 k . Therefore, X k is simply connected and has H ( X k ) (cid:39) Z and With a slight abuse of language, we assume that each X k not only a simplicial set butalso its algorithmic representation with a specified size such as explained in Section 3.
110 90 1 2 33 4 5 06 7 8 69 6 7 89 1011 + + + ++ ++ +++ −−−−−− − − − A BC + + −− Figure 2: The M¨obius band is the mapping cylinder of a degree 2 map S → S .The triangulation has four layers because starting from the boundary, which is atriangle, we first need to pass to a hexagon in order to cover the middle triangletwice, obtaining the desired degree 2 map. Connecting k copies of the M¨obiusband creates a mapping cylinder of a degree 2 k map, using only linearly (in k )many simplices. Gluing the full triangles A and B to the ends of this mappingcylinder finishes the construction of X k . The red coefficients exhibit a generator ξ of H ( X ) = Z ( X ) (cid:39) Z given as a formal sum of 2-simplices.any homology generator will contain the 2-simplex A with coefficient ± B with coefficient ± k .Similarly for d >
2, the simplicial complex X k is obtained by glueing k copies of a triangulated mapping cylinder of a degree 2 map S d − → S d − ,and the two open ends are filled in with two triangulated d -balls.2. For every k ≥ X k to have one vertex ∗ ,no non-degenerate simplices up to dimension d − k non-degenerate( d − σ , . . . , σ k that are all spherical (that is, for all i, j , d i σ j = ∗ is the degeneracy of the only vertex of X k ), and k + 1 d -simplices A, C , C , . . . , C k − , B such that • d A = σ , d j A = ∗ for j > • d C i = σ i , d C i = σ i +1 , d C i = σ i and d j C i = ∗ for j >
2, and • d B = σ k , d j B = ∗ for j > X k does not have any non-degenerate simplices of dimension larger than d . The relations of a simplicial set are satisfied, because d i d j is trivial inall cases. 23he boundary operator in the associated normalised chain complex C ∗ ( X i )acts on basis elements as • ∂A = σ • ∂C i = 2 σ i − σ i +1 , and • ∂B = σ k .To see that X k is ( d − d >
2, it is enough to prove that H d − ( X k ) is trivial (by 1-reduceness and Hurewicz theorem). This is true,because σ is the boundary of A and for i > σ i is the boundary of thechain 2 i − A − i − C − . . . − C i − − C i − . In the case d = 2, X k is not 1-reduced, but we can show 1-connectednesssimilarly as in the proof of the first part: up to homotopy, X k consists oftwo discs with boundaries together via a map of degree 2 k − .There are no non-degenerate ( d + 1)-simplices, so H d ( X k ) (cid:39) Z d ( X k ) anda simple computation shows that every cycle is a multiple of2 k − A − k − C − k − C − . . . − C k − − B. The computer representation of X k has size that grows linearly with k ,but the coefficients of homology generators grow exponentially with k , sothey grow exponentially with size( X k ). Discussion on optimality. If d = 2 and X is a 1-reduced simplicial set,then generators of H ( X ) can be computed via the Smith normal form of thedifferential ∂ : C ( X ) → C ( X ). Using canonical bases, the matrix of ∂ = d − d + d − d satisfies that the sum of absolute values over each column is atmost 4. We were not able to find any infinite family of such matrices so that thesmallest coefficient in any set of homology generating cycles grows exponentiallywith the size of X (that is, the size of the matrix). However, if a set of homology-generating cycles with subexponential coefficients always exists and can be foundalgorithmically in polynomial time, our main algorithm given as Theorem A isoptimal in this case as well. This is because the exponential complexity of thealgorithm only appears in the geometric realisation of an element of GX sph withlarge (exponential) exponents (see “Arrow 3” in Section 6), and the only sourceof such exponents is the homology H ( AX ) (cid:39) H ( X ). Here we will prove Theorem 4.2 by directly describing the algorithm proposedin [3] and analysing its running time. 24 efinition 6.1.
Let G be a simplicial group. Then the Moore complex ˜ G isa (possibly non-abelian) chain complex defined by ˜ G i := G i ∩ ( (cid:84) j> ker d j ) en-dowed with the differential d : ˜ G i → ˜ G i − . It can be shown that d d = 1 in ˜ G and that Im( d ) is a normal subgroupof ker d so that the homology H ∗ ( ˜ G ) is well defined. Definition 6.2.
Let F be a -reduced simplicial set, GF the associated simpli-cial group from Def. 3.3, and (cid:103) GF its Moore complex. We define AF to be theAbelianization of GF and (cid:103) AF to be the Moore complex of AF . The simplicialgroup AF is also endowed with a chain group structure via ∂ = (cid:80) j ( − j d j . If σ ∈ F k , we will denote by σ the corresponding simplex in GF i − , resp. AF i − . Note that, following Def. 3.3, the “last” differential d k σ in AF k equals d k σ − d k +1 σ . Clearly, the Abelianization map p : GF → GF/ [ GF, GF ] = AF takes (cid:103) GF into (cid:103) AF .Kan showed in [26] that for d > d − F , theHurewicz isomorphism can be identified with the map H d − ( (cid:103) GF ) → H d − ( (cid:103) AF )induced by Abelianization, whereas these groups are naturally isomorphic to π d ( F ) and H d ( F ), respectively. Our strategy is to construct maps representingthe isomorphisms 1 , , π d ( F ) H d ( F ) h − (cid:111) (cid:111) (cid:15) (cid:15) H d − ( (cid:103) GF ) (cid:79) (cid:79) H d − ( (cid:103) AF ) . (cid:111) (cid:111) (3)Here h stands for the Hurewicz isomorphism, 1 is induced by a homotopy equiv-alence of chain complexes, 2 is the inverse of H d − ( p ) where p is the Abelian-ization, and 3 represents an isomorphism between the ( d − (cid:103) GF (that models the loop space of F ) and π d ( F ). The algorithms representing1 , , d → F where | Σ d | = S d . Inwhat follows, we will explicitly describe the effective versions of 1 , , Arrow 1.
Let F be a 0-reduced simplicial set, C ∗ ( F ) be the (unreduced) chain complex of F and AF ∗− the shifted chain complex of AF defined by ( AF ∗− ) i := AX i − .As a chain complex, AF ∗− is a subcomplex of C ∗ ( F ) generated by all simplicesthat are not in the image of the last degeneracy. Let (cid:103) AF ∗− be the Moorecomplex of AF ∗− . 25 emma 6.3. There exists a polynomial-time strong chain deformation retrac-tion ( f, g, h ) : C ∗ ( F ) → (cid:103) AF ∗− . That is, f : C ∗ ( F ) → (cid:103) AF ∗− , g : (cid:103) AF ∗− → C ∗ ( F ) are polynomial-time chain-maps and h : C ∗ ( F ) → C ∗ +1 ( F ) is a polyno-mial map such that f g = id and gf − id = h∂ + ∂h .Proof. First we will define a chain deformation retraction from C ∗ ( F ) to AF ∗− represented by f : C ∗ ( F ) → AF ∗− , g : AF ∗− → C ∗ ( F ) and h : C ∗ ( F ) → C ∗ +1 ( F ).The chain complex AF ∗− consists of Abelian groups AF k − freely generatedby k -simplices in F that are not in the image of the last degeneracy s k − . Ongenerators, we define f ( σ ) = σ whenever σ is a k -simplex not in Im( s k − ) and f ( x ) = 0 otherwise. Deciding whether σ is in the image of s k − amounts todeciding σ = s k − d k σ which can be done in time polynomial in size( I ) + size( σ ),the polynomial depending on k . It follows that f is a locally polynomial map.The remaining maps are defined by g ( σ ) := σ − s k − d k σ and h ( σ ) :=( − k s k σ . These maps are locally polynomial as well and it is a matter ofstraight-forward computations to check that f and g are chain maps, f g = idand g f − id = h ∂ + ∂h .Further, we define another chain deformation retraction from AF to (cid:103) AF .For each p ≥
0, let A p be a chain subcomplex of AF defined by( A p ) k := { x ∈ AF k : d i x = 0 for i > max { k − p, } } that is, the kernel of the p last face operators, not including d ( d i refers hereto the face operators in AF ). Then A p +1 is a chain subcomplex of A p andwe define the maps f p +1 : ( A p ) k → ( A p +1 ) k by f p +1 ( x ) = x − s k − p − d k − p x whenever k − p >
0, and f p +1 ( x ) = x otherwise; g p +1 : A p +1 → A p will bean inclusion, and h p +1 : ( A p ) k → ( A p ) k +1 via h p +1 ( x ) = ( − k − p s k − p x if k − p > f p +1 , g p +1 arechain maps, f p +1 g p +1 = id, g p +1 f p +1 − id = h p +1 ∂ + ∂h p +1 and it is clear that f p +1 , g p +1 , h p +1 are polynomial-time maps.By definition, the Moore complex (cid:103) AF = ∩ p> A p . The strong chain defor-mation retraction ( f, g, h ) from C ∗ ( F ) to ˜ AF ∗− is then defined by the infinitecompositions f := . . . f k +1 f k . . . f f g := g g . . . g k g k +1 . . . and the infinite sum h = h + g h f + ( g g ) h ( f f ) + . . . which are all well-defined, because when applying them to an element x , onlyfinitely many of f j , g j differ from the identity map and only finitely many h j arenonzero. These are the maps f, g, h from the lemma and we need to show thatif the degree k is fixed, then we can evaluate f, g, h on C k ( F ) resp. ˜ AF k − intime polynomial in the input size. However, for fixed k , the definition of f, g, h f i , g i , h i for i < k . Then f, g are composed of k polynomial-timemaps and h is a sum of k polynomial-time maps.The polynomial-time version of arrow 1 is then induced by applying the map f from Lemma 6.3. Arrow 2.
Lemma 6.4 (Boundary certificate) . Let d > be fixed and let F be a ( d − -connected simplicial set with polynomial-time homology. There is an algorithmthat, for j < d − and a cycle z ∈ Z j ( (cid:103) AF ) , computes an element c A ( z ) ∈ (cid:103) AF j +1 such that d c A ( z ) = z . The running time is polynomial in size( z ) + size( I ) .Proof. First note that the ( d − F implies that H j +1 ( F ) (cid:39) H j ( (cid:103) AF ) are trivial for j < d −
1, so each cycle in these dimensions is a boundary.By assumption, F has a polynomial-time homology, which means that thereexists a globally polynomial-time chain complex E ∗ F , a locally polynomial-timechain complex Y and polynomial-time reductions from Y to C ∗ ( F ) and E ∗ FE ∗ F P ⇐⇐ Y P ⇒⇒ C ∗ ( F ) . Let ( f (cid:48) , g (cid:48) , h (cid:48) ) be chain homotopy equivalence of Y and (cid:103) AF ∗− defined as thecomposition of Y ⇒⇒ C ∗ ( F ) and the chain homotopy equivalence of C ∗ ( F ) and (cid:103) AF ∗− described in Lemma 6.3. Further, let f (cid:48)(cid:48) , g (cid:48)(cid:48) , h (cid:48)(cid:48) be the maps definingthe reduction Y ⇒⇒ E ∗ F : all of these maps are polynomial-time.Let j < d − z ∈ Z j ( (cid:103) AF ), z = (cid:80) j k j y j . Then the element f (cid:48)(cid:48) g (cid:48) ( z ) is acycle in E j +1 F and can be computed in time polynomial in size( z ) + size( I ). Inparticular, the size of f (cid:48)(cid:48) g (cid:48) ( z ) is bounded by such polynomial. The number ofgenerators of E j +2 F and E j +1 F is polynomial in size( I ) and we can compute,in time polynomial in size( I ), the boundary matrix of ∂ : E j +2 F → E j +1 F withrespect to the generators.Next we want to find an element t ∈ E j +2 F such that ∂t = f (cid:48)(cid:48) g (cid:48) ( z ). Usinggenerating sets for E j +2 F , E j +1 F , this reduces to a linear system of Diophantineequations and can be solved in time polynomial in the size of the ∂ -matrix andthe right hand side f (cid:48)(cid:48) g (cid:48) ( z ) [28].Finally, we claim that c A ( z ) := f (cid:48) g (cid:48)(cid:48) ( t ) − f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) is the desired elementmapped to z by the differential in (cid:103) AF . This follows from a direct computation ∂c A ( z ) := ∂f (cid:48) g (cid:48)(cid:48) ( t ) − ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) == f (cid:48) g (cid:48)(cid:48) ( ∂t ) − ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) == f (cid:48) g (cid:48)(cid:48) f (cid:48)(cid:48) g (cid:48) ( z ) − ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) == f (cid:48) ( h (cid:48)(cid:48) ∂ + ∂h (cid:48)(cid:48) + id) g (cid:48) ( z ) − ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) == f (cid:48) h (cid:48)(cid:48) g (cid:48) ∂z + ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) + f (cid:48) g (cid:48) ( z ) − ∂f (cid:48) h (cid:48)(cid:48) g (cid:48) ( z ) ==0 + f (cid:48) g (cid:48) ( z ) = z The computation of t as well as all involved maps are polynomial-time, hencethe computation of c A ( z ) is polynomial too.27he next lemma will be needed as an auxiliary tool later. Lemma 6.5.
Let S be a countable set with a given encoding, G be the free (non-abelian) group generated by S , and define size( (cid:81) j s k j j ) := (cid:80) j (size( s j )+size( k j )) .Let G (cid:48) := [ G, G ] be its commutator subgroup.Then there exists a polynomial-time algorithm that for an element g = (cid:81) j s k j j in G (cid:48) ⊆ G , computes elements a i , b i ∈ G such that g = (cid:81) j [ a j , b j ] . In other words, we can decompose commutator elements into simple com-mutators in polynomial-time at most.
Proof.
Let us choose a linear ordering on S and let g = (cid:81) j s k j j be in G (cid:48) : thatis, for each j , the exponents { k j (cid:48) : s j (cid:48) = s j } sum up to zero. We will presenta bubble-sort type algorithm for sorting elements in g . Going from the left toright, we will always swap s k j j and s k +1 j +1 whenever s j +1 < s j . Such swap alwayscreates a commutator, but that will immediately be moved to the initial segmentof commutators.More precisely, assume that Init is the initial segment, x = s k j j and y = s k j +1 j +1 should be swapped, Rest represent the segment behind y , and Commutators isa final segment of commutators. The swapping will consists of these steps:Init x y Rest Commutators (cid:55)→
Init y x [ x − , y − ] Rest Commutators (cid:55)→ Init y x
Rest (cid:0) [ x − , y − ] [[ y − , x − ] , Rest − ] Commutators (cid:1) where the parenthesis enclose a new segment of commutators. Before the paren-thesis, x and y are swapped. Each such swap requires enhancing the commutatorsection by two new commutators of size at most size( g ), hence each such swaphas complexity linear in size( g ).Let as call everything before the commutator section a “regular section”.Going from left to right and performing these swaps will ensure that the largestelement will be at the end of the regular section. But no later then that, thelargest element y largest disappears from the regular section completely, becauseall of its exponents add up to 0. Again, starting from the left and performinganother round of swaps will ensure that the second-largest elements disappearfrom the regular section; repeating this, all the regular section will eventuallydisappear which will happen in at most size( g ) swaps in total. Each swap hascomplexity linear in size( g ) and the overall time complexity is not worse thancubic. Lemma 6.6.
Assume that F is a parametrized simplicial set with polynomiallycontractible loops. Let k > , γ ∈ GF k be spherical and α ∈ GF k is arbitrary.There is a polynomial-time algorithm that computes δ ∈ GF (cid:48) k +1 such that d δ =[ α, γ ] and d i δ = 1 for all i > . In other words, a simple commutator of a spherical element with anotherelement can always be “contracted” in GF (cid:48) in polynomial time. Our proofroughly follows the construction in Kan [26, Sec. 8]28 roof. For x ∈ GF , we will denote by c x the element of (cid:103) GF such that d c x = x : this can be computed in polynomial-time by the assumption on poly-nomial loop contractions. For the simplex α ∈ GF k , we define ( k + 1)-simplices β , . . . , β k by β k := s k c d k α and inductively β j − := ( s j d j β j ) · ( s j α − ) · ( s j − α )for j < k . Then the following relations hold: • d β = α . • d j β j = d j β j − , 1 ≤ j ≤ k • d k +1 β k = 1.The second and third equations are a matter of direct computation, while thefirst follows from the more general relation d j +10 β j = d j α which can be provedby induction. If k is fixed, then all β , . . . , β k can be computed in polynomialtime.The desired element δ is then the alternating product δ := [ β , s γ ] [ β , s γ ] − . . . [ β k , s k γ ] ± . Lemma 6.7.
Under the assumptions of Theorem 4.2, there exist homomor-phisms c j : GF j → GF j +1 for ≤ j < d − such that1. d c j = id ,2. d i c j = c j − d i − , < i ≤ j + 1 , and3. c j s i = s i +1 c j − for < j < d − and ≤ i < j ,4. d c ( x ) = 1 for all x ∈ GF .If d is fixed and x ∈ GF j , j < d − , then c j ( x ) can be computed in polynomialtime.Proof. The homomorphism c can be constructed directly from the assumptionon polynomial contractibility of loops. We have a canonical basis of GF con-sisting of all non-degenerate 1-simplices of F . For σ ∈ F , we denote by σ thecorresponding generator of GF . The we define c ( (cid:81) σ k j j ) to be (cid:81) b k j j where b j is the element of GF such that d b j = σ j and d b j = 1.In what follows, assume that 1 ≤ k < d − c i have been defined for all i < k . We will define c k in the following steps. Step 1.
Contractible elements.Let x ∈ GF k . We will say that x is contractible and y ∈ GF k +1 is a contrac-tion of x , if d y = x and d i y = c k − d i − x for all i > c k will be to find a contraction h for eachbasis element (( k + 1)-simplex) g ∈ GF k and define c k ( g ) := h . This will enforce Kan uses a slightly different convention in [26] but the resulting properties are the same.The sequence β , . . . , β k can be interpreted as a discrete path from α to the identity element. g is degenerate, the contraction willbe chosen in such a way that property 3 holds too; otherwise it holds vacuously.Property 4 only deals with c and is satisfied by the definition of loop contraction(a polynomial-time c is given as an input in Theorem 4.2). Step 2.
Contraction of degenerate elements.Let g = s i y be a basis element of GF k , y ∈ GF k − . Then g can be uniquelyexpressed as s j z where j is the maximal i such that g ∈ Im( s i ). We then define c k ( g ) := s j +1 c k − ( z ). Note that d c k ( g ) = d s j +1 c k − ( z ) = s j d c k − ( z ) = s j z = g, so property 1 is satisfied. To verify property 2, first assume that i ∈ { j +1 , j +2 } .Then we have d i c k ( g ) = d i s j +1 c k − ( z ) = c k − ( z ) = c k − d i − s j z = c k − d i − g. This fully covers the case k = 1, because then the only possibility is j = 0 and i ∈ { , } . Further, let k >
1. If i ≤ j , then we have d i c k g = d i c k s j z = d i s j +1 c k − ( z ) = s j d i c k − ( z ) = s j c k − d i − z == c k − s j − d i − z = c k − d i − s j z = c k − d i − g and if i > j +2, then the computation is completely analogous, using the relation d i s j +1 = s j +1 d i − instead.So far, we have shown that c k ( g ) := s j +1 c k − g is a contraction of g . Itremains to show property 3. That is, we have to show that if g = s j z can alsobe expressed as s i y , then c k ( s i y ) = s i +1 c k − y .The degenerate element g has a unique expression g = s i u . . . s i s i v where i < i < . . . < i u = j and is expressible as s i x iff i = i j for some j = 0 , , . . . , u .Choosing such i < j , we can rewrite g as g = s j s i v for some v and then g = s i s j − v , so that y = s j − v and z = s i v . Then we again use induction toshow c k ( s i y ) = s j +1 c k − ( z ) = s j +1 c k − s i v = s j +1 s i +1 c k − v == s i +1 s j c k − v = s i +1 c k − s j − v = s i +1 c k − y as required. Step 3.
Decomposition into spherical and conical parts.We will call an element ˆ x ∈ GF k to be conical , if it is a product of elementsthat are either degenerate or in the image of c k − . Let x ∈ GF k be arbitrary.We define x k := x and inductively x i − := x i ( s i − d i x i ) − . In this way we obtain x , . . . , x n such that x i is in the kernel of d j for j > i and x = x y where y isa product of degenerate simplices. Further, let x s := x ( c k − d x ) − . A simplecomputation shows that x s is spherical , that is, d i x s = 1 for all i . We obtainan equation x = x s ˆ x where ˆ x = ( c k − ( d x ) y ; this is a decomposition of x intoa spherical part x s and a conical element ˆ x .We will define c k on non-degenerate basis elements g = σ by first decom-posing g = g S ˆ g into a spherical and conical part, finding contractions h of30 S and h of ˆ g , and defining c k ( g ) := h h . Then c k ( g ) is a contraction of g and hence satisfies properties 1 and 2: property 3 is vacuously true once g isnon-degenerate. Step 4.
Contraction of the conical part.Let ˆ x := c k − ( d x ) y be the conical part defined in the previous step. Byconstruction, x ∈ ˜ GF k and y is a product of degenerate elements s i u . . . s i l u l .We define the contraction of c k − ( d x ) to be˜ c k ( c k − ( d x )) := s c k − ( d x ) . Note that this satisfies property 1 as d ˜ c k c k − ( d x ) = c k − ( d x ). For property2, we first verify d ˜ c k c k − ( d x ) = d s c k − ( d x ) = c k − ( d x ) = c k − d c k − ( d x ) . Not let i ≥
2. If k = 1, then the remaining face operator is d and we have d ˜ c c ( d x ) = d s c ( d x ) = s d c ( d x ) = 1 = c d c ( d x )using axiom 4 for c . Finally, if i ≥ k ≥
2, we have d i ˜ c k c k − ( d x ) = d i s c k − ( d x ) = s d i − c k − ( d x ) = s c k − d i − d x == s c k − d d i − x = s c k − d c k − c k − d d i − x == c k − c k − d i − d x = c k − d i − c k − ( d x ) , where we exploited the fact that x ∈ (cid:103) GF k and hence d u x = 1 for u ≥ y has already been defined in Step2, so we can define a contraction of c k − ( d x ) y to be s c k − ( d x ) c k ( y ). Step 5.
Contraction of commutators.Let g (cid:48) ∈ GF (cid:48) k be an element of the commutator subgroup. By Lemma 6.5,we can algorithmically decompose g (cid:48) into a product of simple commutators, soto find a contraction of g (cid:48) , it is sufficient to find a contraction of each simplecommutator [ x, y ] in this decomposition.Let x = x S ˆ x and y = y S ˆ y be the decompositions into spherical and conicalparts described in Step 3. Using the notation b a := bab − , we can decompose[ x, y ] as follows [3, p. 60]:[ x, y ] = ([ x, y ][ˆ y, x ]) ([ x, ˆ y ][ˆ y, ˆ x ]) [ˆ x, ˆ y ] = [ xy x − , xy ( y − ˆ y )] [ x ˆ y, x ( x − ˆ x )] [ˆ x, ˆ y ] . (4)Both x − ˆ x and y − ˆ y are spherical simplices and so are their conjugations. Itfollows that equation (4) can be rewritten to [ x, y ] = [ α , γ ] [ α , γ ] [ˆ x, ˆ y ] where γ and γ are spherical. All of these decompositions are done by elementaryformulas and are polynomial-time in the size of x and y .By Lemma 6.6 we can find an elements λ i ∈ (cid:103) GF k +1 such that d λ i = [ α i , γ i ], i = 1 ,
2, in polynomial time. Further, both ˜ x and ˜ y are conical and they arein the form ˜ x = c ( d x ) x deg where x ∈ (cid:103) GF k and x deg is degenerate; similardecomposition holds for y . In Step 4 we showed how to compute elements c x c y such that c x , c y is a contraction of ˆ x , ˆ y , respectively. Then [ c x , c y ] is acontraction of [ˆ x, ˆ y ] and λ λ [ c x , c y ] is a contraction of [ x, y ]. Step 6.
Contraction of spherical elements.The last missing step is to compute a contraction of the spherical element g S where g S is the spherical part of a basis element g ∈ GF k .Let us denote by p the projection GF p → AF . The projection z := p ( g S ) isin the kernel of all face operators and hence a cycle in (cid:103) AF k . By Lemma 6.4,we can compute t := c Ak ( z ) ∈ (cid:103) AF k +1 such that d t = z , in polynomial time.Let h ∈ GF k +1 be any p -preimage of t . Let h k := h and inductively define h j − := h j ( s j − d j h j ) − for j < k . Then h is in the kernel of all faces except d , that is, h ∈ (cid:103) GF k +1 . It follows that p ( h ) ∈ (cid:103) AF k +1 is in the kernel of allfaces except d . We claim that p ( h ) = t .This can be shown as follows: assumethat p ( h j ) = t , then p ( h j − ) = p ( h j ) + p ( s j − d j h − j ) = t + s j − d j t = t + 0 = t .We have the following commutative diagram: h (cid:31) (cid:47) (cid:47) t (cid:103) GF (cid:48) k +1 (cid:31) (cid:127) (cid:47) (cid:47) d (cid:15) (cid:15) (cid:103) GF k +1 p (cid:47) (cid:47) (cid:47) (cid:47) d (cid:15) (cid:15) (cid:103) AF k +1 d (cid:15) (cid:15) (cid:103) GF (cid:48) k (cid:31) (cid:127) (cid:47) (cid:47) (cid:103) GF k p (cid:47) (cid:47) (cid:47) (cid:47) (cid:103) AF k g S (cid:31) (cid:47) (cid:47) z Both g S and d h are mapped by p to the same element z : it follows that g S ( d h ) − is mapped by p to zero and hence is an element of the commutatorsubgroup. Let ˜ h be the contraction of g S ( d h ) − , computed in Step 5, andfinally let h := ˜ hh . Then h is an element of (cid:103) GF k +1 and a direct computationshows that d h = g S as desired.This completes the construction of c k : for each non-degenerate basis element g of GF k , c k ( g ) is defined to be the product of the contraction of g S and thecontraction of ˆ g .All the subroutines described in the above steps are polynomial-time. Thuswe showed that if there exists a polynomial-time algorithm for c k − , then therealso exists a polynomial-time algorithm for c k . The existence of a polynomial-time c follows from the assumption on polynomial loop contractibility and d isfixed, thus there exists a polynomial-time algorithm that for x ∈ GF j computes c j ( x ) for each j < d − Lemma 6.8 (Construction of arrow 2) . Under the assumption of Theorem 4.2,let z ∈ Z d − ( (cid:103) AF ) be a cycle. Then there exists a polynomial-time algorithmthat computes a cycle x ∈ Z d − ( (cid:103) GF ) such that the Abelianization of x is z . For t = (cid:80) j k j σ j , we may choose h = (cid:81) j σ k j j (choosing any order of the simplices). The connectivity assumption on F was exploited in the existence of the contraction c Aj on the Abelian part. z (cid:55)→ x is hence an effective inverse of the isomorphism H d − ( (cid:103) GF ) → H d − ( (cid:103) AF )on the level of representatives. Proof.
Let c d − be the contraction from Lemma 6.7 and z ∈ Z d − ( (cid:103) AF ) be acycle. First choose y ∈ GF d − such that p ( y ) = z . Creating the sequence y n := y , y j − := y j s j − d j y − j for decreasing j , yields an element y ∈ (cid:103) GF d − that is still mapped to z by p , similarly as in Step 4 of Lemma 6.7. The equation pd ( y ) = d p ( y ) = d z = 0 shows that d y is in the commutator subgroup (cid:103) GF (cid:48) d − . We define x := y c d − ( d y ) − : this is already a cycle in (cid:103) GF d − and p ( x ) = p ( y ) = z . Arrow 3.
The construction of map 3 is one of the main results from [4] and involves furtherdefinitions. Here, we describe the main points of the construction only whiledetails are given in later sections.Given a 0-reduced simplicial set F , there exists a simplicial group Ω F that isa discrete analog of a loopspace of F i.e. π d − (Ω F ) ∼ = π d ( F ). Further, there is ahomomorphism of simplicial groups t : GF → Ω F that induces an isomorphismon the level of homotopy groups. This is described in [4, Proposition 3.3].The homomorphism t is given later by formula (6) and the simplicial setΩ F is described in the next section. Here, we remark that the size of t ( g ) isexponential in size of g .Finally, Lemma 6.13 describes an algorithm that for a spherical element γ ∈ Ω F d − constructs a simplicial map γ sph : Σ d ( γ ) → F such that π d − (Ω F ) (cid:51) [ γ ] (cid:39) [ γ sph ] ∈ π d ( F ).The size of γ sph is polynomial in size( γ ). Hence, given a spherical g ∈ (cid:103) GF d − ,the algorithm produces t ( g ) sph : Σ d ( t ( g )) → F that is exponential with respectto size( g ). Berger’s model of the loop space.Definition 6.9 (Oriented multigraph on X n ) . Let X be a 0-reduced simplicialset. We define a directed multigraph M X n = ( V n , E n ) , where the set of vertices V n = X n and the set of edges E n is given by E n = { [ x, i ] (cid:15) | x ∈ X n +1 , ≤ i ≤ n, (cid:15) ∈ { , − }} . We define maps source , target : E n → V n by setting source [ x, i ] = d i +1 x , target [ x, i ] = d i x and source [ x, i ] − = target [ x, i ] and target [ x, i ] − = source [ x, i ] .An edge [ x, i ] (cid:15) ∈ E n is called compressible , if x = s i x (cid:48) for some x (cid:48) ∈ X n . Definition 6.10 (Paths) . Let X ∈ sSet . A sequence of edges in M X n γ = [ x , i ] (cid:15) [ x , i ] (cid:15) · · · [ x k , i k ] (cid:15) k (5)33 s called an n - path , if target [ x j , i j ] (cid:15) j = source [ x j +1 , i j +1 ] (cid:15) j +1 , ≤ j < k .Moreover, for every x ∈ V n = X n we define a path of length zero x with theproperty source x = x = target x and relations a x = a whenever target a = x and x b = b whenever source b = x .The set of paths on M X n is denoted by IX n . Let γ ∈ IX n by as in (5). Wedefine source γ = source [ x , i ] (cid:15) and target γ = target [ x k , i k ] (cid:15) k . The inverse of γ , denoted γ − , is defined as γ − = [ x k , i k ] − (cid:15) k · · · [ x , i ] − (cid:15) . if γ = x , then γ − = γ . Note that each path is either equal to x for some x or can be represented in a form such as (5), without any units. For algorithmic purposes, we assume that a path γ = [ x , i ] (cid:15) [ x , i ] (cid:15) · · · [ x k , i k ] (cid:15) k is represented as a list of triples ( x j , i j , (cid:15) j ) and has sizesize( γ ) := (cid:88) j size( x j ) + size( i j ) + size( (cid:15) j ) , which is bounded by a linear function in (cid:80) j size( x j ).Given an edge [ x, i ] (cid:15) ∈ M X n , we define operators d , . . . d n : E n → IX n − and s , . . . , s n : E n → IX n +1 called face and degeneracy operators, respectively. These are given as follows d j [ x, i ] (cid:15) = [ d j x, i − (cid:15) , j < i ;1 d i d i +1 x , i = j ;[ d j +1 x, i ] (cid:15) , j > i. s j [ x, i ] (cid:15) = [ s j x, i + 1] (cid:15) , j < i ;[ s i x, i + 1][ s i +1 x, i ]) (cid:15) , i = j ;[ s j +1 x, i ] (cid:15) , j > i. One can now extend the definition of face and degeneracy operators to paths,i.e. we define operators d , . . . d n : IX n → IX n − and s , . . . , s n : IX n → IX n +1 d j γ = (cid:26) d j ([ x , i ] (cid:15) ) d j ([ x , i ] (cid:15) ) · · · d j ([ x k , i k ] (cid:15) k ) if γ = [ x , i ] (cid:15) [ x , i ] (cid:15) · · · [ x k , i k ] (cid:15) k , d j x if γ = 1 x , x ∈ X n .s j γ = (cid:26) s j ([ x , i ] (cid:15) ) s j ([ x , i ] (cid:15) ) · · · s j ([ x k , i k ] (cid:15) k ) if γ = [ x , i ] (cid:15) [ x , i ] (cid:15) · · · [ x k , i k ] (cid:15) k s j x if γ = 1 x , x ∈ X n . With the operators defined above, one can see that IX is in fact a simplicialset.For any γ, γ (cid:48) ∈ IX such that target γ = source γ (cid:48) , we define a composition γ · γ (cid:48) in an obvious way.If the simplicial set X is 0-reduced, we denote the unique basepoint ∗ ∈ X . Abusing the notation, we denote the iterated degeneracy of the basepoint s · · · s ∗ (cid:124) (cid:123)(cid:122) (cid:125) k − times ∈ X k by ∗ as well. With that in mind, we define simplicial subsets P X , Ω X of IX as follows: P X = { γ ∈ IX | target γ = ∗} Ω X = { γ ∈ IX | source γ = ∗ = target γ } . We remark that simplicial sets
P X, Ω X intuitively capture the idea of pathspaceand loopspace in a simplicial setting. 34 efinition 6.11. A path γ = [ x , i ] (cid:15) [ x , i ] (cid:15) · · · [ x k , i k ] (cid:15) k ∈ IX is called re-duced , if for every ≤ j < k the following condition holds: ( x j = x j +1 & i j = i j +1 ) ⇒ (cid:15) j = (cid:15) j +1 . e.g. an edge in the path γ is never followed by its inverse.An edge [ x, i ] (cid:15) ∈ E n is called compressible , if x = s i x (cid:48) for some x (cid:48) ∈ X n . Apath is compressed if it does not contain any compressible edge. We define relation ∼ R on IX (or rather on each IX n ) as a relation generatedby [ x, i ] (cid:15) [ x, i ] − (cid:15) ∼ R source ([ x,i ] (cid:15) ) , n ∈ N , [ x, i ] (cid:15) ∈ E n . Similarly, we define ∼ C on IX as a relation generated by[ x, i ] (cid:15) ∼ C source ([ x,i ] (cid:15) ) , if [ x, i ] (cid:15) ∈ E n is compressible . We finally define IX = ( IX/ ∼ C ) / ∼ R . Similarly, one defines P X, Ω X .For γ, γ (cid:48) ∈ IX n , we write γ ∼ γ (cid:48) if they represent the same element in IX n .The symbol γ , denotes the (unique) compressed and reduced path such that γ ∼ γ . One can see IX ( P X, Ω X ) as the set of reduced and compressed pathsin IX ( P X, Ω X ).In a natural way, we can extend the definition of face and degeneracy oper-ators d i , s i on sets IX ( P X ,Ω X ) by setting d i γ = d i γ and s i γ = s i γ . One cancheck that this turns IX , P X and Ω X into simplicial sets.Similarly, we define operation · : Ω X n × Ω X n → Ω X n by γ · γ (cid:48) (cid:55)→ γγ (cid:48) , i.e. wefirst compose the loops and then assign the appropriate compressed and reducedrepresentative. With the operation defined as above, Ω X is a simplicial group. Homomorphism t : GX → Ω X . We first describe how to any given x ∈ X n assign a path γ x ∈ P X n with the property source γ x = x and target γ x = ∗ :For x ∈ X n , n >
0, the 0-reducedness of X gives us d i d i · · · d i n x = ∗ , here i j ∈ { , . . . , j } , 0 < j ≤ n . In particular, d d · · · d n − x = ∗ . Using this, wedefine γ x = [ s n x, n − s n s n − d n − x, n − · · · [ s n s n − · · · s d d · · · d n − x, . Ignoring the degeneracies, one can see the sequence of edges as a path x → d n − x → d n − d n − x → · · · → d d · · · d n − x. We define the homomorphism t on the generators of GX n , i.e. on the ele-ments x , where x ∈ X n +1 as follows: t ( x ) = γ − d n +1 x [ x, n ] γ d n x . (6)This is an element of Ω X n . 35he algorithm representing the map t has exponential time complexity dueto the fact that an element σ k with size size( σ ) + size( k ) is mapped to γ − d n +1 x [ x, n ] γ d n x . . . γ − d n +1 x [ x, n ] γ d n x (cid:124) (cid:123)(cid:122) (cid:125) k times which in general can have size proportional to k . Assuming an encoding ofintegers such that size( k ) (cid:39) ln( k ), this amounts to an exponential increase. Universal preimage of a path.
Intuitively, one can think of the simplicialset IX of paths as of a discretized version of space of continuous maps | X | [0 , .In particular, γ ∈ IX d − is a walk through a sequence of d -simplices in X that connect source γ with target γ . However, in the continuous case an element µ ∈ | X | [0 , corresponds to a continuous map µ : [0 , → | X | . We want topush the parallels further, namely, given any nontrivial γ ∈ IX d − , we aim todefine a simplicial set Dom ( γ ) and a simplicial map γ map : Dom ( γ ) → X withthe following properties:1. | Dom ( γ ) | = D d .2. γ map maps Dom ( γ ) to the set of simplices contained in the path γ .We will utilize the following construction given in [4]. Definition 6.12.
Let γ ∈ IX d − . We define Dom ( γ ) and γ map as follows.Suppose, that γ = [ y , i ] (cid:15) [ y , i ] (cid:15) · · · [ y k , i k ] (cid:15) k . For every edge [ y j , i j ] (cid:15) j , let α j be the simplicial map ∆ d → y j sending the nondegenerate d simplex in ∆ d to y j . We define Dom ( γ ) as a quotient of the disjoint union of k copies of ∆ d : Dom ( γ ) = k (cid:71) i =1 ∆ d / ∼ where each copy of ∆ d corresponds to a domain of a unique α j and the relationis given by ( α j ) − target ([ y j , i j ] (cid:15) j ) ∼ ( α j +1 ) − source ([ y j +1 , i j +1 ] (cid:15) j +1 ) . The map γ map is induced by the collection of maps α , . . . , α k : (cid:70) ki =1 ∆ d α ,...,α k (cid:39) (cid:39) (cid:15) (cid:15) (cid:15) (cid:15) Dom ( γ ) γ map (cid:47) (cid:47) X. We recall that simplicial set IX was defined as the set of “reduced andcompressed” paths in IX . Similarly, one introduces a reduced and compressedversions of the construction Dom . As a final step we then get By nontrivial we mean that γ (cid:54) = 1 x for any x ∈ X d − . emma 6.13 (Section 2.4 in [4]) . Let γ ∈ Ω X d − such that d i γ = 1 ∈ Ω X for all i . Then the map γ map : Dom ( γ ) → X factorizes through a simplicial setmodel of the sphere Σ d ( γ ) as follows: Dom ( γ ) γ map (cid:39) (cid:39) (cid:15) (cid:15) (cid:15) (cid:15) Σ d ( γ ) γ sph (cid:47) (cid:47) X. Further, π d − (Ω X ) (cid:51) [ γ ] (cid:39) [ γ sph ] ∈ π d ( X ) . We will not give the proof of correctness of Lemma 6.13 (it can be found in[4]). Instead, in the next section, we only describe the algorithmic constructionof γ sph : Σ d ( γ ) → X and give a running time estimate. Algorithm from Lemma 6.13.
The algorithm accepts an element γ ∈ Ω X d − such that d i γ = 1 ∈ Ω X for all i , a spherical element. We divide the algorithm into four steps that correspondto the four step factorization in the following diagram: Dom ( γ ) γ map (cid:31) (cid:31) (cid:15) (cid:15) (cid:15) (cid:15) Dom ( γ ) γ c (cid:39) (cid:39) (cid:15) (cid:15) (cid:15) (cid:15) Dom ( γ ) γ cr (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) X Σ d ( γ ) γ sph (cid:55) (cid:55) Dom ( γ ): We interpret γ as an element in IX and construct γ map : Dom ( γ ) → X .This is clearly linear in the size of γ . Dom ( γ ): The algorithm checks, whether an edge [ y, j ] (cid:15) in d i d i . . . d i (cid:96) γ , where 0 ≤ i < i < . . . < i (cid:96) < ( d − (cid:96) −
2) is compressible , i.e. y = s j d j y . Ifthis is the case, add a corresponding relation on the preimages: α − ( y ) ∼ s j d j α − ( y ). Factoring out the relations, we get a map γ c : Dom ( γ ) → X .Although the number of faces we have to go through is exponential in d ,this is not a problem, since d is deemed as a constant in the algorithm andso is 2 d . Hence the number of operations is again linear in the size of γ . Dom ( γ ): Let k < d . We know that d k γ = 1 ∗ , so after removing all compressible ele-ments from the path d k γ , it will contain a sequence of pairs ([ y i , j i ] (cid:15) i , [ y i , j i ] − (cid:15) i )such that, after removing all [ y u , j u ] ± for all u < v , then [ y v , j v ] (cid:15) v and37 y v , j v ] − (cid:15) v are next to each other. Each such pair ([ y i , j i ] (cid:15) i , [ y i , j i ] − (cid:15) i )corresponds to a pair of indices ( l i , m i ) corresponding to the positions ofthose edges in d k γ . These sequences are not unique, but can be easilyfound in time linear in length( γ ). Then we glue α − l i ( y i ) with α − m i ( y i ) forall i . Performing such identifications for all k defines the new simplicialset Dom ( γ ).Σ d ( γ ): It remains to identify α − ( source γ ) and α − ( target γ ) with the appropriatedegeneracy of the (unique) basepoint. The resulting space | Σ d ( γ ) | is a d -sphere. F d In this section, we show that simplicial sets F k , 2 ≤ k ≤ n constructed algo-rithmically in Section 4 have polynomial-time contractible loops, thus provingLemma 4.3. We first give the contraction on F and show that the contraction F i , i > F . The majority of the effort in thissection is then concentrated on the description of the contraction c on F . Notation.
We will further use the following shorthand notation: For a 0-reduced simplicial set X we will denote the iterated degeneracy s · · · s ∗ of itsunique basepoint ∗ by ∗ and we set π i = π i ( X ). For any Eilenberg-Maclanespace K ( π i , i − i ≥
2, we denote its basepoint and its degeneracies by 0.From the context, it will always be clear which simplicial set we refer to.
Loop contraction on F . Assuming that X is a 0-reduced, 1-connectedsimplicial set with a given algorithm that computes the contraction on loops c : ( GX ) → ( GX ) , the contraction c on F is automatically defined, as X = F . Loop contraction on F i , i > Suppose we have defined the contraction onthe generators of G ( F ). i.e. for any ( x, k ) ∈ ( X × τ (cid:48) K ( π , we have c (( x, k )) = ( x , k ) (cid:15) · · · ( x n , k n ) (cid:15) n ( x j , k j ) ∈ ( F ) , (cid:15) j ∈ Z , ≤ j ≤ n such that d c (( x, k )) = ( x, k ) and d c (( x, k )) = 1. In detail, we get thefollowing:( x, k ) = d c (( x, k )) = ( d x , d k ) (cid:15) · · · ( d x n , d k n ) (cid:15) n (7)1 = d c (( x, k )) = (cid:0) ( d x , τ (cid:48) ( x ) d k ) − · ( d x , d k ) (cid:1) (cid:15) · · · (8) (cid:0) ( d x n , τ (cid:48) ( x n ) d k n ) − · ( d x n , d k n ) (cid:1) (cid:15) n We now aim to give a reduction on the generators of G ( F i ), i >
3. Simplicialset F i is an iterated twisted product of the form (cid:0) (( X × τ (cid:48) K ( π , × τ (cid:48) K ( π , × τ (cid:48) · · · × τ (cid:48) K ( π i − , i − (cid:1) × τ (cid:48) K ( π i − , i − For example, [ a, b, b, − [ a, − can be split into a sequence([ b, , [ b, − ) , ([ a, , [ a, − ).
38s simplicial sets K ( π i − , i −
2) are 1-reduced for i >
3, we can identify elementsof ( F i ) with vectors ( x, k, , . . . , k ∈ K ( π , , x ∈ X . We furthershorthand the series of i − . Hence generators G ( F i )are of the form ( x, k, ). The 1-reducedness also implies that τ (cid:48) ( α ) = 0 whenever α ∈ ( F i ) , i > c (( x, k, )) = ( x , k , ) (cid:15) · · · ( x n , k n , ) (cid:15) n ( x j , k j , ) ∈ ( F i ) , (cid:15) j ∈ Z , ≤ j ≤ n The (almost) freeness of G ( F i ), the fact that K ( π i − , i −
2) are 1-reduced for i > d c (( x, k, )) = ( x, k, ) and d c (( x, k, )) =1. Before the definition of contraction on simplicial set F , we remind the basicfacts involving the simplicial model of Eilenberg-MacLane spaces we are using. Eilenberg–MacLane spaces.
As noted in Section 3, given a group π and aninteger i ≥ K ( π, i ) is a space satisfying π j ( K ( π, i )) = (cid:26) π for j = i, . In the rest of this section, by K ( π, i ) we will always mean the simplicial modelwhich is defined in [36, page 101] K ( π, i ) q = Z i (∆ q ; π ) , where ∆ q ∈ sSet is the standard q -simplex and Z i denotes the cocycles. Thismeans that each q –simplex is regarded as a labeling of the i –dimensional facesof ∆ q by elements of π such that they add up to 0 ∈ π on the boundary of every( i + 1)-simplex in ∆ q , hence elements of K ( π, q ) q are in bijection with elementsof π . The boundary and degeneracy operators in K ( π, k ) are given as follows:For any σ ∈ K ( π, i ) q , d j ( σ ) ∈ K ( π, k ) q − is given by a restriction of σ ∈ K ( π, i )to the j -th face of ∆ q . To define the degeneracy we first introduce mapping η j : { , , . . . , q + 1 } → { , , . . . , q } given by η j ( (cid:96) ) = (cid:26) (cid:96) for (cid:96) ≤ j,(cid:96) − (cid:96) > j. Every mapping η j defines a map C ∗ ( η j ) : C ∗ (∆ q ) → C ∗ (∆ q +1 ).The degeneracy s j σ is now defined to be C ∗ ( η j )( σ ) (see [36, 23]).It follows from our model of Eilenberg-MacLane space, that elements of K ( π , can be identified with labelings of 1-faces of a 2-simplex by elementsof π that sum up to zero.As π is an Abelian group, we use the additive notation for π . We identifythe elements of K ( π , with triples ( k , k , k ), k i ∈ π , 0 ≤ i ≤
2, such that k − k + k = 0 ∈ π . 39 oop contraction on F . Let X be a 0-reduced, 1-connected simplicial setwith a given algorithm that computes the contraction on loops c : ( GX ) → ( GX ) .In the rest of the section, we will assume x ∈ X . Then by our assumptions c x = y (cid:15) · · · y n(cid:15) n , where y i ∈ X , (cid:15) i ∈ Z , 1 ≤ i ≤ n . Let k i = τ (cid:48) ( y i ).We first show that in order to give a contraction on elements of the form( x,
0) and ( x, k ), it suffices to have the contraction on elements of the form( ∗ , k ): Contraction on element ( x, Let ( x, ∈ G ( F ). We define c ( x,
0) = n (cid:89) i =1 (cid:0) c ( ∗ , k i ) − ( s d y i , ( k i , k i , · ( y i , (cid:1) (cid:15) i . Contraction on element ( x, k ). Suppose ( x, k ) ∈ ( GF ) . The formula forthe contraction is given using the formulae on contraction on ( x,
0) and ( ∗ , k )as follows c ( x, k ) = ( s x, ( k, , − k )) · s ( x, − · s ( ∗ , − k ) · c (( ∗ , − k )) − · c (( x, Contraction on element ( ∗ , k ). We formalize the existence of the contractionas Proposition 7.4 given at the end of this section. Due to the fact that the proofis rather technical, we need to define and prove some preliminary results first:
Definition 7.1.
Let Z = { z ∈ ( GF ) | d z = 1 } and let W = { d z | z ∈ Z } We define an equivalence relation ∼ on the elements of W in the following way:We say that w ∼ w (cid:48) if there exists z ∈ Z , α, β ∈ ( GF ) such that d z = w , αzβ ∈ Z and d ( αzβ ) = w (cid:48) . Lemma 7.2.
Let w ∈ W such that1. w = ( x, k ) (cid:15) · α , where α ∈ ( GF ) Then w = ( x, k ) (cid:15) · α ∼ α · ( x, k ) (cid:15) = w (cid:48) .2. w = ( ∗ , k ) (cid:15) · α , where α ∈ ( GF ) . Then w ∼ w (cid:48) = ( ∗ , − k ) − (cid:15) · α .3. w = ( ∗ , − k ) − ( x, · α , where α ∈ ( GF ) . Then w ∼ w (cid:48) = ( x, k ) · α .4. w = ( x, − ( x, k ) · α , where α ∈ ( GF ) . Then w ∼ w (cid:48) = ( ∗ , k ) · α .5. w = ( ∗ , − l ) − ( ∗ , k ) · α , where α ∈ ( GF ) . Then w ∼ w (cid:48) = ( ∗ , k + l ) · α .Proof. In all cases, we assume z ∈ Z such that d z = w and we give a formulafor z (cid:48) ∈ Z with d z (cid:48) = w (cid:48) :1. z (cid:48) = s ( x.k ) − (cid:15) · z · s ( x, k ) (cid:15) .2. z (cid:48) = ( ∗ , ( k, , − k )) (cid:15) · ( s ( ∗ , k )) − (cid:15) · z .40. z (cid:48) = ( s ( x, k )) · ( s x, ( k, , − k )) − · z .4. z (cid:48) = ( s ( ∗ , k ))( s x, ( k, k, − · z .5. z (cid:48) = ( s ( ∗ , k + l ))( ∗ , ( k + l, k, − l )) − · z . Lemma 7.3.
Let z ∈ ( GF ) , z ∈ Z with d z = w = ( ∗ , − k ) − · ( x , (cid:15) · · · ( ∗ , − k n ) − · ( x n , (cid:15) n where x (cid:15) · · · x n(cid:15) n = 1 in GX , x i ∈ X , k i ∈ π ( X ) , (cid:15) i ∈ { , − } , ≤ i ≤ n .Then w ∼ ( (cid:80) ni =1 k i , ∗ ) .Proof. We achieve the proof using a sequence of equivalences given in Lemma 7.2.Without loss of generality we can assume that x = x − and (cid:15) , (cid:15) = 1 (If this isnot the case, we can use rule (1) and/or relabel the elements). Using (1) givesus w =( ∗ , − k ) − · ( x , − · ( ∗ , − k ) − · ( x , · · · ( ∗ , − k n ) − · ( x n , (cid:15) n ∼ ( ∗ , − k ) − · ( x , · · · ( ∗ , − k n ) − · ( x n , (cid:15) n · ( ∗ , − k ) − · ( x , − . Then successive use of (3),(1),(4), (1) and finally (5) gives us w ∼ ( x , k ) · · · ( ∗ , − k n ) − · ( x n , (cid:15) n · ( ∗ , − k ) − · ( x , − . ∼ ( x , − · ( x , k ) · · · ( ∗ , − k n ) − · ( x n , (cid:15) n · ( ∗ , − k ) − ∼ ( ∗ , k ) · · · ( ∗ , − k n ) − · ( x n , (cid:15) n · ( ∗ , − k ) − ∼ ( ∗ , k + k ) · ( ∗ , − k ) − · ( x , · · · ( ∗ , − k n ) − · ( x n , (cid:15) n multiple use or rules (2) and (1) and gives us w ∼ ( ∗ , − k − k − k ) − · ( x , · · · ( ∗ , − k n ) − · ( x n , (cid:15) n So far, we have produced some element z (cid:48) ∈ Z ⊆ ( GF ) such that d z (cid:48) = 1, d z (cid:48) = ( ∗ , − k − k − k ) − · ( x , · · · ( ∗ , − k n ) − · ( x n , (cid:15) n and further x (cid:15) · · · x n(cid:15) n = 1 in GX .It follows that the construction described above can be applied iterativelyuntil all elements ( x i ,
0) are removed and we obtain w ∼ ( − (cid:80) ni =1 k i , ∗ ) − ∼ ( (cid:80) ni =1 k i , ∗ ). Proposition 7.4.
Let k ∈ π ( X ) . Then there is an algorithm that computesan element z ∈ ( GF ) such that d z = ( ∗ , k ) and d z = 1 . roof. Given an element k ∈ π ∼ = H ( X ), one can compute a cycle γ ∈ Z ( X )such that [ γ ] = k ∈ π ( X ) ∼ = H ( X ) ∼ = H ( K ( π , ∼ = π ( K ( π , , were the middle isomorphism is induced by ϕ and the other isomorphismsfollow from Hurewicz theorem.If one considers γ ∈ (cid:103) AX then by Lemma 6.8 one can algorithmicallycompute a spherical element γ (cid:48) = y (cid:15) · · · y n(cid:15) n ∈ (cid:103) GX where y i ∈ X and τ (cid:48) y i = k i ∈ π ( X ), such that d γ (cid:48) = 1 = d γ (cid:48) and (cid:80) ni =1 (cid:15) i · k i = k .We define z (cid:48) ∈ ( GF ) by z (cid:48) = ( n (cid:89) i =1 ( s d y i , ( k i , , − k i )) (cid:15) i ) · ( n (cid:89) i =1 ( y i , ( k i , , − k i )) (cid:15) i ) − . Observe that d ( z (cid:48) ) = 1 and d z (cid:48) = (cid:0) ( ∗ , − k ) − · ( d y , (cid:1) (cid:15) · · · (cid:0) ( ∗ , − k n ) − · ( d y n , (cid:1) (cid:15) n . We apply Lemma 7.3 on z (cid:48) and get an element z (cid:48)(cid:48) ∈ ( GF ) with the property d z (cid:48)(cid:48) = 1 and d z (cid:48)(cid:48) = ( ∗ , k ). We define z = s ( ∗ , k ) · ( z (cid:48)(cid:48) ) − . Thus d z = ( ∗ , k )and d z = 1. Computational complexity.
We first observe that that formulas for c ona general element ( x, k ) depend polynomially on the size of c ( x ) and the sizeof contractions on ( ∗ , k ). Hence it is enough to analyse the complexity of thealgorithm described in Proposition 7.4:The computation of γ (cid:48) is obtained by the polynomial-time Smith normalform algorithm presented in [28] and the polynomial-time algorithm in Lemma 6.8.The size of z (cid:48) depends polynomially (in fact linearly) on size of γ (cid:48) . The algorithmdescribed in Lemma 7.3 runs in a linear time in the size of z (cid:48) .To sum up, the algorithm computes the formula for contraction on the ele-ments of GF i in time polynomial in the input (size X + size c ( GX )). This section contains the proof of Lemma 4.5.
Edgewise subdivision of simplicial complexes.
In [12], the authors present,for k ∈ N , the edgewise subdivision Esd k (∆ m ) of an m -simplex ∆ m that gen-eralizes the two-dimensional sketch displayed in Figure 3. This subdivision hasseveral nice properties: in particular, the number of simplices of Esd k (∆ m )grows polynomially with k . Explicitly, the subdivision can be represented asfollows. 424 , ,
0) (3 , ,
0) (2 , ,
0) (1 , ,
0) (0 , , , , , , , ,
3) (0 , ,
4) (0 , , , , , , , ,
1) (1 , , , , k = 4. In this case, thereexists a copy of the 2-simplex completely in the “interior”, defined by vertices(2 , , , ,
1) and (1 , , • The vertices of Esd k (∆ m ) are labeled by coordinates ( a , . . . , a m ) suchthat a j ≥ (cid:80) j a j = k . • Two vertices ( a , . . . a m ) and ( b , . . . , b m ) are adjacent , if there is a pair j < k such that | b j − a j | = | b k − a k | = 1 and a i = b i for i (cid:54) = j, k . • Simplices of Esd k (∆ m ) are given by tuples of vertices such that each vertexof a simplex is adjacent to each other vertex.We define the distance of two vertices to be the minimal number of edges be-tween them. An edgewise k -subdivision of ∆ m induces an edgewise k -subdivisionof all faces, hence we may naturally define an edgewise subdivision of any sim-plicial complex. Constructing the map Esd k (Σ) → X sc . Let R be a chosen root in the tree T . We denote the tree-distance of a vertex W from R by dist T ( W ). Let l := max { dist T ( V ) : V is a vertex of X sc } be the maximal tree-distance of some vertex from R . For each vertex V of X sc ,there is a unique path in the spanning tree that goes from R into V . Further, wedefine the maps M ( j ) : ( X sc ) (0) → ( X sc ) (0) from vertices of X sc into verticesof X sc such that • M ( j )( V ) := V if j ≥ dist T ( V ), and • M ( j )( V ) is the vertex on the unique tree-path from R to V that hastree-distance j from R , if j < dist T ( V ).43f, for example, R − U − V − W is a path in the tree, then M (0)( W ) = R , M (1)( W ) = U etc. Clearly, M ( l ) = M ( l + 1) = . . . is the identity map, as l equals the longest possible tree-distance of some vertex.Assume that d is the dimension of Σ and k := l ( d + 1) + 1. We will define f (cid:48) : Esd k (Σ) → X sc simplexwise. Let τ ∈ Σ be an m -simplex and f ( τ ) = ˜ σ ∈ X be its image in the simplicial set X . If σ is the degeneracy of the base-point ∗ ∈ X , then we define f (cid:48) ( x ) := R for all vertices x of Esd k ( τ ): in other words, f (cid:48) will be constant on the subdivision of τ . Otherwise, ˜ σ is not the degeneracyof a point and has a unique lift σ ∈ X ss . (Recall that X := X ss /T .) Let( V , . . . , V m ) be the vertices of σ (order given by orientation): these vertices arenot necessarily different, as σ may be degenerate.In the algorithm, we will need to know which faces of σ are in the tree T . Weformalize this as follows: let S ⊆ m be the family of all subsets of { , , . . . , m } such that • For each { i , . . . , i j } ∈ S , { V i , . . . , V i j } is in the tree (that is, it is eitheran edge or a single vertex), • Each set in S is maximal wrt. inclusion.Elements of S correspond to maximal faces of σ that are in the tree, in otherwords, to faces of ˜ σ that are degeneracies of the base-point. Definition 8.1.
Let ∆ m be an oriented m -simplex, represented as a sequenceof vertices ( e , . . . , e m ) . For any face s ⊆ { e , . . . , e m } , we define the extendedface E ( s ) in Esd k (∆ m ) to be the set of vertices ( x , . . . , x m ) in Esd k (∆ m ) thathave nonzero coordinates only on positions i such that e i ∈ S . The geometric meaning of this is illustrated by Figure 4.
Definition 8.2.
For S ⊆ m , we define the extended tree E ( T ) to be the unionof the extended faces E ( s ) in Esd k (∆ m ) for all s ∈ S . The edge-distance ofa vertex x in Esd k (∆ m ) from E ( T ) will be denoted by dist ET ( x ) . In words, E ( T ) it is the union of all vertices in parts of the boundary ofEsd k (∆ m ) that correspond to the faces of σ that are in the tree, see Fig. 4.The number dist ET ( x ) is the distance to x from those boundary parts thatcorrespond to faces of σ that are in the tree.To define a simplicial map from Esd k ( τ ) to X sc , we need to label verticesof Esd k ( τ ) by vertices of X sc such that the induced map takes simplices inEsd k ( τ ) to simplices in X sc . Recall that V , . . . , V m are the vertices of σ . For x = ( x , . . . , x m ), we denote by arg max x the smallest index of a coordinate of x among those with maximal value (for instance, arg max (4 , , , ,
0) = 0, asthe first 4 is on position 0). The geometric meaning of V arg max x is illustratedby Figure 5.Now we are ready to define the map f (cid:48) : Esd k ( τ ) → X sc . It is defined onvertices x with coordinates ( x , . . . , x m ) by f (cid:48) ( x , . . . , x m ) := M (dist ET ( x ))( V arg max x ) . (9)44 s E ( s ) E ( s ) 1 1 1211Figure 4: Illustration of extended faces. Here S = { s , s } corresponds tothe lower- and left-face of a 2-simplex. The extended faces E ( s ) and E ( s )are sets of vertices of Esd k (∆ ) that are on the lower- and left- boundary. Thecorresponding extended tree E ( T ) is the union of all these vertices. The integersindicate edge-distances dist ET of vertices in Esd k (∆ ) from E ( T ). V V V V V V V V V V V V V V V Figure 5: Labelling vertices of Esd k (∆ ) by V arg max x .45eometrically, most vertices x will be simply mapped to V j for which the j ’thcoordinate of x is dominant. In particular, a unique m -simplex “most in theinterior of Esd k ( τ )” with coordinates j + 1 j. . .jj + 1 . . .j + 1 T , jj + 1 . . .jj + 1 . . .j + 1 T , . . . , jj. . .j + 1 j + 1 . . .j + 1 T , jj. . .jj + 2 . . .j + 1 T , . . . , jj. . .jj + 1 . . .j + 2 T (10)for suitable j will be labeled by V , V , . . . , V m ; in other words, it will be mappedto σ . However, vertices x close to those boundary parts of Esd k ( τ ) that correspondto the tree-parts of σ , will be mapped closer to the root R and all the extendedtree E ( T ) will be mapped to R . One illustration is in Figure 6. Computational complexity.
Assuming that we have a given encoding ofΣ , f, X, X sc and a choice of T and R , defining a simplicial map f (cid:48) : Esd k (Σ) → X sc is equivalent to labeling vertices of Esd k (Σ) by vertices of X sc . Clearly,the maximal tree-distance l of some vertex depends only polynomially on thesize of X sc and can be computed in polynomial time, as well as the maps M (0) , . . . , M ( l ). Whenever j > l , we can use the formula M ( j ) = id. Further, k = l ( d + 1) + 1 is linear in l , assuming the dimension d is fixed. If τ ∈ Σis an m -simplex, then the number of vertices in Esd k ( τ ) is polynomial in k ,and their coordinates can be computed in polynomial time. Finding the lift σ of f ( τ ) = ˜ σ is at most a linear operation in size( X sc ) + size(˜ σ ). Converting σ ∈ X ss into an ordered sequence ( V , V , . . . , V m ) amounts to computing itsvertices d d . . . ˆ d i . . . , d m σ , where d i is omitted. Collecting information on facesof σ that are in the tree and the set of vertices E ( T ) is straight-forward: note thatassuming fixed dimensions, there are only constantly many faces of each simplexto be checked. If s = { i , . . . , i j } is a face, then the edge-distance of a vertex x from E ( s ) equals to (cid:80) u x i u . Applying formula (9) to x requires to compute theedge-distance of x from E ( T ): this equals to the minimum of the edge-distancesof x from E ( s ) for all faces s of σ that are in the tree. Computing arg max x isa trivial operation. Finally, the number of simplices τ of Σ is bounded by thesize of Σ, so applying (9) to each vertex of Esd k (Σ) only requires polynomiallymany steps in size(Σ , f, X sc , T, X ). Correctness.
What remains is to prove that formula (9) defines a well-definedsimplicial map and that | Esd k (Σ) | → | X sc | → | X | is homotopic to | Σ | → | X | . Lemma 8.3.
The above algorithm determines a well-defined simplicial map
Esd(Σ) → X sc . If dim( τ ) = d is maximal, then j = l and this most-middle simplex has particularly nicecoordinates ( l + 1 , l, . . . , l ) , . . . , ( l, . . . , l, l + 1). Here the assumption on the fixed dimension d is crucial. V V V V V V V V V V V V V V V V V R R R R R R R RR R R R R R R V V V V V V V In the tree In the treeTree in X Figure 6: Example of the labeling induced by formula (9). We assume that f ( τ ) = ˜ σ where σ is a simplex of X sc with three different vertices V V V . Inthis example, the tree connects R − V − V as well as R − V − V and the edge V V is not in the tree. On the right, we give the induced labeling of verticesof Esd k ( τ ) which determines a simplicial map to X sc . The bottom and leftfaces of σ are in the tree, hence the bottom and left extended faces in Esd k ( τ )are all mapped into R . The right face of σ is the edge V V that is not in thetree: the corresponding right extended face in Esd k ( τ ) is mapped to a loop R − V − V − V − V − R , where V V is the only part that is not in the tree.The most interior simplex in Esd k ( τ ) is highlighted and is the only one mappedto σ . Proof.
First we claim that formula (9) defines a global labeling of vertices ofEsd k (Σ) by vertices of X sc . For this we need to check that if τ (cid:48) is a face of τ ,then (9) maps vertices of Esd k ( τ (cid:48) ) compatibly. This follows from the followingfacts, each of them easily checkable: • If τ (cid:48) is spanned by vertices of τ corresponding to s ⊆ { , . . . , m } , then avertex x (cid:48) := ( x , . . . , x j ) in Esd k ( τ (cid:48) ) has coordinates x in Esd k ( τ ) equalto zero on positions { . . . , m } \ s and to x , . . . , x j on other positions,successively. • If V (cid:48) k := V i k for s = ( i , . . . , i j ) are the vertices of the corresponding faceof σ , then V (cid:48) arg max x (cid:48) = V arg max x • The extended tree E (cid:48) ( T ) in Esd k ( τ (cid:48) ) equals the intersection of the extendedtree in Esd k ( τ ) with E ( τ (cid:48) ) • The distance dist ET ( x (cid:48) ) in Esd k ( τ (cid:48) ) equals dist ET ( x ) in Esd k ( τ ).47urther, we need to show that this labeling defines a well-defined simplicialmap, that is, it maps simplices to simplices. We claim that each simplex inEsd k ( τ ) is mapped either to some subset of { V , . . . , V m } or to some edge in thetree T , or to a single vertex.We will show the last claim by contradiction. Assume that some simplex is not mapped to a subset of { V , . . . , V m } , and also it is not mapped to an edgeof the tree and not mapped to a single vertex. Then there exist two vertices x and y in this simplex that are labeled by U and W in X sc , such that either U or W is not in { V , . . . , V m } , U W is not in the tree, and U (cid:54) = W .The fact that at least one of { U, W } does not belong to { V . . . , V m } , impliesthat dist ET ( x ) < l or dist ET ( y ) < l (as M ( j ) maps each V arg max x to itself for j ≥ l ).Without loss of generality, assume that arg max x = 0 and arg max y = 1.Then the coordinates of x and y are either x = ( j + 1 , j, x , . . . , x m ) , y = ( j, j + 1 , x , . . . , x m )such that x i ≤ j + 1 for all i ≥
3, or x = ( j, j, x , . . . , x m ) , y = ( j − , j + 1 , x , . . . , x m )for some j such that x i ≤ j for all i ≥ V (cid:54) = V and that the edge V V is not in the tree. This isbecause there exists a tree-path from R via U to V and also a tree-path from R via W to V (and U (cid:54) = W ): both V = V as well as a tree-edge V V would createa circle in the tree. In coordinates, this means that vertices ( ∗ , ∗ , , , . . . ,
0) arenot contained in E ( T ), apart of ( k, , , . . . ,
0) and (0 , k, , . . . , E ( T ) has a zero on either the zeroth or the first coordinate. This immediatelyimplies that dist ET ( x ) ≥ j and dist ET ( y ) ≥ j . Keeping in mind that coordinatesof x (and y ) has to sum up to k = l ( d + 1) + 1, the smallest possible value of j is j = l (if m = d is maximal), in which case x = ( l + 1 , l, l, . . . , l ) and y = ( l, l + 1 , . . . , l ). This choice, however, would contradict the fact that eitherdist ET ( x ) < l or dist ET ( y ) < l . Therefore we have a strict inequality j > l. Finally, we derive a contradiction having either dist ET ( x ) ≥ j > l > dist ET ( x ),or a similar inequality for y .This completes the proof that each simplex is either mapped to a subsetof { V , . . . , V m } or to an edge in the tree or to a single vertex: the image isa simplex in X sc in either case. Lemma 8.4.
The geometric realisations of pf (cid:48) : Esd k (Σ) → X and f : Σ → X are homotopic.Proof. First we reduce the general case to the case when all maximal simplicesin Σ (wrt. inclusion) have the same dimension d . If this were not the case, wecould enrich any lower-dimensional maximal simplex τ = { x , . . . , x j } ∈ Σ bynew vertices y τj +1 , . . . , y τd and produce a maximal d -simplex˜ τ = { x , . . . , x j , y τj +1 , . . . , y τd } . ⊇ Σ with the required property. When-ever f ( τ ) is mapped to ˜ σ where σ = ( V , . . . , V j ), we define f (˜ τ ) to be s d − jj ˜ σ ,a degenerate simplex with lift ( V , . . . , V j , V j , . . . , V j ). The map f (cid:48) : ˜Σ → X sc is constructed from f : ˜Σ → X as above and if we prove that | f | is homotopicto | pf (cid:48) | as maps | ˜Σ | → | X | , it immediately follows that their restrictions arehomotopic as maps | Σ | → | X | as well.Further, assume that all maximal simplices have dimension d . Let τ ∈ Σ bea d -dimensional simplex and let τ int be the simplex in Esd k ( τ ) spanned by thevertices ( l + 1 , l, . . . , l ) , . . . , ( l, . . . , l, l + 1) , that is, the simplex in the interior of τ that is mapped by pf (cid:48) to ˜ σ . Let H τ ( · ,
1) : | τ | → | τ | be a linear map that takes | τ | linearly to | τ int | via mapping the i ’thvertex to ( l, . . . , l + 1 , . . . , l ) where the l + 1 is on position i . Further, let H τ be a linear homotopy | τ | × [0 , → | τ | between the identity H τ ( · ,
0) = id and H τ ( · , | pf (cid:48) | H τ then gives a homotopy | τ | × [0 , → | X | between the restrictions ( | pf (cid:48) | ) | | τ | and ( | f | ) | | τ | . For a general x ∈ | Σ | , thereexists a maximal d -simplex | τ | such that x ∈ | τ | and we define a homotopy( x, t ) (cid:55)→ | pf (cid:48) | H τ ( x, t ) . It remains to show that this map is independent on the choice of τ .Let as denote the (ordered) vertices of τ by { v , v , . . . , v d } and let δ ⊆ τ beone of its faces: further, let w i be the vertex of τ int with barycentric coordinates( l, . . . , l, l + 1 , l, . . . , l ) /k in | τ | such that the l + 1 is in position i . The homotopy H τ sends points in | δ | onto the span of points w i for which v i ∈ δ . For y ∈ | δ | ,the j -th barycentric coordinate of H τ ( y, t ) is equal to t ( l/k ) for each j / ∈ δ . Inparticular, the j -th coordinate of H τ ( y, t ) is between 0 and l/k for j / ∈ δ , andhence it is not the “dominant” coordinate. It follows that each z := H τ ( x, t ) iscontained in the interior of a unique simplex ∆ of Esd k ( τ ) such that v arg max x ∈ δ for all vertices x of ∆.Let i < i . . . < i k be the indices such that v i j ∈ δ and j < . . . < j d − k be the remaining indices. Let τ (cid:48) = ( v (cid:48) , . . . , v (cid:48) d ) be another d -simplex containing δ as a face. Assume, for simplicity, that the vertices of τ (cid:48) are ordered so thatvertices of δ have orders i , . . . , i k —such as it is in τ . Let σ, σ (cid:48) be the lift of f ( τ ), f ( τ (cid:48) ) respectively, and V i , V (cid:48) i the i -th vertex of σ , σ (cid:48) respectively.We define a “mirror” map m : | τ | → | τ (cid:48) | , which to a point with barycentriccoordinates ( x , . . . , x d ) with respect to τ assigns a point in | τ (cid:48) | with the samebarycentric coordinates with respect to τ (cid:48) . Clearly, H τ (cid:48) ( y, t ) = m ( H τ ( y, t ))for y ∈ | τ | and whenever z is in the interior of a simplex ∆ ∈ Esd k ( τ ), then m ( z ) is in the interior of m (∆), where vertices of ∆ and m (∆) have the samebarycentric coordinates with respect to τ and τ (cid:48) , respectively. If, moreover, ∆ issuch that each of its vertices r have coordinates ≤ l/k on positions j , . . . , j d − k ,then V arg max r = V (cid:48) arg max m ( r ) .To summarize these properties, H τ ( y, t ) and H τ (cid:48) ( y, t ) satisfy that In general, vertices of δ may have different order in τ and τ (cid:48) and the assumption on V V V V V V V V V V W yV V V V zz Figure 7: The homotopy H τ takes y linearly into z and H τ (cid:48) takes y into z (cid:48) . Dueto the symmetry represented by the horizontal line, | pf (cid:48) | maps H τ ( y, t ) into thesame point of X as | pf (cid:48) | H τ (cid:48) ( y, t ). • they have the same coordinates wrt. τ , τ (cid:48) , respectively, • they are in the interior of simplices ∆ ∈ Esd k ( τ ), ∆ (cid:48) ∈ Esd k ( τ (cid:48) ) whosevertices have the same coordinates wrt. τ , τ (cid:48) , respectively, • the arg max labeling induces the same labeling of vertices of ∆, ∆ (cid:48) byvertices of δ , respectively.The map pf (cid:48) takes each m -simplex ∆ in Esd k ( τ ) with vertices t u labeled by V arg max t u onto p ( V arg max t , . . . , V arg max t m ) and it follows from the above prop-erties that m (∆) is mapped to the same simplex. We conclude that | pf (cid:48) | H τ ( y, t ) = | pf (cid:48) | H τ (cid:48) ( y, t ) for each y ∈ | δ | and t ∈ [0 , Acknowledgements.
We would like to thank Marek Krˇc´al, Luk´aˇs Vokˇr´ınekand Sergey Avvakumov for helpful conversations and comments.
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