Extendability of Continuous Maps Is Undecidable
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Abstract
We consider two basic problems of algebraic topology: the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.
The extension problem is the following: Given topological spaces X and Y, a subspace A⊆X, and a (continuous) map f:A→Y, decide whether f can be extended to a continuous map \(\bar{f}\colon X\to Y\). All spaces are given as finite simplicial complexes, and the map f is simplicial.
Recent positive algorithmic results, proved in a series of companion papers, show that for (k−1)connected Y, k≥2, the extension problem is algorithmically solvable if the dimension of X is at most 2k−1, and even in polynomial time when k is fixed.
Here we show that the condition \(\mathop{\mathrm{dim}}\nolimits X\leq 2k1\) cannot be relaxed: for \(\mathop{\mathrm{dim}}\nolimits X=2k\), the extension problem with (k−1)connected Y becomes undecidable. Moreover, either the target space Y or the pair (X,A) can be fixed in such a way that the problem remains undecidable.
Our second result, a strengthening of a result of Anick, says that the computation of π _{ k }(Y) of a 1connected simplicial complex Y is #Phard when k is considered as a part of the input.
Keywords
Simplicial Complex Cell Complex Homotopy Class Homotopy Group Extension Problem1 Introduction
One of the central themes in algebraic topology is to understand the structure of all continuous maps X→Y, for given topological spaces X and Y (all maps between topological spaces in this paper are assumed to be continuous). For topological purposes, two maps f,g:X→Y are usually considered equivalent if they are homotopic, i.e., if one can be continuously deformed into the other;^{1} thus, the object of interest is [X,Y], the set of all homotopy classes of maps X→Y.
Many of the celebrated results throughout the history of topology can be cast as information about [X,Y] for particular spaces X and Y. In particular, one of the important challenges propelling the research in algebraic topology has been the computation of the homotopy groups of spheres ^{2} π _{ k }(S ^{ n }), for which only partial results have been obtained in spite of an enormous effort (see, e.g., [13, 22]).
A closely related question is the extension problem: given A⊂X and a map f:A→Y, can it be extended to a map X→Y? For example, the famous Brouwer fixed point theorem can be restated as nonextendability of the identity map S ^{ n }→S ^{ n } to the ball D ^{ n+1} bounded by the sphere S ^{ n }. See [33] for a very clear and accessible introduction to the extension problem, including further examples and applications and covering the earlier developments until the late 1950s.
Computational Homotopy Theory
In this paper, we consider the (theoretical) computational complexity of homotopytheoretic questions such as the extension problem, the homotopy classification of maps, and the computation of homotopy groups. More precisely, we prove hardness and undecidability results that complement recent positive algorithmic results obtained in a series of companion papers [5, 6, 12]. To put our results into context, we first give more background.
By classical uncomputability results in topology (see, e.g., the survey [31]), most of these problems are algorithmically unsolvable if we place no restriction on the space Y. Indeed, by a result of Adjan and of Rabin, it is undecidable whether the fundamental group π _{1}(Y) of a given finite simplicial complex Y is trivial, even if Y is assumed to be 2dimensional. The triviality of π _{1}(Y) is equivalent to [S ^{1},Y] having only one element, represented by the constant map, and so [S ^{1},Y] is uncomputable in general. Moreover, by the Boone–Novikov theorem, it is undecidable whether a given pointed map f:S ^{1}→Y is homotopic to a constant map, and this homotopic triviality is equivalent to the extendability of f to the 2dimensional ball D ^{2}. Therefore, the extension problem is undecidable as well.^{3}
In these results, the difficulty stems from the intractability of the fundamental group of Y. Thus, a reasonable restriction is to assume that π _{1}(Y) is trivial (which in general cannot be tested, but in many cases of interest it is known), or more generally, that Y is kconnected, meaning that π _{ i }(Y) is trivial for all i≤k (equivalently, every map S ^{ i }→Y, i≤k, can be extended to D ^{ i+1}). A basic and important example of a (k−1)connected space is the sphere S ^{ k }.
For a long time, the only positive result concerning the computation of [X,Y] was that of Brown [4], who showed that [X,Y] is computable under the assumption that Y is 1connected and that all the higher homotopy groups π _{ k }(Y), \(2\leq k\leq \mathop{\mathrm{dim}}\nolimits X\), are finite (the second assumption is rather strong and not satisfied if Y is a sphere, for example). Brown also gave an algorithm that, given k≥2 and a finite 1connected simplicial complex Y, computes π _{ k }(Y).
In the 1990s, three independent collections of works appeared with the goal of making various more advanced methods of algebraic topology effective (algorithmic): by Schön [28], by Smith [30], and by Sergeraert, Rubio, Dousson, and Romero (e.g., [24, 25, 26, 29]; also see [27] for an exposition). New algorithms for computing higher homotopy groups follow from these methods; see Real [23] for an algorithm based on Sergeraert et al.
An algorithm that computes π _{ k }(Y) for a given 1connected simplicial complex Y in polynomial time for every fixed k≥2 was recently presented in [6], also relying on [12] and on the methods of effective homology developed earlier by Sergeraert et al.
The problem of computing [X,Y] was addressed in [5], where it was shown that its structure is computable assuming that Y is (k−1)connected and \(\mathop{\mathrm{dim}}\nolimits (X)\leq2k2\), for some integer k≥2. These assumptions are sometimes summarized by saying that X and Y are in the stable range.
As observed in [6], the methods of [5] can also be used to obtain an algorithmic solution of the extension problem. Here \(\mathop{\mathrm{dim}}\nolimits X\) can even be 1 beyond the stable range;^{4} thus, given finite simplicial complexes A⊆X and Y and a simplicial map f:A→Y, where Y is (k−1)connected and \(\mathop{\mathrm{dim}}\nolimits X\leq2k1\), k≥2, it can be decided algorithmically whether f can be extended to a continuous map X→Y. The algorithm again runs in polynomial time for k fixed, and the same holds for the algorithm mentioned above for computing [X,Y] in the stable range.
New Undecidability Results
For the algorithms for homotopy classification and extendability, we have two types of assumptions: The first is that the dimension of X is suitably bounded in terms of the connectivity of Y (in the stable range or at most one more). This is essential for the algorithms to work at all.^{5} The second assumption is that the relevant dimensional parameter k is fixed, which guarantees that the algorithm runs in polynomial time.
Our main result is that, for the extension problem, the first assumption is necessary and sharp.
Theorem 1.1
 (a)
(Fixed Target) There is a fixed (k−1)connected finite simplicial complex Y=Y _{ k } such that the following problem is algorithmically unsolvable: Given finite simplicial complexes A⊆X with \(\mathop{\mathrm{dim}}\nolimits X=2k\) and a simplicial map f:A→Y, decide whether there exists a continuous map X→Y extending f. For k even, we can take Y _{ k } to be the sphere S ^{ k }.
 (b)
(Fixed Source) There exist fixed finite simplicial complexes A=A _{ k } and X=X _{ k } with A⊆X and \(\mathop{\mathrm{dim}}\nolimits X=2k\) such that the following problem is algorithmically unsolvable: Given a (k−1)connected finite simplicial complex Y and a simplicial map f:A→Y, decide whether there exists a continuous map X→Y extending f.
The theorem is stated in terms of simplicial complexes since these are a standard input model for topological spaces in computational topology that we assume may be most familiar to most readers. For the purposes of our reductions, we actually work with simplicial sets (see Sect. 3.2), which offer a more flexible, but still purely combinatorial, way of representing topological spaces. The simplicial sets are then converted into simplicial complexes by a suitable subdivision.
When constructing A, X, and Y as simplicial sets, we can furthermore ensure that Y has a certain additional property, namely that it is (k−1)reduced, which provides an immediate certificate that Y is (k−1)connected; this is proved in Appendix B. Thus, in particular, the difficulty of the extension problem does not lie in verifying the (k−1)connectedness of Y.
On the Hardness of Computing [X,Y]
When \(\mathop{\mathrm{dim}}\nolimits X=2k\) and Y is (k−1)connected, we can no longer equip [X,Y] with the group structure of the stable range. Thus, it is not clear in what sense the potentially infinite set [X,Y] could be computed in general. A natural computational problem in this setting is to decide whether [X,Y]>1; in other words, whether there is a homotopically nontrivial map X→Y for given simplicial complexes X and Y as above.
We can prove that this problem is NPhard for every even k≥2; in order to keep this paper reasonably concise, the proof is to be presented in the Ph.D. thesis of the second author. The reduction is very similar to that of Theorem 1.1. We can show that the problem is at least as hard as deciding the existence of a nonzero integral solution of the quadratic system (QSYM) defined in Sect. 2 below with all the constant terms b _{ q } equal to zero. This problem may well be undecidable, but as far as we know, the best known lower bound is that of NPhardness.
#PHardness
Our second result concerns the problem of computing the higher homotopy groups π _{ n }(Y)≅[S ^{ n },Y] for a simply connected space Y, if n is not considered fixed but part of the input (n is given in unary encoding).^{6} Anick [2] proved that this problem is #Phard,^{7} where Y can even be assumed to be a 4dimensional space.^{8}
However, Anick’s hardness result has the following caveat: It assumes that the input space Y is given in a very concise form, as a cell complex with the degrees of the attaching maps encoded in binary (see Sect. 5.3 for a review of the construction). A straightforward way of converting this cell complex to a simplicial complex yields a 4dimensional simplicial complex with an exponential number of simplices, which renders the hardness result meaningless for simplicial complexes. In Sect. 6.3, we provide a different way of converting Anick’s concise encoding of the input space Y into a homotopy equivalent^{9} simplicial complex that can be constructed in polynomial time, and in particular, has only polynomially many simplices. This yields the following result.
Theorem 1.2
It is #Phard to compute the rank of π _{ n }(Y) (i.e., the number of summands of π _{ n }(Y) isomorphic to \(\mathbb {Z}\)) for a given number \(n\in \mathbb {N}\) (encoded in unary) and a given simply connected 4dimensional simplicial complex Y.
Outline of the Proof of Theorem 1.1
We aim to present our results in a way that makes the statements of the results and the main steps and ideas accessible while assuming only a moderate knowledge of topology on the side of the reader.
We thus review a number of basic topological concepts and provide proofs for various assertions and facts that may be rather elementary for topologists. On the other hand, some of the proofs assume a slightly stronger topological background, since reviewing every single notion and fact would make the paper too lengthy.
For proving Theorem 1.1, we present an algorithm that converts a given system of Diophantine equations into an instance of the extension problem; i.e., it constructs simplicial complexes A, X, and Y and a map f:A→Y such that there is an extension of f to all of X iff the given system of equations is solvable. Moreover, as stated in the theorem, there are actually two versions of the reduction: The first uses a fixed target space Y=Y _{ k } and encodes the equations into A, X, and f. The second uses a fixed pair (X _{ k },A _{ k }) of source complexes and encodes the equations into f and Y.
We will actually work only with quadratic Diophantine equations of a slightly special form (which is sufficient; see Sect. 2). The unknowns are represented by the degrees of restrictions of the desired extension \(\bar{f}\) to suitable kdimensional spheres. The quadratic terms in the equations are obtained using the Whitehead product, which is a binary operation that, for a space Z, assigns to elements α∈π _{ k }(Z) and β∈π _{ ℓ }(Z) an element [α,β]∈π _{ k+ℓ−1}(Z); see Sect. 4.2.

The spaces X and A are simplest to describe as cell complexes. The subcomplex A is a union of r spheres S ^{2k−1}, which intersect only at a single common point. This union is called a wedge sum and denoted by A=S ^{2k−1}∨⋯∨S ^{2k−1}. The space X is homotopy equivalent to another wedge sum, of s spheres S ^{ k }; i.e., X≃S ^{ k }∨⋯∨S ^{ k }.

The fixed (k−1)connected target space Y is the ksphere S ^{ k }.

Maps X→S ^{ k } can be described completely by their restrictions to the kspheres in the wedge sum. Each such restriction is characterized, uniquely up to homotopy, by its degree—this can be an arbitrary integer. Thus, a potential extension \(\bar{f}\) can be encoded into a vector x=(x _{1},…,x _{ r }) of integers.

Similarly, the map f:A→S ^{ k } can be described by its restrictions to the (2k−1)spheres in the wedge sum. Crucially for our construction, the homotopy group π _{2k−1}(S ^{ k }) has an element of infinite order, namely, the Whitehead square [ι,ι], where ι is the identity S ^{ k }→S ^{ k } (we still assume k even). We will work with maps f whose restriction to the qth sphere is (homotopic to) an integral multiple b _{ q }[ι,ι], for some (unique) integer b _{ q }. Thus, f is specified by the vector b=(b _{1},…,b _{ s }) of these integers.

Given arbitrary integers \(a_{ij}^{(q)}\), 1≤i<j≤r, q=1,2,…,s, we construct the pair (X,A) in such a way that, taking \(\bar{f}\colon X\to Y\) specified by x as above, the restriction of \(\bar{f}\) to the qth sphere of A is homotopic to \(\sum_{i< j} a_{ij}^{(q)}x_{i}x_{j}[\iota,\iota]\) (here the addition and multiplication by integers are performed in π _{2k−1}(S ^{ k })). Since [ι,ι] is an element of π _{2k−1}(S ^{ k }) of infinite order, \(\bar{f}\) is an extension of f iff \(\sum_{i< j} a_{ij}^{(q)}x_{i}x_{j}=b_{q}\) for all q=1,2,…,s.

In this way, we can simulate an arbitrary system of quadratic equations by an extension problem. Some more work is still needed to describe X and A as finite simplicial complexes and f as a simplicial map.

For Theorem 1.1(a) with k odd, the Whitehead square [ι,ι] as above no longer has infinite order. Instead, we use Y=S ^{ k }∨S ^{ k } and replace [ι,ι] by the Whitehead product [ι _{1},ι _{2}] of the inclusions of the two spheres into Y. This leads to skewsymmetric systems of quadratic equations, and showing that these are still undecidable needs some work (see Sect. 2).
In Theorem 1.1(b) with k even, the (fixed) source space X is homotopy equivalent to S ^{ k } and A=S ^{2k−1}. Under the homotopy equivalence X≃S ^{ k }, the inclusion A↪X becomes the Whitehead square [ι,ι], and for k odd it is replaced by [ι _{1},ι _{2}]. In both cases, the system of quadratic equations is encoded into the structure of the cell complex Y and the map f:A→Y.
2 Diophantine Equations and Undecidability
Lemma 2.1
The solvability of the system (QSYM), as well as that of (QSKEW), in the integers is algorithmically undecidable.
Proof
First, it is well known and easy to see that the solvability of a general quadratic system of Diophantine equations is no easier than the solvability of an arbitrary Diophantine system (1), and thus undecidable.^{10}
It is clear that each solution of (QSYM) yields a solution of the new system. Conversely, supposing that the new system has a solution, we claim that it also has a solution with \(x_{0}=y_{0}'=1\) and \(y_{0}=x_{0}'=0\). Once we have a solution satisfying these additional conditions, it is easy to check that x _{1},…,x _{ r } form a solution of the original system.
To verify the claim, for notational convenience, let us index the x and y variables in the new system by the set I={0,1,…,r,0′,1′,…,r′}, where \(x_{i'}=x'_{i}\) and \(y_{i'}=y'_{i}\). We suppose that (x _{ i },y _{ i }:i∈I) form a solution of the new system. Since \(x_{0}y_{0}'x_{0}'y_{0}=1\), the 2×2 matrix \(\big( \begin{array}{c@{\ }c} \scriptstyle x_{0} & \scriptstyle x_{0}' \\[3pt] \scriptstyle y_{0} & \scriptstyle y_{0}' \end{array} \big)\) has determinant 1 and thus an integral inverse matrix, which we denote by T.
3 Cell Complexes and Simplicial Sets
This section and the next one primarily present known material from topology. In several cases we need to adapt results from the literature to our needs, which is sometimes best done by reproving them. Readers may want to skim these two sections quickly and return to them later when needed.
Here we review two basic ways of building topological spaces from simple pieces: cell complexes and simplicial sets. Cell complexes, also known as CW complexes, are fairly standard in topology, and we will use them for a simple description of the various spaces in our proofs. Simplicial sets are perhaps less well known, and for us, they will mainly be a convenient device for converting cell complexes into simplicial complexes. Moreover, they are of crucial importance in the algorithmic results mentioned in the introduction. For a thorough discussion of simplicial complexes, simplicial sets, cell complexes, and the connections between the three, we refer to [7].
3.1 Cell Complexes
In the case of cell complexes, the building blocks are topological disks of various dimensions, called cells, which can be thought of as being completely “flexible” and which can be glued together in an almost arbitrary continuous fashion. Essentially the only condition is that each ndimensional cell must be attached along its boundary to the (n−1)skeleton of the space, i.e., to the part that has already been built, inductively, from lower dimensional cells. The formal definition is as follows.
We recall that if X and Y are topological spaces and if f:A→Y is a map defined on a subspace A⊆X, then the space X∪_{ f } Y obtained by attaching X to Y via f is defined as the quotient of the disjoint union X⊔Y under the equivalence relation generated by the identifications a∼f(a), a∈A.
A closed or open ncell is a space homeomorphic to the closed ndimensional unit disk D ^{ n } in ndimensional Euclidean space or its interior \(\mathring{D}^{n}\), respectively; a point is regarded as both a closed and an open 0cell.
 1.
X ^{(0)} is a discrete set of points (possibly infinite) that are regarded as 0cells.
 2.
Inductively, the nskeleton X ^{(n)} is formed by attaching closed ncells \(D^{n}_{i}\) (where i ranges over some arbitrary index set) to X ^{(n−1)} via attaching maps \(\varphi_{i}\colon S^{n1}_{i}=\partial D^{n} \to X^{(n1)}\). Formally, we can consider all attaching maps together as defining a map φ=⊔_{ i } φ _{ i } from the disjoint union \(\bigsqcup_{i} S_{i}^{n1}\) to X ^{(n−1)} and form \(X^{(n)}= (\bigsqcup_{i}D_{i}^{n} ) \cup_{\varphi} X^{(n1)}\).
For every closed cell \(D_{i}^{n}\), one has a characteristic map ^{12} \(\varPhi_{i}\colon D_{i}^{n} \to X^{(n)}\subseteq X\), which restricts to an embedding on the interior \(\mathring{D}_{i}^{n}\). The image \(\varPhi _{i}(\mathring{D}_{i}^{n})\) is commonly denoted by \(e_{i}^{n}\), and it follows from the construction that every point of X is contained in a unique open cell (note that these are generally not open subsets of X, however).
As a basic example, the nsphere is a cell complex with one ncell and one 0cell, obtained by attaching D ^{ n } to a point e ^{0} via the constant map that maps all of S ^{ n−1} to e _{0}.
Subcomplexes
A subcomplex A⊆X is a subspace that is closed and a union of open cells of X. In particular, for each cell in A, the image of its attachment map is contained in A, so A is itself a cell complex (and its cell complex topology agrees with the subspace topology inherited from X).
The Homotopy Extension Property
An important fact is that cell complexes have the homotopy extension property: Suppose that X is a cell complex and that A⊆X is a subcomplex. If we are given a map f _{0}:A→Y into some space Y, an extension \(\bar{f}_{0}\colon X\to Y\) of f _{0} and a homotopy H:A×[0,1] between f _{0} and some other map f _{1}:A→Y, then H can be extended to a homotopy \(\bar{H}\colon X\times[0,1]\to Y\) between \(\bar{f}_{0}\) and some extension \(\bar{f}_{1}\colon X\to Y\) of f _{1}. The following is an immediate consequence.
Corollary 3.1
Cellular Maps and Cellular Approximation
A map f:X→Y between cell complexes is called cellular if it maps skeletons to skeletons, i.e., f(X ^{(n)})⊆Y ^{(n)} for every n.
The cellular approximation theorem (see [9, Theorem 4.8]) states that every continuous map f:X→Y between cell complexes is homotopic to a cellular one; moreover, if the given map f is already cellular on some subcomplex A⊆X, then the homotopy can be taken to be stationary on A (i.e., the image of every point in A remains fixed throughout).
3.2 Simplicial Sets
For certain constructions it is advantageous to use a special type of cell complex with an additional structure that allows for a purely combinatorial description; the latter also facilitates representing and manipulating the objects in question, simplicial sets, on a computer. We refer to [8] for a very friendly and thorough introduction to simplicial sets.
Formally, a simplicial set X is given by a sequence (X _{0},X _{1},X _{2},…) of mutually disjoint sets, where the elements of X _{ n } are called the nsimplices of X (we note that, unlike the case of simplicial complexes, a simplex in a simplicial set need not be determined by the set of its vertices; indeed, there can be many simplices with the same vertex set). The 0simplices are also called vertices.
For every n≥1, there are n+1 mappings ∂ _{0},…,∂ _{ n }:X _{ n }→X _{ n−1} called face operators; the intuitive meaning is that for a simplex σ∈X _{ n }, ∂ _{ i } σ is the face of σ opposite to the ith vertex. Moreover, there are n+1 mappings s _{0},…,s _{ n }:X _{ n }→X _{ n+1} called the degeneracy operators; the approximate meaning of s _{ i } σ is the degenerate simplex which is geometrically identical to σ, but with the ith vertex duplicated. A simplex is called degenerate if it lies in the image of some s _{ i }; otherwise, it is nondegenerate. We write X ^{ndg} for the set of all nondegenerate simplices of X. A simplicial set is called finite if it has only finitely many nondegenerate simplices (if X is nonempty, there are always infinitely many degenerate simplices, at least one for every positive dimension).
There are natural axioms that the ∂ _{ i } and the s _{ i } have to satisfy, but we will not list them here, since we won’t really use them. Moreover, the usual definition of simplicial sets uses the language of category theory and is very elegant and concise; see, e.g., [7, Sect. 4.2].
If A and X are simplicial sets such that A _{ n }⊆X _{ n } for every n and the face and degeneracy operators of A are the restrictions of the corresponding operators of X, then we call A a simplicial subset of X.
Examples
Here we sketch some basic examples of simplicial sets; again, we won’t provide all details, referring to [8]. Let Δ^{ p } denote the standard pdimensional simplex regarded as a simplicial set. For p=0, (Δ^{0})_{ n } consists of a single simplex, denoted by 0^{ n }, for every n=0,1,…; 0^{0} is the only nondegenerate simplex. The face and degeneracy operators are defined in the only possible way.
For p=1, Δ^{1} has two 0simplices (vertices), say 0 and 1, and in general there are n+2 simplices in (Δ^{1})_{ n }; we can think of the ith one as containing i copies of the vertex 0 and n+1−i copies of the vertex 1, i=0,1,…,n+1. For p arbitrary, the nsimplices of Δ^{ p } can be thought of as all nondecreasing (n+1)term sequences with entries in {0,1,…,p}; the ones with all terms distinct are nondegenerate.
In a similar fashion, every simplicial complex K can be converted into a simplicial set X in a canonical way; first, however, we need to fix a linear ordering of the vertices. The nondegenerate nsimplices of X are in onetoone correspondence with the nsimplices of K, but many degenerate simplices show up as well.
Geometric Realization
Like a simplicial complex, every simplicial set X defines a topological space X, the geometric realization of X, which is unique up to homeomorphism. More specifically, X is a cell complex with one ncell for every nondegenerate nsimplex of X, and these cells are glued together according to the identifications implied by the face and degeneracy operators (we omit the precise definition of the attachments, since we will not really use it and refer to the literature, e.g., to [8] or [7, Sect. 4.3]).
Simplicial Maps
Simplicial sets serve as a combinatorial way of describing a topological space; in a similar way, simplicial maps provide a combinatorial description of continuous maps.
A simplicial map f:X→Y of simplicial sets X,Y consists of maps f _{ n }:X _{ n }→Y _{ n }, n=0,1,…, that commute with the face and degeneracy operators.
A simplicial map f:X→Y induces a continuous, in fact, a cellular map f:X→Y of the geometric realizations in a natural way (we again omit the precise definition). Often we will take the usual liberty of omitting ⋅ and not distinguishing between simplicial sets and maps and their geometric realizations.
Of course, not all continuous maps are induced by simplicial maps. However, simplicial maps can be used to approximate arbitrary continuous maps up to homotopy. The simplicial approximation theorem (which may be most familiar in the context of simplicial complexes) says that for an arbitrary continuous map φ:X→Y between the geometric realizations of simplicial sets, with X finite, there exist a sufficiently fine subdivision X′ of X and a simplicial map f:X′→Y whose geometric realization is homotopic to φ; see Sect. 3.4 for more details.
Encoding Finite Simplicial Sets
A finite simplicial complex can be encoded in a straightforward way by listing the vertices of each simplex.
For simplicial sets, the situation is a bit more complicated, since the simplices are no longer uniquely determined by their vertices, but if X is finite, then we can encode X by the set X ^{ndg} of its nondegenerate simplices (which we assume to be numbered from 1 to N, where N is the total number of nondegenerate simplices), plus a little bit of additional information.
The simple but crucial fact (see, e.g. [7, Theorem 4.2.3]) we need is that every simplex σ can be written uniquely as σ=sτ, where τ is nondegenerate and s is a degeneracy, i.e., a composition \(s=s_{i_{k}}\ldots s_{i_{1}}\) of degeneracy operators where \(k=\mathop{\mathrm{dim}}\nolimits \sigma\mathop{\mathrm{dim}}\nolimits \tau\) (in particular, σ is nondegenerate itself if σ=τ and s is the identity). Thus, as mentioned above, degenerate simplices σ do not need to be encoded explicitly but can be represented by sτ when needed, where the degeneracy s can be encoded by the sequence (i _{ k },…,i _{1}) of indices of its components.^{14} The extra information we need to encode X, in addition to the list of its nondegenerate simplices, is how these fit together. Specifically, for \(\sigma\in X_{n}^{\mathrm{ndg}}\) and 0≤i≤n, the ith face can be written uniquely as ∂ _{ i } σ∈X _{ n−1}=sτ with τ nondegenerate, and for each σ, we record the (n+1)tuple of pairs (τ,s).
Similarly, if f:X→Y is a simplicial map between finite simplicial sets, then given the encodings of X and Y, we can encode f by expressing, for each \(\sigma\in X_{n}^{\mathrm{ndg}}\), the image f(σ)=sτ, with \(\tau\in Y_{m}^{\mathrm{ndg}}\) and recording the list of triples (σ,τ,s).
For a finite simplicial set X, we define \(\operatorname {\mathsf {size}}(X)\) as the number of nondegenerate simplices. If the dimension of X is bounded by some number d, then the number of bits in the encoding of X described above is bounded by \(O(\operatorname {\mathsf {size}}(X)\log \operatorname {\mathsf {size}}(X))\), with the constant of proportionality depending only on d.
The notion of size will be a convenient tool that allows us to ensure that our reductions can be carried out in polynomial time, without analyzing the running time in complete detail, which we feel would be cumbersome and not very enlightening.
More specifically, our reductions will be composed of a sequence of various basic constructions of simplicial sets, which will be described in the next subsection.
For each of these basic constructions, it is straightforward to check^{15} that when we apply them to finite simplicial sets of bounded dimension, both the running time of the construction (the number of steps needed to compute the encoding of the output from the encoding of the input) as well as the size of the output simplicial set are polynomial in the size of the input. Thus, to ensure that the overall reduction is polynomial, it will be enough to take care that we combine only a polynomial number of such basic constructions, that the size of every intermediate simplicial set constructed during the reduction remains polynomial in the initial input, and that the dimension remains bounded.
3.3 Basic Constructions
In this subsection, we review several basic constructions for cell complexes and simplicial sets. (One advantage of simplicial sets over simplicial complexes is that various operations on topological spaces, in particular Cartesian products and quotients, have natural counterparts for simplicial sets. This is where the degeneracy operators and degenerate simplices turn out to be necessary.) For more details, we refer to [7, 9].
Pointed and kReduced Simplicial Sets and Cell Complexes
Several of the constructions are defined for pointed spaces. We recall that a pointed space (X,x _{0}) is a topological space X with a choice of a distinguished point x _{0}∈X (the basepoint). If X is a cell complex or a simplicial set, then we will always assume the basepoint to be a vertex (i.e., a 0cell or 0simplex, respectively). A pointed map (X,x _{0})→(Y,y _{0}) of pointed spaces (cell complexes, simplicial sets) is a continuous (cellular, simplicial) map sending x _{0} to y _{0}. Homotopies of pointed maps are also meant to be pointed; i.e., they must keep the image of the basepoint fixed. The reader may recall that, for example, the homotopy groups π _{ k }(Y) are defined as homotopy classes of pointed maps. The set of pointed homotopy classes of pointed maps X→Y will be denoted by [X,Y]_{∗}.
A simplicial set X is called kreduced, k≥0, if it has a single vertex and no nondegenerate simplices in dimensions 1 through k. Similarly, a cell complex X is kreduced if it has a single vertex and no cells of dimensions 1 up to k. It is then necessarily kconnected.
If (Y,y _{0}) is a 0reduced cell complex (or simplicial set), then any cellular (or simplicial) map from a pointed complex (X,x _{0}) into Y is automatically pointed. Moreover, if Y is 1reduced, then every homotopy is pointed, too, and thus [X,Y] is canonically isomorphic to [X,Y]_{∗}.
Products
If X and Y are cell complexes, then their Cartesian product X×Y has a natural cell complex structure whose ncells are products e ^{ p }×e ^{ q }, where p+q=n and e ^{ p } and e ^{ q } range over the pcells of X and the qcells of Y, respectively.
Furthermore, if X and Y are simplicial sets, then there is a formally very simple way to define their product X×Y: one sets (X×Y)_{ n }:=X _{ n }×Y _{ n } for every n, and the face and degeneracy operators work componentwise; e.g., ∂ _{ i }(σ,τ):=(∂ _{ i } σ,∂ _{ i } τ). As one would expect from a good definition, the product of simplicial sets corresponds to the Cartesian product of their geometric realizations, i.e., X×Y≅X×Y.^{16} The apparent simplicity of the definition hides some intricacies, though, as one can guess after observing that, for example, the product of two 1simplices is not a simplex—so the above definition must imply some canonical way of triangulating the product.
Remark 3.2
A pair (sσ,tτ) of degenerate simplices in the factors may yield a nondegenerate simplex in the product, if the degeneracies s and t are composed of different degeneracy operators s _{ i }. However, \(\mathop{\mathrm{dim}}\nolimits (X\times Y)=\mathop{\mathrm{dim}}\nolimits X +\mathop{\mathrm{dim}}\nolimits Y\), so the product contains no nondegenerate simplices of dimension larger than \(\mathop{\mathrm{dim}}\nolimits X+\mathop{\mathrm{dim}}\nolimits Y\), and hence \(\operatorname {\mathsf {size}}(X\times Y)\) is at most \(\operatorname {\mathsf {size}}(X)\times \operatorname {\mathsf {size}}(Y)\) times some factor that depends only on the dimension^{17} \(\mathop{\mathrm{dim}}\nolimits (X\times Y)\).
Moreover, if the dimensions are bounded, the product can be constructed in polynomial time.
Quotients and Attachments
If X, Y, and A are cell complexes with A⊆X and if f:A→Y is a cellular map, then the space X∪_{ f } Y obtained by attaching X to Y along f is also a cell complex in a natural way (see, e.g., [7, Sect. 2.3]). In particular, X/A is a cell complex, with cells corresponding to the cells of X not contained in A, plus one additional 0cell (corresponding to the image of A under the quotient map).
Similarly, if X is a simplicial set and if ∼ is an equivalence relation on each X _{ n } that is compatible with the face and degeneracy operators, then the quotient X/∼ is also a simplicial set. In particular, this includes simplicial attachments X∪_{ f } Y of simplicial sets along a simplicial map f:A→Y defined on a simplicial subset A⊆X, and quotients X/A by simplicial subsets. These constructions are compatible with geometric realizations, e.g., X∪_{ f } Y≅X∪_{f}Y.
Moreover, the size of X∪_{ f } Y is at most the size of X plus the size of Y, and in bounded dimension, the attachment can be constructed in polynomial time.
Wedge Sum (or Wedge Product)
If X _{1},…,X _{ m } are pointed spaces, then their wedge sum X _{1}∨⋯∨X _{ m } is simply the disjoint union of the X _{ i } with the basepoints identified (this is a very special type of attachment). If the X _{ i } are cell complexes or simplicial sets, then so is their wedge sum.
Mapping Cylinder and Mapping Cone
For a map f:X→Y, the mapping cylinder of f is the space \(\operatorname {Cyl}(f)\) defined as the quotient of (X×[0,1])⊔Y under the identifications (x,0)∼f(x) for each x∈X. The mapping cone \(\operatorname {Cone}(f)\) is defined as the quotient \(\operatorname {Cyl}(f)/(X\times\{1\})\) of \(\operatorname {Cyl}(f)\) with the subspace X×{1} collapsed into a point.
We will use the mapping cylinder in our construction to replace an arbitrary map f:X→Y by an inclusion \(X\hookrightarrow \operatorname {Cyl}(f)\), which has the same homotopy properties as f. A more precise statement is given in the following lemma (see, e.g., [9, Corollary 0.21]).
Lemma 3.3
 (a)
Y is a strong deformation retract ^{19} of \(\operatorname {Cyl}(f)\).
 (b)
X (considered as a subspace via i _{ X }) is a strong deformation retract of \(\operatorname {Cyl}(f)\) iff f is a homotopy equivalence.
 (c)
i _{ X }∼i _{ Y } f are homotopic as maps \(X\to \operatorname {Cyl}(f)\).
 (d)
If f:X→Y is a homotopy equivalence and if g:Y→X is a homotopy inverse for f, then i _{ X } g∼i _{ Y } as well.
Reduced Mapping Cone and Mapping Cylinder
If X and Y and f are pointed, with basepoints x _{0} and y _{0}, it will be technically convenient, particularly in Sect. 6, to consider the spaces \(\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (f)\) and \(\mathop{\mathrm{\widetilde {\operatorname {Cone}}}}\nolimits (f)\), called the reduced mapping cylinder and the reduced mapping cone, respectively, that are obtained from \(\operatorname {Cyl}(f)\) and \(\operatorname {Cone}(f)\) by collapsing the segment x _{0}×[0,1] (whose lower end is identified with y _{0}) to a single point. We will apply this construction only to cellular or simplicial mapping cylinders and cones, in which case contracting the subcomplex x _{0}×[0,1] is a homotopy equivalence.
Moreover, if f is a homotopy equivalence, then we may assume that its homotopy inverse g is pointed as well and that the homotopies fg≃id_{ Y } and gf≃id_{ X } keep the basepoints fixed (see [9, Corollary 0.19]). It follows that Lemma 3.3 remains true if we take \(C=\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (f)\) as the reduced mapping cylinder (the inclusions are given as those into \(\operatorname {Cyl}(f)\), followed by the quotient map \(\operatorname {Cyl}(f)\to \mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (f)\), which does not make any identifications within X or within Y).
By the remarks concerning the size of simplicial products and attachments, the size of the (reduced or unreduced) simplicial mapping cylinder or cone is at most the size of X plus the size of Y, times a factor depending only on \(\mathop{\mathrm{dim}}\nolimits X\).
3.4 Subdivisions and Simplicial Approximation
For simplicial complexes, there is the wellknown notion of barycentric subdivision (see, e.g., [18, §15]). An analogous notion of subdivision, called normal subdivision, can also be defined for simplicial sets. Informally speaking, the normal subdivision \(\operatorname {Sd}(X)\) of a simplicial set X is defined by barycentrically subdividing each simplex of X and then gluing these subdivided simplices together according to the identifications implied by the face and degeneracy operators of X. We refer to [7, Sect. 4.6] for the precise formal definition and just state the facts that we will need in what follows.
For the standard simplex Δ^{ p }, the nondegenerate ksimplices of \(\operatorname {Sd}(\Delta^{p})\) correspond to chains of proper inclusions of nondegenerate simplices (faces) of Δ^{ p }. It follows that \(\operatorname {Sd}(\Delta^{p})\) has (p+1)! nondegenerate psimplices and, in general, at most 2^{ p+1}(p+1)! nondegenerate simplices of any dimension. Consequently, for any simplicial set X, the size of \(\operatorname {Sd}(X)\) is at most the size of X times a factor that depends only on \(\mathop{\mathrm{dim}}\nolimits X\) and which can be bounded from above by \(2^{\mathop{\mathrm{dim}}\nolimits X+1}(\mathop{\mathrm{dim}}\nolimits X+1)!\). Moreover, if the dimension is bounded, \(\operatorname {Sd}(X)\) can be constructed in time polynomial in \(\operatorname {\mathsf {size}}(X)\).
If f:X→Y is a simplicial map, then subdivision also induces a map \(\operatorname {Sd}(f)\colon \operatorname {Sd}(X)\to \operatorname {Sd}(Y)\), and this is compatible with compositions, i.e., \(\operatorname {Sd}(fg)= \operatorname {Sd}(f) \operatorname {Sd}(g)\).
For each simplicial set X, there is a simplicial map \(\mathop{\mathrm{lastv}}\nolimits _{X}\colon \operatorname {Sd}(X)\to X\), called the last vertex map,^{20} which is a homotopy equivalence that is compatible with simplicial maps f:X→Y, i.e., \(f \mathop{\mathrm{lastv}}\nolimits _{X}=\mathop{\mathrm{lastv}}\nolimits _{Y} \operatorname {Sd}(f)\).^{21} ^{,} ^{22}
There is also a simplicial approximation theorem for simplicial sets, which uses iterated normal subdivisions. Specifically, the tfold iterated normal subdivision of a simplicial set is defined inductively as \(\operatorname {Sd}^{t}(X):=\operatorname {Sd}(\operatorname {Sd}^{t1}(X))\), where \(\operatorname {Sd}^{0}(X):=X\).
Theorem 3.4
[7, Theorem 4.6.25]
Let X and Y be simplicial sets such that X has only finitely many nondegenerate simplices, and let f:X→Y be a continuous map. Then there exist a finite integer t (which depends on f) and a simplicial map \(g\colon \operatorname {Sd}^{t}(X)\to Y\) such that g is homotopic to the composition \(f\mathop{\mathrm{lastv}}\nolimits _{X}^{t}\) of f with the iterated last vertex map \(\mathop{\mathrm{lastv}}\nolimits _{X}^{t}\colon \operatorname {Sd}^{t}(X)\to Y\).
To convert arbitrary simplicial sets into homotopy equivalent (in fact, homeomorphic) simplicial complexes, another subdivisionlike operation is needed (see, e.g., [11]). Given a simplicial set Z, one can define a simplicial complex B _{∗}(Z) inductively, by introducing a new vertex v _{ σ } for every nondegenerate simplex σ, and then replacing σ by the cone with apex v _{ σ } over B _{∗}(∂σ). If the simplicial set Z has a certain regularity property—which is satisfied, for instance, if \(Z=\operatorname {Sd}(X)\)—then B _{∗}(Z) and Z are homotopy equivalent (in fact, homeomorphic).^{23} We summarize the properties that we need in the following proposition (for completeness, we provide a proof in Appendix A).
Proposition 3.5
If X is a simplicial set, then the twofold subdivision \(B_{*}(\operatorname {Sd}(X))\) is a simplicial complex. Moreover, there is a simplicial map \(\gamma_{X}\colon B_{*}(\operatorname {Sd}(X))\to X\), which is a homotopy equivalence. For a simplicial subset A⊆X, \(B_{*}(\operatorname {Sd}(A))\) is a subcomplex of \(B_{*}(\operatorname {Sd}(X))\) and γ _{ X }_{ A }=γ _{ A }.^{24}
If X is finite and of bounded dimension, there are algorithms that construct the simplicial complex \(B_{*}(\operatorname {Sd}(X))\) and evaluate the map γ _{ X }, both in polynomial time.
4 Homotopy Groups
We review some further facts about homotopy groups that we will need. For more details see, e.g., [9, Sect. 4.1].
4.1 Basic Facts
So far, we used the definition of the nth homotopy group π _{ n }(X,x _{0}) of a pointed space (X,x _{0}) as the set of homotopy classes of pointed maps (S ^{ n },p _{0})→(X,x _{0}), where p _{0}∈S ^{ n } is an arbitrarily chosen basepoint, and the homotopies are required to keep the basepoint fixed. Equivalently, the elements of π _{ n }(X,x _{0}) can be viewed as homotopy classes [f] of maps f:(D ^{ n },∂D ^{ n })→(X,x _{0}) sending all of ∂D ^{ n } to x _{0}, modulo homotopies that keep the image of ∂D ^{ n } fixed (as before, we will often drop the basepoint from the notation).^{25}
In what follows, we will also need the Abelian group operation in π _{ n }(X,x _{0}), n≥2, which can be defined as follows: Suppose f _{1},…,f _{ m } are maps (D ^{ n },∂D ^{ n })→(X,x _{0}). Suppose we have a cellular decomposition of D ^{ n } as a cell complex Open image in new window with ncells \(e_{1}^{n},\ldots,e_{m}^{n}\) (in Sect. 6.1 we will provide a concrete geometric construction of Open image in new window ). Then we can define a map f from Open image in new window to (X,x _{0}) representing the homotopy class [f _{1}]+…+[f _{ m }] by sending the (n−1)skeleton of Open image in new window to x _{0}, and by defining the restriction of f to each open cell \(e_{i}^{n}\) to be f _{ i }.

The sphere S ^{ n } is (n−1)connected.

For all n≥1, π _{ n }(S ^{ n }) is isomorphic to \(\mathbb {Z}\) and generated by the homotopy class ι of the identity. For each map φ:S ^{ n }→S ^{ n }, there is a unique integer \(a\in \mathbb {Z}\) such that [φ]=aι; it is called the degree of φ and denoted by degφ. The degree is obviously invariant under homotopy.

We have \(\pi_{3}(S^{2})\cong \mathbb {Z}\). The group is generated by the famous Hopf map ^{26} η:S ^{3}→S ^{2}.
We will also need the following simple fact.
Lemma 4.1
Let g:S ^{ n }→S ^{ n } be a map of degree \(b\in \mathbb {Z}\). Then, for any map f:S ^{ n }→X, we have [fg]=b⋅[f]∈π _{ n }(X).
Proof
Consider the ndimensional unit cube I ^{ n }≅D ^{ n }, where I=[0,1] is the unit interval. We identify S ^{ n } with the quotient I ^{ n }/∂I ^{ n }. From the map of sets (I ^{ n },∂I ^{ n })→(I ^{ n },∂I ^{ n }) given by (s _{1},…,s _{ n−1},s _{ n })↦(s _{1},…,s _{ n−1},bs _{ n }mod1) we obtain g _{0}:S ^{ n }→S ^{ n } by passing to quotients. By the definition of the addition of homotopy classes, on the one hand, [g _{0}] is the bfold sum of the identity, and hence a particular example of a map of degree b. On the other hand, fg _{0} is a representative of b⋅[f], the bfold sum of [f]. Since [fg] depends only on the homotopy class [g], which is uniquely determined by the degree of g, the lemma follows. □
Let X be a cell complex and A⊆X a subcomplex. Then the homotopy groups of the spaces A, X, and X/A in a certain range are connected by an exact sequence.
Theorem 4.2
Proof
In the proof of Theorem 1.1 we will need a description of the nth homotopy group of a cell complex Y obtained from T by attaching (n+1)cells e _{ ℓ }, 1≤ℓ≤m, by attaching maps φ _{ ℓ }:S ^{ n }→T.
Proposition 4.3
Proof
4.2 Whitehead Products and Wedge Sums of Spheres
Whitehead Products
There is another type of operation on elements of homotopy groups that we will need. Consider two spheres S ^{ k } and S ^{ ℓ } with their standard structures as cell complexes (one vertex and one cell of the top dimension). Then the product S ^{ k }×S ^{ ℓ } is also a cell complex, with one vertex, one respective cell e ^{ k } and e ^{ ℓ } in dimensions k and ℓ, and one cell e ^{ k }×e ^{ ℓ } in dimension k+ℓ. In particular, the (k+ℓ−1)skeleton of the product is a wedge S ^{ k }∨S ^{ ℓ }, to which the (k+ℓ)cell is attached via a map φ:S ^{ k+ℓ−1}≅∂(D ^{ k+ℓ })→S ^{ k }∨S ^{ ℓ }.
Now, if f:S ^{ k }→X and g:S ^{ ℓ }→X are (pointed) maps, we can combine them into a map f∨g:S ^{ k }∨S ^{ ℓ }→X. If we compose this with the attachment map φ discussed before, we get a map [f,g]:S ^{ k+ℓ−1}→X, called the Whitehead product of f and g. The homotopy class of this product clearly depends only on the homotopy classes of the factors, so we get a welldefined product π _{ k }(X)×π _{ ℓ }(X)→π _{ k+ℓ−1}(X), again denoted by [⋅,⋅]. As a quite trivial but nonetheless useful example, if X=S ^{ k }×S ^{ ℓ }, then the attachment map φ itself equals the Whitehead product \([\iota_{S^{k}},\iota_{S^{\ell}}]\) of the two inclusions \(\iota _{S^{k}}\colon S^{k}\hookrightarrow S^{k}\vee S^{\ell}\) and \(\iota_{S^{\ell}} \colon S^{\ell}\hookrightarrow S^{k}\vee S^{\ell}\).
In the proof of Theorem 1.1 we will need some facts about the homotopy groups of spheres and their wedge sums.
Theorem 4.4
[9, Corollary 4B.2], [34, XI, Theorem 2.5]
There is a homomorphism (called the Hopf invariant) \(H\colon \pi_{2k1}(S^{k})\to\mathbb{Z}\) such that for d even H([ι,ι])=±2.
Let us note that for k odd the Whitehead product [ι,ι]∈π _{2k−1}(S ^{ k }) is of order 2, i.e., 2[ι,ι]=0. Whitehead products play a crucial role in Hilton’s theorem, which converts the computation of homotopy groups of a wedge of spheres to the computations of homotopy groups of spheres. We do not need this theorem in its full generality as it was proved in [10], and so we restrict ourselves to a special case.
Theorem 4.5
[10, Theorem A]
We will say that some elements x _{1},…,x _{ r } of an Abelian group are integrally independent if the only valid relation a _{1} x _{1}+⋯+a _{ r } x _{ r }=0 with coefficients \(a_{i}\in \mathbb {Z}\) is that with all a _{ i } zero. The following statement is an immediate consequence of Theorems 4.5 and 4.4.
Corollary 4.6
If k≥2 is odd, then the elements μ _{ q }, 1≤q≤s, and [ν _{ i },ν _{ j }], 1≤i<j≤r are integrally independent in π _{2k−1}(T).
If k≥2 is even, then the elements μ _{ q }, 1≤q≤s and [ν _{ i },ν _{ j }], 1≤i≤j≤r are integrally independent in π _{2k−1}(T).
Proof
The reason is that every element in the list comes from a different direct summand and is of infinite order. □
In the case k=2 and s=0 we can say even more.
Corollary 4.7
[9, Example 4.52]
The homotopy group \(\pi_{3}(\bigvee_{i=1}^{r}S_{i}^{2})\) is a free Abelian group generated by the Whitehead products [ν _{ i },ν _{ j }], 1≤i<j≤r, and homotopy classes ν _{ i } η, where η:S ^{3}→S ^{2} is the Hopf map.
5 The Constructions for Theorem 1.1 Presented as Cell Complexes
Here we present the essence of the proof of Theorem 1.1. Namely, for every system of quadratic Diophantine equations of the form (QSYM) (for k even) or (QSKEW) (for k odd), we construct cell complexes A, X, Y, and a continuous map f:A→Y, where Y is (k−1)connected and \(\mathop{\mathrm{dim}}\nolimits X=2k\), such that f is extendable to X iff the Diophantine system has a solution. Moreover, one of (X,A) and Y can be assumed to be fixed, as in Theorem 1.1(a) and (b). We will also see the role of Whitehead products and Hilton’s theorem in the proof.
What remains for the next section is to convert X, A, Y into finite simplicial complexes and f into a simplicial map, so that the solvability of the extension problem remains unchanged. Moreover, the construction has to be algorithmic.
While discussing the cellular constructions of X, A, Y, it is also natural to describe the cell complex used by Anick in the proof of his #Phardness result. Indeed, his construction uses tools very similar to those employed in our constructions.
The Generalized Extension Problem
Let i _{ A } and i _{ W } be the inclusions of A and W into X. On the one hand, given a solution \(\bar{f}\colon X\to Y\) of the extension problem, i.e., \(\bar{f}i_{A}=f\), we can define \(h:=\bar{f}i_{W}\) as the restriction of \(\bar{f}\) to W. Then \(hg=\bar{f}i_{W} g\sim\bar {f}i_{A}=f\), so h is a solution to the generalized extension problem.
On the other hand, given a solution h for the generalized extension problem (GEP), let r:X→W be the retraction from X to W and define \(\bar{f}:=hr\). Then \(\bar{f}i_{A}=hri_{A}\sim hg\sim f\), so \(\bar{f}\) is an extension of a map homotopic to f, and since extendability depends only on the homotopy class of a map (Corollary 3.1), f can be extended as well.
Thus, we are free to consider the generalized extension problem with \(\mathop{\mathrm{dim}}\nolimits W\le2k\) and \(\mathop{\mathrm{dim}}\nolimits A\le2k1\).
5.1 Fixed Target
We describe an instance of the generalized extension problem for part (a) of Theorem 1.1, where the target Y is fixed. The system of equations will be encoded into cell complexes A, W and the maps g:A→W, f:A→Y.
Fixed Target with k Even
Proposition 5.1
Let the maps f:A→S ^{ k } and g:A→W be as in (9) and (10) above. Then f can be extended to \(X=\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (g)\) if and only if the system (QSYM) has a solution.
Fixed Target with k Odd
5.2 Fixed Source
The idea for the first step of the proof of Theorem 1.1(b) is similar; however, the constructions involve attaching cells, and also the usage of Hilton’s theorem is more substantial.
Fixed Source with k Even
Lemma 5.3
Proof
The statement is a consequence of Proposition 4.3 and Corollary 4.6. □
Fixed Source with Odd k
Summarizing our findings, for k both even and odd we get the following.
5.3 Anick’s 4Dimensional Cell Complexes
Here we introduce complexes constructed by Anick [2, p. 42] for his hardness result. These are compact 4dimensional cell complexes which arise from the wedge \(W =S_{1}^{2}\vee \cdots\vee S_{r}^{2}\) of r copies of S ^{2} by attaching s 4cells. According to Corollary 4.7, the homotopy class of a general attaching map must be an integral combination of the homotopy classes ν _{ i } η and the Whitehead products [ν _{ i },ν _{ j }], where η is the homotopy class of the Hopf map S ^{3}→S ^{2} and ν _{ i } is the homotopy class of the inclusion S ^{2}→W on the ith copy of S ^{2}.
In Anick’s #Phardness result, the input complex \(Y^{4}_{ {\boldsymbol {a}}}\) (whose higher homotopy groups are to be computed) is encoded very concisely by the vector a of integers, represented in binary.
Theorem 5.5
(Anick [2])
It is #Phard to compute the rank of \(\pi_{n}(Y^{4}_{ {\boldsymbol {a}}})\) for a given integer n≥2 (encoded in unary) and a given integer vector a (represented in binary).
In Sect. 6, we will show that, given a, we can construct, in polynomial time, a finite 4dimensional simplicial complex homotopy equivalent to \(Y^{4}_{ {\boldsymbol {a}}}\). Together with Anick’s result, this will imply Theorem 1.2.
6 Simplicial Constructions
In this section, we prove that the constructions of cell complexes and cellular maps from the last section can be converted into homotopy equivalent finite simplicial complexes and simplicial maps. The constructions of such simplicial sets and maps involve only simplicial products, attachments (in particular, mapping cylinders), quotients, and subdivisions, which are all algorithmic. It is crucial, that in the case of Anick’s complexes, the algorithms are polynomial time.
We remark that the conversion of the cell complexes and maps for the extension problem (those from Propositions 5.1, 5.2, and 5.4) runs in time polynomial in the size of the binary encoding of the coefficients \(a_{ij}^{q}\) and the unary encoding of the coefficients b _{ q }. The running time could be made polynomial with respect to the binary encoding of all the coefficients including b _{ q }; however, it would complicate the presentation slightly, and it is unnecessary for the undecidability results.
Let us denote by Σ ^{ p } a “model” of the sphere S ^{ p } as a simplicial set with only two nondegenerate simplices, one in dimension 0 and the other in dimension p.
6.1 Constructing the Sum of Several Maps S ^{ p }→Y
In this short section we describe how, given simplicial maps f _{1},…,f _{ m }:Σ ^{ p }→Y, we can construct a simplicial representative of the sum [f _{1}]+⋯+[f _{ m }]∈π _{ p }(Y). To this end, we have to change the domain to a simplicial set with a larger number of simplices.
6.2 Generalized Mapping Cylinders
In the above approach, in order to construct simplicial maps Σ ^{ p }→Y, we replaced the domain Σ ^{ p } by a homotopy equivalent simplicial set. This will be very useful for the proof of part (a) of Theorem 1.1. For part (b), the domain has to be left unchanged, and thus a different construction must be used.^{27} It is roughly “dual” to the previous one: It replaces the target Y by a homotopy equivalent simplicial set.
Definition 6.1
Let M be a pointed simplicial set with two simplicial subsets X,Y⊆M containing the basepoint of M. Let i _{ X }:X→M and i _{ Y }:Y→M be the corresponding inclusion maps, and let f:X→Y be a pointed continuous map. We say that M is a generalized mapping cylinder for f, with upper rim X and lower rim Y, if i _{ Y } is a homotopy equivalence and i _{ X }∼i _{ Y } f. We denote this situation by Open image in new window .
We remark that the above definition depends only on the homotopy class of f; we may thus say that M is a generalized mapping cylinder for the homotopy class [f].
By Lemma 3.3 and the remark following it, the reduced mapping cylinder \(\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (f)\) of f:X→Y is an example of such a generalized mapping cylinder with upper rim X and lower rim Y. Moreover, if f is a homotopy equivalence with a homotopy inverse g, one can easily see from the definition that a generalized mapping cylinder M for f is also a generalized mapping cylinder \(M^{\operatorname{op}}\) for g with upper rim Y and lower rim X (i.e., the roles of upper and lower rim are interchanged).^{28}
The important property of generalized mapping cylinders, which we are going to use heavily, is that they may be used for attaching cells.
Proposition 6.2
Proof
Set \(X=\varSigma^{p}_{1}\vee\cdots\vee\varSigma^{p}_{m}\). Then the space from the statement, obtained from Y by attaching cells, is the mapping cone of f. By [3, Theorem I.14.19] the mapping cone of f:X→Y is homotopy equivalent to that of i _{ Y } f:X→M (since i _{ Y } is a homotopy equivalence). Further, by [3, Theorem I.14.18] it is also homotopy equivalent to the mapping cone of i _{ X }:X→M (since i _{ Y } f∼i _{ X }). By [3, Theorem VII.1.6] the mapping cone of i _{ X } is homotopy equivalent to M/X.
All the involved maps respect Y (which is naturally a subspace of all the mapping cones and also maps by \(\operatorname{proj}i_{Y}\) to the quotient M/X), proving that the resulting homotopy equivalence is indeed an extension of the composition \(\operatorname{proj}i_{Y}\). □
Generalized mapping cylinders can be composed in an obvious way.
Lemma 6.3
Let f:X→Y and g:Y→Z be pointed continuous maps, and let M and N be generalized mapping cylinders for f and g, respectively. Let NM:=N∪_{ Y } M be the simplicial set obtained by identifying the lower rim of M with the upper rim of N. Then NM is a generalized mapping cylinder for gf.
Proof
It is well known (see [9, Theorem 4.5]) that to every inclusion i _{ Y }:Y→M which is a homotopy equivalence there is a deformation retraction r:M→Y. Then the map h:N∪_{ Y } M→N defined as j _{ Y } r on M and as the identity on N is a homotopy inverse to i _{ N }. □
We will also need simplicial maps representing (the homotopy classes of) a constant number of specific maps, such as the Whitehead product [ι,ι]:S ^{2d−1}→S ^{ d } of the identity on S ^{ d } with itself, the Whitehead product \([\iota_{1},\iota_{2}]\colon S^{2d1}\to S^{d}_{1}\vee S_{2}^{d}\) of the inclusions \(S_{i}^{d}\to S^{d}_{1}\vee S^{d}_{2}\), and the Hopf map η:S ^{3}→S ^{2}. In each case, it is possible to construct these explicitly, but we will use the following general lemma.
Lemma 6.4
Let X and Y be finite simplicial sets and let f:X→Y be an arbitrary but fixed pointed continuous map. Then there exists a generalized mapping cylinder Open image in new window . It is of dimension \(\max\{\mathop{\mathrm{dim}}\nolimits X+1,\mathop{\mathrm{dim}}\nolimits Y\}\).
The point here is that, in contrast with Theorem 3.4, we can prescribe the exact triangulations of the upper and lower rims, which will make it easy to compose the resulting generalized mapping cylinders.
Proof
By the simplicial approximation theorem for simplicial sets (Theorem 3.4), there exist an iterated barycentric subdivision \(X'= \operatorname {Sd}^{t}(X)\) of X and a simplicial map g:X′→Y homotopic to fℓ, where ℓ:X′→X is the natural homotopy equivalence (the iterated last vertex map).
Let \(M:=\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (\ell)\) and \(N:=\mathop{\mathrm{\widetilde {\operatorname {Cyl}}}}\nolimits (g)\) be the corresponding reduced simplicial mapping cylinders. Since ℓ is a homotopy equivalence, we can also view M as a generalized mapping cylinder M ^{op} for a homotopy inverse h:X→X′ of ℓ, with upper rim X and lower rim X′. Thus, NM ^{op} is a generalized mapping cylinder for gh∼fℓh∼f. □
The following proposition plays a crucial role in our simplicial constructions.
Proposition 6.5
The proof will be given in a series of lemmas. Before going into the proof, we will generalize this proposition slightly. A homotopy class of a pointed map \(f\colon \varSigma^{p}_{1}\vee\cdots\vee\varSigma^{p}_{s}\to Y\) is determined uniquely by its restrictions \(\varSigma^{p}_{k}\to Y\). When each restriction is expressed as an integral combination of the f _{ i }, we may use Proposition 6.5 together with Lemma 6.7 to provide a generalized mapping cylinder for f.
Lemma 6.6
There is an algorithm that, given an integer of the form c=±2^{ d }, \(d\in \mathbb {N}\), constructs a generalized mapping cylinder Open image in new window for the map of degree c. Moreover, if p is fixed, then \(\operatorname {\mathsf {size}}(N_{c})\) is linear in d+1 and the running time of the algorithm is polynomial in d+1, which is the encoding size of c.
Proof
Consider maps g _{−1},g _{2}:Σ ^{ p }→Σ ^{ p } of degrees −1 and 2, respectively, and choose a generalized mapping cylinder N _{ i } for each g _{ i } according to Lemma 6.4. For p constant, these are fixed simplicial sets.
Moreover, if p is fixed, then we can precompute the generalized mapping cylinders N _{−1}, N _{2}, which leads to \(\operatorname {\mathsf {size}}(N_{c})\) and running time as requested. □
Lemma 6.7
Suppose that X _{1},…,X _{ m } and Y are pointed simplicial sets and that Open image in new window are generalized mapping cylinders for pointed maps f _{ i }:X _{ i }→Y, 1≤i≤m. Then there is an algorithm that constructs a generalized mapping cylinder Open image in new window for the map f:X _{1}∨⋯∨X _{ m }→Y with restrictions \(f_{X_{i}}=f_{i}\).
Moreover, if p is fixed, then \(\operatorname {\mathsf {size}}(N)\) is linear in \(\sum_{i}\operatorname {\mathsf {size}}(M_{i})\) and the running time is polynomial.
Proof
Lemma 6.8
Let Open image in new window be generalized mapping cylinders for [f _{1}],…,[f _{ m }]∈π _{ p }(Y). Then there is an algorithm that constructs a generalized mapping cylinder Open image in new window for the homotopy class [f _{1}]+…+[f _{ m }]∈π _{ p }(Y), in polynomial time if p is fixed.
Proof
Proof of Proposition 6.5
Let Open image in new window , i=1,…,m, be generalized mapping cylinders.
Using the binary expansion of an integer c, Lemma 6.8, and Lemma 6.6, we can construct the generalized mapping cylinder N _{ c } for every map Σ ^{ p }→Σ ^{ p } of degree c in time polynomial (at most quadratic) in the bit length of c. The composition \(M_{i}N_{c_{i}}\) is a generalized mapping cylinder for c _{ i }[f _{ i }]. Lemma 6.8 then constructs a generalized mapping cylinder for the sum c _{1}[f _{1}]+⋯+c _{ m }[f _{ m }]. □
6.3 Proofs of the Main Results
In Sect. 5 we described the relevant spaces as cell complexes. It remains to construct them as finite simplicial complexes and the map f:A→Y as a simplicial map.
Proof of Theorem 1.1(a)
Since both γ _{ X′} and γ _{ A′} are homotopy equivalences, the extendability of f′ is equivalent to that of f′γ _{ A′} by Corollary 3.1. □
Proof of Theorem 1.1(b)
It remains to replace \(\tilde{f}\) by a simplicial map. But since f is a combination of the Whitehead products and the remaining maps i _{ T } and \(\operatorname{proj}\) are simplicial, we may achieve this by replacing M/S further by the generalized mapping cylinder Open image in new window as in Sect. 6.2. We denote the inclusion Σ ^{2k−1}→Y′ by f′. From the definition of the generalized mapping cylinder, \(i_{M/S} \tilde{f}\sim f'\) and i _{ M/S } is a homotopy equivalence, and therefore the extendability of \(\tilde{f}\) is equivalent to that of f′. This finishes the construction of a simplicial replacement of the extension problem.
To make everything into simplicial complexes, we apply \(B_{*}\operatorname {Sd}\) to all the involved simplicial sets A′=Σ ^{2k−1}, X′, Y′ and the simplicial map f′. □
Proof of Theorem 1.2
To get the statement of Theorem 1.2, it is now sufficient to realize that the algorithmic construction above can be carried out in the time polynomial in the binary encoding of the vector \(a_{ij}^{(q)}\), 1≤i≤j≤r, 1≤q≤s. □
Footnotes
 1.
More precisely, f and g are homotopic, in symbols f∼g, if there is a map F:X×[0,1]→Y such that F(⋅,0)=f and F(⋅,1)=g. With this notation, [X,Y]={[f]:f:X→Y}, where [f]={g:g∼f} is the homotopy class of f.
 2.
We recall that the kth homotopy group π _{ k }(Y) of a space Y is defined as the set of all homotopy classes of pointed maps f:S ^{ k }→Y, i.e., maps f that send a distinguished basepoint s _{0}∈S ^{ k } to a distinguished basepoint y _{0}∈Y (and the homotopies F also satisfy F(s _{0},t)=y _{0} for all t∈[0,1]). Strictly speaking, one should write π _{ k }(Y,y _{0}) but for a pathconnected Y, the choice of y _{0} does not matter. Moreover, if Y is simply connected, i.e., if π _{1}(Y) is trivial, then the pointedness of the maps does not matter either and one can identify π _{ k }(Y) with [S ^{ k },Y]. For k≥1, each π _{ k }(Y) is a group, which for k≥2 is Abelian; the definition of the group operation will be reviewed in Sect. 4.
 3.
 4.
In the border case \(\mathop{\mathrm{dim}}\nolimits X=2k1\), the algorithm just decides the existence of an extension, while for \(\mathop{\mathrm{dim}}\nolimits X\leq2k2\) it also yields a classification of all possible extensions up to homotopy.
 5.
We remark that the stable range assumption guarantees that [X,Y] has a canonical Abelian group structure, which we exploit heavily (for instance, it means that [X,Y] has a finite description even when it is an infinite set). In the special case π _{ k }(Y)≅[S ^{ k },Y], by contrast, the group structure has a different origin and is available for all dimensions k.
 6.
Note that with a unary encoding of n, the size of the input is significantly larger than with a binary (or decimal encoding), and hence the hardness result is correspondingly stronger.
 7.
Somewhat informally, the class of #Phard problems consists of computational problems that should return a natural number (as opposed to YES/NO problems) and are at least as hard as counting the number of all Hamiltonian cycles in a given graph, or counting the number of subsets with zero sum for a given set of integers, etc. These problems are clearly at least as hard as NPcomplete problems, and most likely even less tractable.
 8.
Actually, the hardness already applies to the potentially easier problem of computing the rational homotopy groups \(\pi_{n}(Y)\otimes \mathbb {Q}\); practically speaking, one asks only for the rank of π _{ n }(Y), i.e., the number of direct summands isomorphic to \(\mathbb {Z}\).
 9.
Spaces X and Y are homotopy equivalent if there are maps f:X→Y and g:Y→X such that the compositions fg and gf are homotopic to identities. From the point of view of homotopy theory, such X and Y are indistinguishable and, in particular, π _{ k }(X)=π _{ k }(Y) for all k≥0.
 10.
The idea is to represent higher degree monomials in the general system using new variables; e.g., for the monomial x ^{3} y we can introduce new variables t _{1},t _{2}, new quadratic equations t _{1}=x ^{2} and t _{2}=xy, and replace x ^{3} y by t _{1} t _{2}.
 11.
Cell complexes can also be infinite dimensional, in which case some care must be taken in defining their topology, but we will deal with cell complexes that are made of finitely many cells, and are thus finite dimensional.
 12.
The composition of the inclusion \(D_{i}^{n} \hookrightarrow (\bigsqcup_{i}D_{i}^{n} )\sqcup X^{(n1)}\) with the quotient map.
 13.
More precisely, the vertex set of each simplex is equipped with an ordering, and the identifications are required to be weakly orderpreserving maps (not necessarily injective) between the vertex sets.
 14.
Moreover, this sequence is unique, by the simplicial set axioms that we have not specified, if one stipulates i _{ k }<⋯<i _{1}.
 15.
A notable exception are subdivisions, for which we provide more detail in Appendix A.
 16.
To be more precise, the above equality holds literally, with the product topology on the righthand side, only under suitable assumptions on X and Y, e.g., if both X and Y have only countably many simplices. In the general case, one has to interpret the product X×Y differently, in the category of socalled kspaces, and the same subtlety arises for products of cell complexes; see, e.g., the discussion in the respective appendices in [7, 9]. For the spaces we will encounter, however, this issue will not arise, and the product will be the same as the usual product of topological spaces.
 17.
This follows from the fact about realizations mentioned above. Another way of seeing this is that if \(\mathop{\mathrm{dim}}\nolimits \sigma=p\), \(\mathop{\mathrm{dim}}\nolimits \tau =q\), and \(\mathop{\mathrm{dim}}\nolimits (s\sigma)=\mathop{\mathrm{dim}}\nolimits (t\tau)=n> p+q\), then s and t involve n−p and n−q degeneracy operators s _{ i } with i≤n, respectively, so there must be a repetition since n−p+n−q>n. Without further reflection, this immediately implies that \(\operatorname {\mathsf {size}}(X\times Y)\leq \operatorname {\mathsf {size}}(X)\cdot \operatorname {\mathsf {size}}(Y)\cdot(\mathop{\mathrm{dim}}\nolimits X)!(\mathop{\mathrm{dim}}\nolimits Y)!\).
In fact, the factor is only singly exponential in the dimensions. For instance, for a product Δ^{ p }×Δ^{ q } of two standard simplices, the vertices of Δ^{ p }×Δ^{ q } correspond to the grid points in {0,…,p}×{0,…,q}, and the nondegenerate ksimplices correspond to subsets of size k+1 of the grid that are weakly monotone in both coordinates (weakly monotone paths of length k). Thus, the number of nondegenerate simplices of full dimension p+q equals \(\binom{p+q}{p}\), and the number of all nondegenerate simplices is at most 4^{ p+q }, say. Thus, \(\operatorname {\mathsf {size}}(X\times Y)\leq \operatorname {\mathsf {size}}(X)\cdot \operatorname {\mathsf {size}}(Y)\times4^{\mathop{\mathrm{dim}}\nolimits X+\mathop{\mathrm{dim}}\nolimits Y}\), say.
 18.
More precisely, the inclusion maps are given as the composition of the respective inclusions X≅X×{1}⊆X×[0,1]⊔Y and Y⊆X×[0,1]⊔Y with the quotient map \(X\times [0,1]\sqcup Y \to \operatorname {Cyl}(f)\).
 19.
We recall that a deformation retraction of a space X onto a subspace A is a map H:X×[0,1]→X such that H(x,0)=x and H(x,1)∈A for all x∈X and H(a,1)=a for all a∈A. Thus, a deformation retraction witnesses that the inclusion map i _{ A }:A↪X is a homotopy equivalence with a homotopy inverse r=H(⋅,1):X→A that is a retraction, i.e., that restricts to the identity on A.
A deformation retraction is called strong if it keeps A fixed pointwise throughout, i.e., if H(a,t)=a for all a∈A and t∈[0,1] (some authors include this directly in the definition of a deformation retraction).
 20.
On the standard simplex Δ^{ n }, seen as a simplicial set, this map is defined by sending a chain (σ _{0},…,σ _{ k }) (a ksimplex of \(\operatorname {Sd}(\Delta^{n})\)) to the simplex [v _{0},…,v _{ k }], where v _{ i } is the last vertex of the simplex σ _{ i } (recall that the vertices in each simplex are ordered).
 21.
In the language of category theory, \(\operatorname {Sd}\) is a functor and \(\mathop{\mathrm{lastv}}\nolimits \) is a natural transformation between \(\operatorname {Sd}\) and the identity functor on simplicial sets.
 22.
In fact, it is true that X and \(\operatorname {Sd}(X)\) are not only homotopy equivalent but homeomorphic (as one might expect given the terminology “subdivision”). However, for simplicial sets this is a decidedly nontrivial result; see [7, Corollary 4.6.5]. The difficulty is related to the fact that there is no way of defining this homeomorphism for all simplicial sets in such a way that it becomes compatible with simplicial maps. For our purposes, the natural homotopy equivalence \(\mathop{\mathrm{lastv}}\nolimits _{X}\) will be sufficient and more convenient.
 23.
As an illustration that this fails for general simplicial sets, consider the case where Z=Σ ^{ p } is the simplicial set model of the dsphere with only two nondegenerate simplices, one in dimension 0 and one in dimension d. In this case, B _{∗}(Σ ^{ d }) is a 1dimensional simplex.
 24.
In fact, the construction \(B_{*}\operatorname {Sd}\) is functorial and γ is a natural transformation (like the construction \(\operatorname {Sd}\) and the map \(\mathop{\mathrm{lastv}}\nolimits \)), but we will never use this stronger fact.
 25.
The claimed equivalence is obtained by identifying S ^{ n } with the quotient D ^{ n }/∂D ^{ n } of the ndisk by its boundary and p _{0} with the image of ∂D ^{ n } under the quotient map.
 26.
See [9, Example 4.45] for the definition (which is not difficult, but which we will not need).
 27.
There is a further issue with the subdivision—it is not polynomial. The construction of a representative of a multiple b[f] of a map f:Σ ^{ p }→Y requires b simplices, and this number is exponential in the number of bits of b.
 28.
When f is injective, one can use Y as a generalized mapping cylinder for f.
 29.
When b _{ q }=0, we take \(A'_{q}= \operatorname {Sd}^{t}(\varSigma^{2k1})\) and \(f'_{q} \colon \operatorname {Sd}^{t}(\varSigma^{2k1})\to S^{k}\) the constant map onto the basepoint.
 30.
The important ingredients are that every simplex can be expressed uniquely as a degeneracy of a nondegenerate simplex, and that every operator can be written uniquely as a degeneracy of a face.
 31.
This is not too hard to show, but we do not want to dwell into the exact definition of \(\operatorname {Sd}_{*}(\operatorname {Sd}(X))\). The main point is that the preimages under π of the simplex σ ^{0}>⋯>σ ^{ n } are given by the choices of the subsequences φ. The commutativity is then implied by the independence of the operators (σ ^{ i−1})_{0}→(σ ^{ i })_{0} on these choices.
 32.
On the other hand, it is possible to compute these homology groups for any 1connected X. This is another application of the “brute force” version of Theorem B.2 that is explained just prior to its statement.
 33.
Formally, we define \(\operatorname {\mathcal {K}}^{0}(Y):=Y\) and \(\operatorname {\mathcal {K}}^{n}(Y):=\operatorname {\mathcal {K}}(\operatorname {\mathcal {K}}^{n1}(Y))\) for n≥1.
Notes
Acknowledgements
We are grateful to Maurice Rojas for providing us useful information on Diophantine equations.
This research was supported by the ERC Advanced Grant No. 267165. The research of M.Č. was supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. The research by M.K. and J.M. was supported by the Center of Excellence, Inst. for Theor. Comp. Sci., Prague (project P202/12/G061 of GA ČR). The research of L.V. was supported by the Center of Excellence, Eduard Čech Institute (project P201/12/G028 of GA ČR). The research by U.W. was supported by the Swiss National Science Foundation (grants SNSF200020138230 and SNSFPP00P2138948).
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