1 Introduction
How difficult is it to describe an explicit presentation of an abstract mathematical structure? For a particular structure S, what is the degree spectrum of its isomorphism type, i.e., what are the Turing degrees that compute a presentation of a structure that is isomorphic to S? For a class of structures, what are the possible degree spectra of their isomorphism types? These have long been the fundamental questions in computable structure theory, and researchers in this area have obtained a huge number of interesting results on Turing degrees of presentations of isomorphism types of groups, rings, fields, linear orders, lattices, Boolean algebras, and so on [Reference Ash and Knight2, Reference Ershov, Goncharov, Nerode, Remmel and Marek12, Reference Fokina, Harizanov and Melnikov14, Reference Hirschfeldt, Khoussainov, Shore and Slinko20].
In this article we focus on presentations of compact Polish spaces. The notion of a presentation plays a central role, not only in computable structure theory, but also in computable analysis [Reference Brattka and Hertling4, Reference Brattka, Hertling and Weihrauch5, Reference Weihrauch43]. In this area, one of the most crucial problems was how to present large mathematical objects (which possibly have the cardinality of the continuum) such as metric spaces and topological spaces, and then researchers have obtained a number of answers to this question. In particular, the notion of a computable presentation of a Polish space can be traced back to the 1960s; e.g., [Reference Moschovakis34]. Computable Polish spaces have been extensively studied in computable analysis [Reference Brattka and Hertling4, Reference PourEl and Ian Richards38, Reference Weihrauch43] and descriptive set theory [Reference Moschovakis35].
In recent years, several researchers have succeeded in obtaining various results on Turing degrees of isometric isomorphism types of Polish spaces, separable Banach spaces, and Polish groups; see [Reference Clanin, McNicholl and Stull8, Reference Lupini, Melnikov and Nies28, Reference McNicholl and Stull29, Reference Melnikov31]. All these works are devoted to metric structures. Results on Turing degrees of homeomorphism types of Polish spaces have been obtained only very recently, and independently of the present article, in [Reference Downey and Melnikov9, Reference HarrisonTrainor, Melnikov and Ng18, Reference Lupini, Melnikov and Nies28, Reference Melnikov32]. Computable Polish groups were investigated in [Reference Greenberg, Melnikov, Nies and Turetsky15, Reference Melnikov30], and general topological spaces were studied in [Reference Selivanov40] in analogy with the earlier investigation of degrees of isomorphism types of algebraic structures. Some results were also obtained for domains.
In this article, a Polish presentation of a Polish space X is a dense sequence $(s_i)_{i\in \mathbb {N}}$ in X and a complete metric d inducing the topology; for a Turing degree ${\mathbf {d}}$ , such a presentation is ${\mathbf {d}}$ computable if the real numbers $d(s_i,s_j)$ are uniformly ${\mathbf {d}}$ computable. The Polish degree spectrum of X is then the set of Turing degrees that compute a Polish presentation of X. We will also be interested in compact presentations of a compact Polish space X, that carry a Polish presentation together with an enumeration of the finite open covers of X. The compact degree spectrum of X is the set of Turing degrees that compute a compact presentation of X.
One of the first questions to be asked about presentations of Polish spaces is:
Question 1. Does there exist a Polish space with a $\mathbf {0}'$ computable Polish presentation but not computable one?
Observe that there are continuum many homeomorphism types of Polish spaces, but there exist only countably many computable presentations of Polish spaces, so there is a Polish space which is not homeomorphic to any computably presented Polish space.
Countable spaces cannot be used to solve this problem because of the “hyperarithmeticisrecursive” phenomenon, see [Reference Greenberg and Montalbán16]; see also Section 3. In this article we answer Question 1 in the affirmative. We note that HarrisonTrainor, Melnikov, and Ng [Reference HarrisonTrainor, Melnikov and Ng18] have recently obtained an independent solution, and via a different proof. One possible approach to solve this problem is using Stone duality between countable Boolean algebras and zerodimensional compact Polish spaces; see also Section 3.1. Combining this idea with classical results on isomorphism types of Boolean algebras [Reference Knight and Stob27], one can conclude that every low $_4$ presented zerodimensional compact Polish space is homeomorphic to a computable one. This was also independently noticed by HarrisonTrainor, Melnikov, and Ng [Reference HarrisonTrainor, Melnikov and Ng18].
Our next step is to develop new techniques beyond Stone duality. More specifically, the next question is whether there exists a Polish space whose Polish degree spectrum is different from that of a zerodimensional compact space. In particular, it is natural to ask the following:
Question 2. Does there exist a Polish space with a low $_4$ Polish presentation but no computable one?
One of the main results of this article is that for every Turing degree ${\mathbf {d}}$ and every $n\geq 1$ , there exists a compact Polish space $\mathcal {X}_{{\mathbf {d}},n}$ whose compact degree spectrum is $\{\mathbf {x}:{\mathbf {d}}\leq \mathbf {x}^{(n)}\}$ and Polish degree spectrum is $\{\mathbf {x}:{\mathbf {d}}\leq \mathbf {x}^{(n+1)}\}$ (Corollaries 5.2 and 5.12). This result was independently obtained by Melnikov [Reference Melnikov32], Lupini, Melnikov, and Nies [Reference Lupini, Melnikov and Nies28], and Downey and Melnikov [Reference Downey and Melnikov9] using different arguments, namely by proving that Čech cohomology groups of a compact space are computable. For $n=1$ the result presented here is obtained by investigating the computable aspects of Čech homology groups. A part of the analysis of homology groups is a direct adaptation of the analysis of cohomology groups from [Reference Downey and Melnikov9, Reference Lupini, Melnikov and Nies28], then the arguments diverge: Čech homology groups are not computable in general, but they are sufficiently effective for our purpose. More precisely, we show that the nontriviality of Čech homology groups with coefficients in $\mathbb{Z}/2\mathbb{Z}$ is $\Sigma^0_2$ ; along the way, we also show that it is $\Sigma^1_1$ complete when taking coefficients in $\mathbb{Z}$ .
The result gives a solution to Questions 1 and 2: there is a space with a $\mathbf {0}'$ computable low $_3$ Polish presentation but no computable Polish presentation, namely the space $\mathcal {X}_{{\mathbf {d}}",1}$ for any ${\mathbf {d}}\leq \mathbf {0}'$ which is low $_3$ . It also implies the existence of a compact Polish space having a computable Polish presentation but no computable compact presentation (Corollary 5.3).
Another consequence is that for $n\geq 2$ there is a space whose Polish degree spectrum is the set of high $_n$ degrees, namely $\mathcal {X}_{{\mathbf {d}},n1}$ where ${\mathbf {d}}=\mathbf {0}^{(n+1)}$ . This also clarifies substantial differences between zerodimensional compact Polish spaces and infinitedimensional ones since the class of high $_n$ degrees is never the degree spectrum of a Boolean algebra [Reference Jockusch and Soare24].
Another important question is whether a given Polish space has a least Turing degree in its Polish degree spectrum. For instance, it is known that the isomorphism types of linear orders, trees, abelian pgroups, etc. have no least presentation whenever they are not computably presentable, see [Reference Fokina, Harizanov and Melnikov14].
Question 3. Does there exist a compact Polish space with no computable Polish presentation, but whose Polish degree spectrum contains a least Turing degree?
We answer Question 3 in the negative. More precisely, we show the coneavoidance theorem for compact Polish spaces, which states that, for any nonc.e. set $A\subseteq \mathbb {N}$ , every compact Polish space has a Polish presentation that does not enumerate A (Theorem 6.1).
It contrasts with a result proved in [Reference Lupini, Melnikov and Nies28] that for each set $A\subseteq {\mathbb {N}}$ , there exists a compact Polish space whose compact degree spectrum is $\{\textbf {x}:A\text { is} $ x $\text {c.e.}\}$ . If $(p_i)_{i\in \mathbb {N}}$ is the increasing enumeration of the prime numbers, this space is defined as the Pontryagin dual $\hat {G}$ of the subgroup G of $\mathbb {Q}$ generated by the elements $\frac {1}{p_i}$ with $i\in A$ .
We finally prove a precise relationship between the Polish degree spectrum and the compact degree spectrum of compact Polish spaces that are perfect (Corollary 7.2), and show that it fails for some nonperfect space (Proposition 7.5).
The article is organized as follows. In Section 2 we present the necessary background and develop the technical tools that will be used throughout the article. In Section 3 we briefly discuss the effective aspects of the Cantor–Bendixson derivative and of Stone duality. In Section 4 we investigate the computable aspects of the Čech homology groups, which will be used in Section 5, where we show how to realize certain sets of Turing degrees as degree spectra of compact Polish spaces. In Section 5 we show the coneavoidance theorem. In Section 6 we compare degree spectra and compact degree spectra.
2 Preliminaries
Basic terminology and results of computability theory and computable structure theory are summarized in [Reference Ash and Knight2]. For basics of computable analysis, we refer the reader to [Reference Avigad and Brattka3–Reference Brattka, Hertling and Weihrauch5, Reference Weihrauch43]. For the basic definitions and facts of general topology and dimension theory, see [Reference Hurewicz and Wallman22, Reference van Mill33].
2.1 Presentations of Polish spaces
If X is a Polish space, then a Polish presentation (or simply a presentation) of X is $(X,d,S)$ where $S=(s_n)_{n\in \mathbb {N}}$ is a dense sequence in X and d is a complete metric which is compatible with the topology. For a discussion of presentations of Polish spaces, see also [Reference Gregoriades, Kispéter and Pauly17].
For a Turing degree ${\mathbf {d}}$ , a presentation $(X,d,S)$ of X is ${\mathbf {d}}$ computable if the real numbers $d(s_i,s_j)$ are uniformly ${\mathbf {d}}$ computable. A $\mathbf {d}$ computable Polish space is a Polish space which has a $\mathbf {d}$ computable presentation. For a Polish space X, its Polish degree spectrum $\mathrm {Sp}(X)$ is the set of all Turing degrees $\mathbf {d}$ such that X has a $\mathbf {d}$ computable Polish presentation.
Let $(X,d,S)$ be a presentation of a Polish space X. A rational open ball is $B(s_i,r)=\{x\in X:d(s_i,x)<r\}$ where $r>0$ is rational. A rational open set is a finite union of rational open balls. A code of a finite rational open cover of X is a finite set $E\subseteq \mathbb {N}\times \mathbb {Q}_{>0}$ such that $X=\bigcup _{(i,r)\in E}B(s_i,r)$ . If X is compact, then a compact presentation of X is a presentation of X equipped with an enumeration of the codes of all finite rational open covers of X. In particular, a compact presentation contains an information of total boundedness; that is, a function $\ell :\mathbb {N}\to \mathbb {N}$ such that for all n, $\{B(a_i,2^{n}):i<\ell (n)\}$ covers the whole space X.
A compact Polish space X is $\mathbf {d}$ computably compact if it has a $\mathbf {d}$ computable compact presentation. Its compact degree spectrum $\mathrm {Sp}_c(X)$ is the set of all Turing degrees $\mathbf {d}$ such that X has a $\mathbf {d}$ computable compact presentation.
We will often use the next elementary result.
Lemma 2.1. Let ${X}$ be a compact Polish space. If ${X}$ has a $\mathbf {d}$ computable Polish presentation, then ${X}$ has a $\mathbf {d}'$ computable compact presentation.
Proof Assume that X has a ${\mathbf {d}}$ computable Polish presentation $(X,d,S)$ . By compactness of X, one can observe that E is a code of a finite rational open cover of X if and only if there exists $s\in {\mathbb {N}}$ such that for all $x\in X$ we have $d(x,s_i)\leq r2^{s}$ for some $(i,r)\in E$ . The latter is equivalent to the existence of $s\in {\mathbb {N}}$ such that for all $j\in {\mathbb {N}}$ we have $d(s_j,s_i)\leq r2^{s}$ for some $(i,r)\in E$ . As E is finite, this is a $\Sigma ^0_2$ condition relative to ${\mathbf {d}}$ , so it is ${\mathbf {d}}'$ c.e. In other words, X has a $\mathbf {d}'$ computable compact presentation.
In Section 6, we will present a sharp analysis of this relationship between Polish and compact presentations.
2.1.1 Hyperspaces.
Every Polish space embeds in the Hilbert cube, and this fact induces an equivalent definition of Polish and compact presentations. The Hilbert cube $Q=[0,1]^{\mathbb {N}}$ is endowed with the complete metric $d_Q(x,y)=\sum _i2^{i}x_iy_i$ . A point of Q is rational if its coordinates are rational and only finitely many of them are nonzero. The rational points of Q, enumerated in a canonical way, make Q a computable Polish space which is computably compact.
Let $\mathcal {V}(Q)$ be the hyperspace of compact subsets of Q endowed with the lower Vietoris topology. A subbasis is given by $\{K\subseteq Q:K\cap B\neq \emptyset \}$ , where B is a rational ball in Q (technically, we need to add $\{\emptyset \}$ , which is the singleton containing the empty compact set, to make it a subbasis).
Let $\mathcal {K}(Q)$ be the hyperspace of compact subsets of Q endowed with the Vietoris topology. A subbasis is given by a subbasis for the lower Vietoris topology, together with $\{K\subseteq Q:K\subseteq U\}$ , where U is a rational open set in Q. The Hausdorff metric $d_H$ is a complete metric generating the Vietoris topology, and the dense sequence of finite sets of rational points of Q makes $\mathcal {K}(Q)$ a computable Polish space which is computably compact.
We say that a compact set $K\subseteq Q$ is ${\mathbf {d}}$ computably overt if ${\mathbf {d}}$ computes an enumeration of the basic neighborhoods of K in the lower Vietoris topology. Equivalently, K is ${\mathbf {d}}$ computably overt if it contains a dense computable sequence. We say that K is ${\mathbf {d}}$ computably compact if ${\mathbf {d}}$ computes an enumeration of the basic neighborhoods of K in the Vietoris topology. Equivalently, K is ${\mathbf {d}}$ computably compact if K is ${\mathbf {d}}$ computably overt and $K\in \Pi ^0_1({\mathbf {d}})$ , or if ${\mathbf {d}}$ computes a sequence of finite sets $K_n\subseteq Q$ of rationals points such that $d_H(K_n,K)<2^{n}$ .
The next result is folklore. The two parts appear in [Reference Amir and Hoyrup1, Fact 2.11] and [Reference Downey and Melnikov9, Theorem 3.36], respectively.
Proposition 2.2. A compact Polish space ${X}$ has a $\mathbf {d}$ computable Polish presentation if and only if it has a ${\mathbf {d}}$ computably overt copy in Q.
A compact Polish space ${X}$ has a $\mathbf {d}$ computable compact presentation if and only if it has a ${\mathbf {d}}$ computably compact copy in Q.
Proof It is essentially an effective version of the fact that every Polish space embeds in Q.
Let $(X,d,S)$ be a ${\mathbf {d}}$ computable Polish presentation of X. Consider the function $f:X\to Q$ mapping $x\in X$ to $(x_i)_{i\in \mathbb {N}}\in Q$ defined by $x_i=d(x,s_i)/(1+d(x,s_i))$ . It is computable and onetoone, so it is a homeomorphism as X is compact. Its image is ${\mathbf {d}}$ computably overt, because it contains the dense ${\mathbf {d}}$ computable sequence $(f(s_n))_{n\in \mathbb {N}}$ .
If $(X,d,S)$ is moreover ${\mathbf {d}}$ computably compact, then so is $f(X)$ , because the image of a ${\mathbf {d}}$ computably compact set by a computable function is ${\mathbf {d}}$ computably compact.
Conversely, if a compact set $K\subseteq Q$ is ${\mathbf {d}}$ computably overt, then it has a ${\mathbf {d}}$ computable Polish presentation, using the metric $d_Q$ and a dense ${\mathbf {d}}$ computable sequence in K. If K is moreover ${\mathbf {d}}$ computably compact, then this presentation is ${\mathbf {d}}$ computably compact.
2.2 Realizations
We recall usual operations on spaces, such as disjoint unions and wedge sum, and show that they preserve computability notions.
We will implicitly use the fact that $[0,1]^n\times Q$ and $Q\times Q$ are computably homeomorphic to Q, so the results of the next constructions are subsets of Q.

• If $X,Y\subseteq Q$ , then their disjoint union is $X\amalg Y=(\{0\}\times X)\cup (\{1\}\times Y)$ .

• If $X,Y\subseteq Q$ both contain $\overline {0}$ , then their wedge sum is $X\vee Y=(X\times \{\overline {0}\})\cup (\{\overline {0}\}\times Y)$ .

• If $X_n\subseteq Q$ for all $n\in \mathbb {N}$ , then their disjoint union is $\coprod _nX_n=\bigcup _n \{0\}^n\times \{1\}\times X_n$ , and its onepoint compactification is $\alpha _0(\coprod _nX_n)=\{\overline {0}\}\cup \coprod _nX_n$ , where $\overline {0}=(0,0,0,\ldots )\in Q$ .
When $X,Y$ are Polish spaces, their disjoint union $X\amalg Y$ is implicitly defined as the Polish space obtained by embedding X and Y in Q and applying the previous definition, and similarly for the wedge sum and the onepoint compactification of the disjoint union.
Proposition 2.3. If $X,Y,X_n$ are uniformly ${\mathbf {d}}$ computably overt $($ resp. compact $)$ , then $X\amalg Y,X\vee Y$ and $\alpha _0(\coprod _n X_n)$ are ${\mathbf {d}}$ computably overt $($ resp. compact $)$ .
Proof The result is straightforward for finite disjoint unions and wedge sums, because the sets $\{0\}\times X$ , $\{1\}\times Y$ , $X\times \{\overline {0}\}$ and $\{\overline {0}\}\times Y$ easily inherit the computability properties of X and Y, and so do their unions.
Let us consider the onepoint compactification of the disjoint union. The function $f_n:Q\to Q$ sending $x=(x_0,x_1,\ldots )$ to $(0,\ldots ,0,1,x_0,x_1,\ldots )$ , starting with n occurrences of $0$ , is computable uniformly in n. Equivalently, the preimages of rational balls by $f_n$ are effectively open, uniformly in n.
Assume that $X_n$ are uniformly ${\mathbf {d}}$ computably overt. A rational open ball B intersects the set $\alpha _0(\coprod _n X_n)$ iff $f_n^{1}(B)$ intersects $X_n$ for some n, which is ${\mathbf {d}}$ c.e.
Now assume that $X_n$ are uniformly ${\mathbf {d}}$ computably compact. A rational open set U contains $\alpha _0(\coprod _nX_n)$ iff there exists n such that $\{0\}^n\times Q\subseteq U$ and $X_i\subseteq f_i^{1}(U)$ for all $i<n$ , which is ${\mathbf {d}}$ c.e.
The next result will be a building block for constructing spaces that encode information about a set of natural numbers.
Proposition 2.4. Let $X_\infty $ and $(X_n)_{n\in \mathbb {N}}$ be uniformly computably compact subsets of Q satisfying the following conditions $:$

• For all $n\in \mathbb {N}$ , $X_n\subseteq X_{n+1}\subseteq X_\infty $ .

• $d_H(X_n,X_\infty )<2^{n}$ .
To a set $A\subseteq \mathbb {N}$ we associate the compact set
The set $X_A$ is uniformly computably compact relative to A and uniformly computably overt relative to any enumeration of $\mathbb {N}\setminus A$ .
Proof For $s\in \mathbb {N}$ let $X_A[s]=X_s$ if $A\cap [0,s]$ is empty, and $X_A[s]=X_n$ where $n=\min (A\cap [0,s])$ otherwise. The sequence $(X_A[s])_{s\in \mathbb {N}}$ can be computed from A and satisfies $d_H(X_A[s],X_A)<2^{s+1}$ , so $X_A$ is computably compact relative to A.
We show that a rational ball B intersects $X_A$ if and only if there exists $n\in \mathbb {N}$ such that $[0,n1]\cap A=\emptyset $ and B intersects $X_n$ . This condition is c.e. relative to any enumeration of $\mathbb {N}\setminus A$ . If $[0,n1]\cap A=\emptyset $ and B intersects $X_n$ , then $X_A$ contains $X_n$ , so B intersects $X_A$ . Conversely, if B intersects $X_A$ , then either $A=\emptyset $ and any sufficiently large n satisfies the conditions, or $A\neq \emptyset $ and $n=\min A$ satisfies the conditions.
2.3 Good covers
We will often extract information about a compact Polish space from its finite open covers. In order for this extraction to be computable, we need to decide which open sets of such a cover intersect, which is made possible by considering good covers only. The content of this section is essentially folklore. Closely related results have appeared in [Reference Brattka and Presser6, Reference Downey and Melnikov9, Reference Iljazovic23, Reference Kamo25, Reference Pauly, Seon, Ziegler, Fernández and Muscholl37, Reference Yasugi, Mori and Tsujii44].
Let $(X,d,S)$ be a Polish presentation of X. To a rational open ball $B(s_i,r)$ we associate the corresponding rational closed ball $\overline {B}(s_i,r)=\{x\in X:d(s_i,x)\leq r\}$ , and to a rational open set U we associate the corresponding union $\overline {U}$ of rational closed balls. The closed ball always contains the closure of the open ball and is a $\Pi ^0_1$ subset of X, relative to the presentation of X.
A good open cover of X is a family of rational open sets $\kern1pt\mathcal {U}=(U_i)_{i\in I}$ where $I\subseteq \mathbb {N}$ is finite, such that $X=\bigcup _{i\in I}U_i$ and for every $J\subseteq I$ , if $\bigcap _{i\in J}U_i=\emptyset $ then $\bigcap _{i\in J}\overline {U}_i=\emptyset $ . An open cover $(U_i)_{i\in I}$ is a strong refinement of an open cover $(V_j)_{j\in J}$ if for every $i\in I$ , there exists $j\in J$ such that $\overline {U}_i\subseteq V_j$ .
The next result can be found in [Reference Downey and Melnikov9, Theorems 1.1 and 3.16].
Lemma 2.5. Given a compact presentation of X, one can compute a strong refining sequence of good open covers, i.e., a sequence $(\mathcal {U}_s)_{s\in \mathbb {N}}$ of good open covers such that $\kern1pt\mathcal {U}_{s+1}$ strongly refines $\kern1pt\mathcal {U}_s$ .
Proof The mesh of a finite open cover $\kern1pt\mathcal {U}=(U_i)_{i\in I}$ is the maximal diameter of the $U_i$ ’s. We first show that there are good open covers of arbitrarily small meshes, and that if $\kern1pt\mathcal {U}$ is a finite open cover of X then there exists $\epsilon>0$ such that every finite cover of mesh $<\epsilon $ strongly refines $\kern1pt\mathcal {U}$ .
Let $\epsilon>0$ and $(B(s,\epsilon ))_{s\in I}$ be a rational open cover of X, where I is finite. By compactness, let $\delta <\epsilon $ be such that $(B(s,\delta ))_{s\in I}$ is still a cover of X. There are only finitely many values of $r\in [\delta ,\epsilon ]$ such that $(B(s,r))_{r\in I}$ is not a good cover; indeed, these values are $\min _{x\in X}\max _{s\in J}d(x,s)$ , for $J\subseteq I$ ; indeed, when the cover is not good there exists $J\subseteq I$ and a point x at distance $\leq r$ from every $s\in J$ , but no point y at distance $<r$ from every $s\in J$ , so $\max _{s\in J}d(x,s)=r$ and x minimizes this quantity. Therefore, there exists a rational number $r\in [\delta ,\epsilon ]$ avoiding this finite set of values, providing a good open cover of mesh $<2\epsilon $ .
Let $\kern1pt\mathcal {U}$ be a finite open cover of X. By compactness, $\kern1pt\mathcal {U}$ has a Lebesgue number $\delta>0$ , which means that any set of diameter ${\leq} \delta $ is contained in some member of U. If $\mathcal {V}=(V_i)_{i\in J}$ is a good cover of mesh ${\leq} \delta $ , then each $\overline {V_i}$ has diameter ${\leq} \delta $ , so $\mathcal {V}$ strongly refines $\kern1pt\mathcal {U}$ .
Now we are given a compact presentation of X as oracle. Whether a finite rational cover is good is $\Sigma ^0_1$ , and whether it strongly refines another rational cover is $\Sigma ^0_1$ , and whether its mesh is ${<}q$ is $\Sigma ^0_1$ . Therefore, we start with searching for some good open cover $\kern1pt\mathcal {U}_0$ of mesh ${<}1$ and inductively look for a good open cover $\kern1pt\mathcal {U}_{s+1}$ of mesh ${<}2^{s}$ strongly refining $\kern1pt\mathcal {U}_s$ . These objects exist as proved above, and can be effectively found by exhaustive search.
2.4 Clopen subsets
We show that from a compact presentation of X, one can compute a presentation of the Boolean algebra of clopen subsets.
Proposition 2.6. Given a compact presentation of X, one can compute an enumeration $(C_i)_{i\in \mathbb {N}}$ of the clopen subsets of X, such that equality, inclusion, and the finite Boolean operations are computable.
Proof Assume that a compact presentation is given as oracle. One can compute an enumeration $(U_i,V_i)_{i\in \mathbb {N}}$ of all the pairs of rational open sets satisfying $X=U_i\cup V_i=X$ and $\overline {U}_i\cap \overline {V}_i=\emptyset $ , because these conditions are c.e. relative to a compact presentation of X. Let then $C_i=U_i$ for each i. One can then easily compute the Boolean operations: given i, one can compute j such that $(U_j,V_j)=(V_i,U_i)$ , so $C_j=X\setminus C_i$ ; given $i,j$ , one can compute k such that $(U_k,V_k)=(U_i\cap U_j,V_i\cup V_j)$ , so that $C_k=C_i\cap C_j$ , and symmetrically for the union. The set $\{i\in \mathbb {N}:C_i=\emptyset \}$ can be computed, because $C_i=\emptyset \iff X\subseteq V_i\iff U_i=\emptyset $ , which is both c.e. and coc.e. relative to a compact presentation. Therefore, equality and inclusion are decidable because they reduce to emptiness of some Boolean combination.
In particular, whether X is connected is $\Pi ^0_1$ relative to a compact presentation of X.
2.5 Covering dimension
Let us briefly recall the covering dimension of a topological space X. If $\kern1pt\mathcal {U},\mathcal {V}$ are two open covers of X, we say that $\mathcal {V}$ is a refinement of $\kern1pt\mathcal {U}$ if every $V\in \mathcal {V}$ is contained in some $U\in \mathcal {U}$ . An open cover $\kern1pt\mathcal {U}$ has order n if the intersection of any n elements of $\kern1pt\mathcal {U}$ is empty. The covering dimension of X, written $\dim (X)$ , is the least number n such that every open cover $\kern1pt\mathcal {U}$ of X has a refinement $\mathcal {V}$ of order $n+1$ , if it exists.
Proposition 2.7. Let $n\in \mathbb {N}$ . Given a compact presentation of X, the predicate $\dim X\leq n$ is $\Pi ^0_2$ , uniformly in n.
Proof We are given a compact presentation of X as oracle. As X is compact, it is routine to check that in the definition of dimension, one can assume that $\kern1pt\mathcal {U}$ and $\mathcal {V}$ are good covers and that $\mathcal {V}$ strictly refines $\kern1pt\mathcal {U}$ . Therefore, $\dim X\leq n$ iff for every good cover $\kern1pt\mathcal {U}$ , there exists a good cover $\mathcal {V}$ that strongly refines $\kern1pt\mathcal {U}$ , and such that for all $V_0,\ldots ,V_n\in \mathcal {V}$ , one has $\overline {V}_0\cap \cdots \cap \overline {V}_n=\emptyset $ . This is a $\Pi ^0_2$ predicate, because one can computable enumerate the good covers of X, and the strong refinements of a given good cover.
Proposition 2.8. If $\dim (X)\leq n$ , then given a compact presentation of X, one can compute a strong refining sequence of good open covers $(\mathcal {U}_s)_{s\in \mathbb {N}}$ of order $n+1$ .
Proof We proceed as in the proof of Lemma 2.5, but when searching for a good cover $\kern1pt\mathcal {U}_{s+1}$ that strongly refines $\kern1pt\mathcal {U}_s$ , we additionally test whether $\kern1pt\mathcal {U}_{s+1}$ has order $n+1$ , which is also a $\Sigma ^0_1$ predicate. As $\dim (X)\leq n$ , it always exists so it can be effectively found.
Corollary 2.9. If X is zerodimensional, then from any compact presentation of X, one can compute a pruned tree $T\subseteq 2^{<\omega }$ such that X is homeomorphic to $[T]$ .
Proof As X is zerodimensional, one can compute by Proposition 2.8 a strongly refining sequence of good open covers $(\mathcal {U}_s)_{s\in \mathbb {N}}$ of order $1$ . Each $\kern1pt\mathcal {U}_s$ is therefore made of clopen sets. All these clopen sets ordered by inclusion form a finitely branching pruned tree, in which the number of nodes at each level can be computed. By standard arguments, it can be effectively converted into a binary tree.
3 Cantor–Bendixson derivative
Let X be a topological space. The Cantor–Bendixson derivative of X is the subspace $X'$ of all nonisolated points of X. We discuss the computability of $X'$ in comparison with the computability of X. This problem has been investigated in the context of reverse mathematics in [Reference Greenberg and Montalbán16].
In the next result, it is important to note that in order to produce a compact presentation of $X'$ , we only use a Polish presentation of X.
Lemma 3.1. Let ${X}$ be a compact Polish space.
If $X\subseteq Q$ is ${\mathbf {d}}$ computably overt, then $X'$ is $\Pi ^0_1({\mathbf {d}}')$ and ${\mathbf {d}}"$ computably overt.
Therefore, if X has a ${\mathbf {d}}$ computable Polish presentation, then its CantorBendixson derivative $X'$ has a ${\mathbf {d}}"$ computable compact presentation.
Proof Assume that $X\subseteq Q$ is ${\mathbf {d}}$ computably overt, and let $(x_i)_{i\in \mathbb {N}}$ be a ${\mathbf {d}}$ computable dense sequence in X.
Let $B=B(s,r)$ be a rational open ball in Q, and for $k\in \mathbb {N}$ let $B^k=B(s,r2^{k})$ . We claim that B intersects $X'$ if and only if there is k such that $B^k\cap X$ contains infinitely many points. For the forward direction, choose $x\in B\cap X'$ . One has $x\in B^k$ for sufficiently large k. Since $B^k$ is open and x is not isolated in X, $B^k$ contains infinitely many points. For the backward direction, if $B^k\cap X$ contains infinitely many points, then its closure, which is contained in $B\cap X$ , contains a nonisolated point; therefore, B intersects $X'$ . By this claim, the property $B\cap {X}'\not =\emptyset $ is equivalent to
which is $\Sigma ^0_3$ relative to ${\mathbf {d}}$ , or equivalently $\Sigma ^0_1$ relative to ${\mathbf {d}}"$ . This means that $X'$ is ${\mathbf {d}}"$ computably overt.
Next, let $A=\{(i,k):\forall j,x_j=x_i\text { or }d_Q(x_i,x_j)\geq 2^{k}\}$ . The set A is $\Pi ^0_1$ relative to ${\mathbf {d}}$ , hence ${\mathbf {d}}'$ computable. One can easily see that x is isolated in ${X}$ if and only if $x\in B(x_i,2^{k})$ for some $(i,k)\in A$ . Thus, the set of isolated points is a $\Sigma ^0_1({\mathbf {d}}')$ subset of ${X}$ ; hence ${X}'$ is a $\Pi ^0_1({\mathbf {d}}')$ subset of ${X}$ , which is itself $\Pi ^0_1({\mathbf {d}}')$ , so $X'$ is $\Pi ^0_1({\mathbf {d}}')$ .
Therefore, $X'$ is ${\mathbf {d}}"$ computably compact.
3.1 Stone duality
Here we show that spectra of compact zerodimensional spaces are closely related to spectra of Boolean algebras. This follows from an effectivization of Stone duality in [Reference Odintsov and Selivanov36].
Let $\mathbf {B}$ be the category formed by the Boolean algebras as objects and the $\{\vee ,\wedge ,\bar {}\;,0,1\}$ homomorphisms as morphisms. Recall that a Stone space is a compact topological space X such that for any distinct $x,y\in X$ there is a clopen set U with $x\in U\not \ni y$ (i.e., zerodimensional and $T_1$ ). Let $\mathbf {S}$ be the category formed by the Stone spaces as objects and the continuous mappings as morphisms.
The Stone duality states the dual equivalence between the categories $\mathbf {B}$ and $\mathbf {S}$ . More explicitly, the Stone space $s(B)$ corresponding to a given Boolean algebra B is formed by the set of prime filters of B with the base of open (in fact, clopen) sets consisting of the sets $\{F\in s(B)\mid a\in F\}$ , $a\in B$ . (Note that one could equivalently take ideals in place of filters.) Conversely, the Boolean algebra $b(X)$ corresponding to a given Stone space X is formed by the set of clopen sets (with the usual settheoretic operations). By Stone duality, any Boolean algebra B is canonically isomorphic to the Boolean algebra $b(s(B))$ (the isomorphism $f:B\rightarrow b(s(B))$ is defined by $f(a)=\{F\in s(B)\mid a\in F\}$ ), and any Stone space X is canonically homeomorphic to the space $s(b(X))$ .
Restricting the Stone duality to the countable Boolean algebras, we obtain their duality with the class $\mathbf {CP}_0$ of compact zerodimensional countably based spaces, and in fact with the compact subspaces of the Cantor space $2^\omega $ . As the nonempty closed subsets of $2^\omega $ coincide with the sets $[T]$ of infinite paths through a pruned tree $T\subseteq 2^\omega $ , we obtain a close relation between such subspaces and countable Boolean algebras.
Fact 3.2.

(1) A Boolean algebra has a ${\mathbf {d}}$ c.e. $($ resp. ${\mathbf {d}}$ coc.e., ${\mathbf {d}}$ computable $)$ copy if and only if it is isomorphic to the Boolean algebra of clopen subsets of $[T]$ for some ${\mathbf {d}}$ coc.e. $($ resp. ${\mathbf {d}}$ c.e., ${\mathbf {d}}$ computable $)$ pruned tree T.

(2) Every ${\mathbf {d}}$ coc.e. Boolean algebra is isomorphic to a ${\mathbf {d}}$ computable Boolean algebra.

(3) There is a ${\mathbf {d}}$ c.e. Boolean algebra which is not isomorphic to a ${\mathbf {d}}$ computable Boolean algebra.
Proof The first item follows from [Reference Odintsov and Selivanov36, Lemma 3]; see also [Reference Selivanov41]. The second item follows from [Reference Odintsov and Selivanov36]. The third item follows from [Reference Feiner13].
As already noticed by HarrisonTrainer, Melnikov, and Ng [Reference HarrisonTrainor, Melnikov and Ng18], one can use Stone duality to show several results on degree spectra of zerodimensional compacta. For instance, Stone duality can be used to show the following:
Fact 3.3 (see [Reference HarrisonTrainor, Melnikov and Ng18]).

(1) There exists $X\in \mathbf {CP}_0$ which has a $\mathbf {0}'$ computable Polish presentation, but no computable Polish presentation,

(2) If $X\in \mathbf {CP}_0$ has a low $_4$ Polish presentation, then it has a computable Polish presentation.
For $X\in \mathbf {CP}_0$ , Corollary 2.9 implies that X has a ${\mathbf {d}}$ computable compact presentation if and only if there exists a ${\mathbf {d}}$ computable pruned tree $T\subseteq 2^{<\omega }$ such that X is homeomorphic to $[T]$ . Therefore, we obtain the following result.
Corollary 3.4. The degree spectra of countable Boolean algebras coincide with the compact degree spectra of zerodimensional compact Polish spaces.
The Stone dual of the Cantor–Bendixson derivative is known as the Fréchet derivative $B'$ of a Boolean algebra B which is the quotient of B by the ideal generated by atoms (minimal nonzero elements). Since the isolated points x of the space $s(B)$ (realized as $[F]$ above) are precisely the atoms $[\tau ]\cap [F]$ for suitable prefix $\tau \sqsubseteq x$ , we obtain the following.
Proposition 3.5. For any countable Boolean algebra B, $s(B')$ is homeomorphic to $s(B)'$ .
Precise complexity estimations for the Fréchet derivative were obtained in [Reference Odintsov and Selivanov36]: for any Turing degree ${\mathbf {d}}$ , a countable Boolean algebra C is ${\mathbf {d}}"$ computably presentable iff C is isomorphic to $B'$ for some ${\mathbf {d}}$ computable Boolean algebra B, and there is a ${\mathbf {d}}$ computable Boolean algebra B such that $B'$ is not ${\mathbf {d}}'$ computably presentable. The iterated version is also known for any $n>0$ : a countable Boolean algebra C is ${\mathbf {d}}^{(2n)}$ computably presentable iff C is isomorphic to the nth derivative $B^{(n)}$ for some ${\mathbf {d}}$ computable Boolean algebra B, and there is a ${\mathbf {d}}$ computable Boolean algebra B such that the nth derivative $B^{(n)}$ is not ${\mathbf {d}}^{(2n1)}$ computably presentable.
These results have an immediate consequence in terms of compact degree spectra of spaces in $\mathbf {CP}_0$ , by Corollary 3.4 and Proposition 3.5.
Theorem 3.6. Let ${\mathbf {d}}$ be any Turing degree. For any $n\geq 1$ , a space $Y\in \mathbf {CP}_0$ has a ${\mathbf {d}}^{(2n)}$ computable compact presentation if and only if Y is homeomorphic to the nth derivative $X^{(n)}$ for some ${\mathbf {d}}$ computable compact $X\in \mathbf {CP}_0$ , and there is a ${\mathbf {d}}$ computable compact $X\in \mathbf {CP}_0$ such that the nth derivative $X^{(n)}$ does not have a ${\mathbf {d}}^{(2n1)}$ computable compact presentation.
It implies in particular that Lemma 3.1 is almost optimal.
3.2 Countable spaces
We next note that countable topological spaces cannot be used to construct nontrivial degree spectra inside the hyperarithmetical hierarchy.
Mazurkiewicz–Sierpiński’s theorem states that every countable compact Polish space is homeomorphic to the ordinal space $\omega ^\alpha \cdot n+1$ for some $\alpha <\omega _1$ and $n\in \omega $ , endowed with the order topology.
Let $\omega _1^x$ be the least ordinal which is not computable in x.
Proposition 3.7. For any countable ordinal $\alpha $ , the compact and Polish degree spectrum of the space $\omega ^\alpha +1$ are both $\{x:\alpha <\omega _1^x\}$ .
Proof We first show that if $\alpha $ is $\mathbf {x}$ computable, then $\omega ^\alpha +1$ has an $\mathbf {x}$ computable compact presentation.
Given an $\mathbf {x}$ computable wellordering $\preceq $ of $\mathbb {N}$ of order type $\omega ^\alpha +1$ , we build a copy X of $\omega ^\alpha +1$ in $[0,1]$ as follows. We can assume w.l.o.g. that $0$ is minimal and $1$ is maximal for $\preceq $ . We embed $(\mathbb {N},\preceq )$ in $([0,1],\leq )$ by inductively defining a rational number $x_n\in [0,1]$ for each $n\in \mathbb {N}$ . We need some care to make sure that we will obtain a topological embedding, and that the copy X will be $\mathbf {x}$ computably compact. We start with $x_0=0$ and $x_1=1$ . Let $n\geq 2$ and assume that $x_0,\ldots ,x_{n1}$ have been defined. Let i and j be, respectively, the predecessor and the successor of n in $(\{0,\ldots ,n\},\preceq )$ , and define $x_{n}=2^{n}x_i+(12^{n})x_j$ . Note that $x_jx_n\leq 2^{n}$ . Let $X=\{x_n:n\in \mathbb {N}\}$ with the topology inherited from $\mathbb {R}$ .
The function $n\mapsto x_n$ is by construction an order embedding of $(\mathbb {N},\preceq )$ in $([0,1], \leq)$ . We need to show that it is a topological embedding, i.e., that it is continuous. It is sufficient to check that if $n=\sup _\preceq \{m:m\prec n\}$ , then $x_n=\sup _\leq \{x_m:m\prec n\}$ . There exist infinitely many m’s such that $m\geq n$ , $m\prec n$ and m is the predecessor of n in $(\{0,\ldots ,m\},\preceq )$ . When defining $x_m$ for any such m, the successor of m is $j=n$ , so $x_nx_m\leq 2^{m}$ , which is arbitrarily small, so $x_n=\sup _\leq \{m:m\prec n\}$ as wanted.
Therefore, the set $X=\{x_n:n\in \mathbb {N}\}$ is a copy of $\omega ^\alpha +1$ . The sequence $(x_n)_{n\in \mathbb {N}}$ is an $\mathbf {x}$ computable sequence of rational numbers. Let $X_n=\{x_0,\ldots ,x_n\}\subseteq X$ and observe by construction that $d_H(X_n,X_{n1})\leq 2^{n}$ , so $d_H(X_n,X)\leq 2^{n}$ . As the compact sets $X_n$ are uniformly $\mathbf {x}$ computable and converge fast to X, X is $\mathbf {x}$ computable as well.
Conversely, if $\omega ^\alpha +1$ has an $\mathbf {x}$ computable Polish presentation, then by Lemma 2.1, it has an $\mathbf {x}'$ computable compact presentation. In particular, there is a countable $\Pi ^0_1(\mathbf {x}')$ class $P\subseteq 2^\omega $ which is homeomorphic to $\omega ^\alpha +1$ . Since the Cantor–Bendixson rank of $\omega ^\alpha +1$ is $\alpha $ , and the Cantor–Bendixson rank is a topological invariant, the rank of P is also $\alpha $ . However, as noted by Kreisel, the Spector boundedness principle implies that the rank of a countable $\Pi ^0_1(\mathbf {x}')$ class must be $\mathbf {x}'$ computable; see also [Reference Cenzer7, Section 4]. As an $\mathbf {x}$ hyperarithmetic ordinal is always $\mathbf {x}$ computable, this implies that $\alpha <\omega _1^{\mathbf {x}}$ .
The same arguments show that the compact and Polish degree spectra of $\omega ^\alpha \cdot n+1$ are both $\{x:\alpha <\omega _1^x\}$ . Therefore, if a countable Polish space has an hyperarithmetical compact (or even Polish) presentation, then it has a computable compact presentation.
For more details, see also [Reference Ash and Knight2, Reference Greenberg and Montalbán16].
4 Computable aspects of Čech homology groups
4.1 Background on Čech homology groups
All the details about the definitions and results mentioned in this section can be found in the textbook of Hurewicz and Wallman [Reference Hurewicz and Wallman22]. We will refer to the corresponding sections in that reference.
Simplicial complexes. We refer to [Reference Hurewicz and Wallman22, Section V.9]. Let V be a finite set, whose elements are called the vertices. A simplicial complex K on V is a collection of subsets of V such that if $A\subseteq B\subseteq V$ and $B\in K$ , then $A\in K$ . An element $A\in K$ is called a simplex, and an nsimplex if the cardinality of A is $n+1$ .
To a simplicial complex K corresponds a topological space $X_K$ , called its geometric realization. Assuming that $V=\{0,\ldots ,k\}$ , let $\Delta _V=\{(x_0,\ldots ,x_k):\forall i, x_i\in [0,1]\text { and }\sum _i x_i=1\}$ . For $x=(x_0,\ldots ,x_k)\in \Delta _V$ , let $\mathrm {supp}(x)=\{i:x_i>0\}$ . The geometric realization of K is the subset of $\Delta _V$ defined by
Let $K,L$ be simplicial complexes on vertices $V_K$ and $V_L$ respectively. A simplicial map, written $f:K\to L$ , is a map $f:V_K\to V_L$ such that if $A\in K$ then $f(A)=\{f(v):v\in A\}\in L$ . In other words, a simplicial map sends each simplex of K to a simplex of L. Note that if A is an nsimplex of K, then $f(A)$ is a psimplex of L for some $p\leq n$ , and $f(A)$ is an nsimplex if and only if the images of the vertices of A under f are pairwise distinct.
Simplicial homology groups. We refer to [Reference Hurewicz and Wallman22, Sections VIII.1, VIII.3]. To a simplicial complex K are associated its homology groups $H_n(K;G)$ where G is an abelian group, called the coefficient group. Here we will only consider the coefficient group $G={{\mathbb {Z}}_2}:={\mathbb {Z}}/2{\mathbb {Z}}$ .
Let K be a simplicial complex on a finite set V of vertices. An nchain of K is a set of nsimplices of K, orequivalently a formal sum of nsimplices with coefficients in ${{\mathbb {Z}}_2}$ . Let $C_n(K)$ be the finite abelian group of nchains, endowed with the sum operation. The boundary operator is the group homomorphism $\partial _n:C_{n}(K)\to C_{n1}(K)$ that sends an nsimplex $\Delta =\{v_0,\ldots ,v_{n}\}$ to $\sum _{i=0}^{n}\Delta \setminus \{v_i\}$ .
An ncycle is an nchain whose boundary is $0$ , i.e. is an element of $\ker \partial _{n}$ . An nboundary is an nchain which is the boundary of some $(n+1)$ chain, i.e. is an element of ${\mathrm {im}} \partial _{n+1}$ . One has $\partial _{n}\circ \partial _{n+1}=0$ , in other words ${\mathrm {im}}\partial _{n+1}\subseteq \ker \partial _{n}$ , i.e. every nboundary is an ncycle. The nth homology group of K is the finite abelian group $H_n(K;{{\mathbb {Z}}_2})=\ker \partial _{n}/{\mathrm {im}} \partial _{n+1}$ .
Let $K,L$ be simplicial complexes and $f:K\to L$ a simplicial map. For each $n\in {\mathbb {N}}$ , f induces group homomorphisms $f_\#:C_n(K)\to C_n(L)$ defined as follows. If $A\in K$ is an nsimplex, then
Said differently, $f_\#(A)=f(A)$ if the images of the vertices of A are pairwise distinct. The homomorphism $f_\#$ commutes with $\partial _{n+1}$ , i.e. $f_\#\circ \partial _{n+1}=\partial _{n+1}\circ f_\#$ , which implies that $f_\#$ induces a group homomorphism $f_*:H_n(K;{{\mathbb {Z}}_2})\to H_n(L;{{\mathbb {Z}}_2})$ .
Čech homology groups. We refer to [Reference Hurewicz and Wallman22, Section VIII.4]. There are several ways to define homology groups of more general topological spaces. We will use the Čech definition, whichapproximates the space by simplicial complexes.
Let X be a compact Polish space. To a finite open cover ${\mathcal {U}}=(U_0,\ldots ,U_k)$ of X is associated its nerve $K_{\mathcal {U}}$ , which is a simplicial complex. Its set of vertices is $V_{\mathcal {U}}=\{0,\ldots ,k\}$ , and a simplex $A\subseteq V_{\mathcal {U}}$ belongs to $K_{\mathcal {U}}$ if and only if $\bigcap _{i\in A}U_i\neq \emptyset $ .
Let $({\mathcal {U}}_s)_{s\in {\mathbb {N}}}$ be a sequence of finite open covers of X such that ${\mathcal {U}}_{s+1}$ refines ${\mathcal {U}}_s$ , i.e. each open set of ${\mathcal {U}}_{s+1}$ is contained in some open set of ${\mathcal {U}}_s$ , and such that the maximal diameter of elements of ${\mathcal {U}}_s$ tends to $0$ as s grows.
Let $K_s$ be the nerve of ${\mathcal {U}}_s$ . As ${\mathcal {U}}_{s+1}$ refines ${\mathcal {U}}_s$ , one can define a simplicial map $f_{s}:K_{s+1}\to K_s$ by sending a vertex corresponding to $U\in {\mathcal {U}}_{s+1}$ to a vertex corresponding to some $V\in {\mathcal {U}}_s$ containing U. There are several possible choices for $f_s$ , but in the end they will lead to the same result.
The nth Čech homology group $\check {H}_n(X;{{\mathbb {Z}}_2})$ of X is defined as the inverse limit of the inverse system $f_{s,*}:H_n(K_{s+1};{{\mathbb {Z}}_2})\to H_n(K_{s};{{\mathbb {Z}}_2})$ . In other words, $\check {H}_n(X;{{\mathbb {Z}}_2})$ is the set of sequences $(c_s)_{s\in {\mathbb {N}}}$ with $c_s\in H_n(K_s;{{\mathbb {Z}}_2})$ and $c_s=f_{s,*}(c_{s+1})$ , with componentwise addition. This group does not depend on the choice of open covers and simplicial maps.
If X is itself homeomorphic to the geometric realization of a simplicial complex, then the Čech homology group $\check {H}_n(X;{{\mathbb {Z}}_2})$ is isomorphic to the simplicial homology group $H_n(X;{{\mathbb {Z}}_2})$ [Reference Hurewicz and Wallman22, Theorem VIII.4.E].
4.2 Computable aspects
Lupini, Melnikov, Nies [Reference McNicholl and Stull29] and Downey, Melnikov [Reference Downey and Melnikov9] proved that the Čech cohomology groups $\check {H}^n(X)$ of a compact Polish space X are computable, i.e. can be presented as a computable set of relations over a countable set of generators, relative to any compact presentation ofX. We will use part of their argument to investigate the computability of the Čech homology groups $\check {H}_n(X;{{\mathbb {Z}}_2})$ . However, these groups do not behave so well in terms of computability. First, they are not countable in general, so they have no countable set of generators and their computability does make immediatesense. Second, as we will show, their nontriviality is $\Sigma ^1_1$ complete when taking coefficients in ${\mathbb {Z}}$ , which contrasts with the nontriviality of the Čech cohomology groups with coefficients in ${\mathbb {Z}}$ , which is $\Sigma ^0_2$ . However, we now show that when taking coefficients in a finite group such as ${{\mathbb {Z}}_2}$ , the nontriviality of the Čech homology groups is $\Sigma ^0_2$ .
Given a combinatorial description of a finite simplicial complex K, each group $H_n(K;{{\mathbb {Z}}_2})$ can be fully computed: it is a finite group whose cardinality and operation can be computed, for instance under the form of a Cayley table. Given a combinatorial description of $K,L$ and of a simplicial map $f:K\to L$ , the group homomorphisms $f^*_n:H_n(K;{{\mathbb {Z}}_2})\to H_n(L;{{\mathbb {Z}}_2})$ can be computed, as functions between finite sets. It makes the inverse limit $\check {H}_n(X;{{\mathbb {Z}}_2})$ a coc.e. profinite group, in the sense of La Roche [Reference Lupini, Melnikov and Nies28] and Smith [Reference Selivanov41], i.e. the inverse limit of a computable inverse system of finite groups.
In the next statement, a finite group is represented in the strongest possible way by a Cayley table, which encodes the cardinality of the group and the group operation. A group homomorphism is also represented in thestrongest possible way by arranging its graph in a finite list.
Proposition 4.1. Given an inverse system of finite groups $f_n:G_{n+1}\to G_{n}$ , whether their inverse limit is nontrivial is $\Sigma ^0_2$ .
Proof Let G be the inverse limit. We claim that G is nontrivial if and only if there exist $n\in {\mathbb {N}}$ and $g\in G_n$ such that $g\neq 0$ and $g\in {\mathrm {im}}(f_n\circ \ldots \circ f_p)$ for all $p\geq n$ . This predicate is $\Sigma ^0_2$ , because the groups are finite. In particular, the predicate $g\in {\mathrm {im}}(f_n\circ \ldots \circ f_p)$ can be decided in finite time, by testing $f_n\circ \ldots \circ f_p(g')=g$ for each $g'$ in the finite set $G_{p+1}$ .
If G is nontrivial, then let $(g_n)_{n\in {\mathbb {N}}}$ be a nonzero element of G. As it is nonzero, there exists n such that $g_n\neq 0$ . Moreover, for every $p\geq n$ , $g_n=f_n\circ \ldots \circ f_p(g_{p+1})$ belongs to ${\mathrm {im}}(f_n\circ \ldots \circ f_p)$ .
Conversely, assume the existence of n and g satisfying the conditions. As the groups are finite, by König’s lemma, there exists a sequence $(g_p)_{p\geq n}$ such that $g_n=g$ and $g_p=f_p(g_{p+1})$ for all $p\geq n$ . For $m<n$ , let $g_m=f_m\circ \ldots \circ f_{n1}(g)$ . The sequence $(g_n)_{n\in {\mathbb {N}}}$ is an element of G, and is nonzero as its nth coordinate is $g\neq 0$ .
We now have all the ingredients to show that the nontriviality of Čech homology groups with coefficients in ${{\mathbb {Z}}_2}$ has limited complexity.
Corollary 4.2. Whether the nth Čech homology group $\check {H}_n(X;{{\mathbb {Z}}_2})$ of a compact Polish space X is nontrivial is $\Sigma ^0_2$ , uniformly in n and a compact presentation of X.
Proof Given a compact presentation of X and a good open cover ${\mathcal {U}}=(U_0,\ldots ,U_k)$ of X, its nerve $K_{\mathcal {U}}$ on the vertices $V=\{0,\ldots ,k\}$ can be computed, in the sense that one can decide whether a set $J\subseteq V$ belongs to $K_{\mathcal {U}}$ . Indeed, $J\in K_{\mathcal {U}}$ iff $\bigcap _{j\in J}U_j\neq \emptyset $ iff $\bigcap _{j\in J}\overline {U}_j\neq \emptyset $ . The first inequality is $\Sigma ^0_1$ and the second inequality is $\Pi ^0_1$ , so it is decidable.
If ${\mathcal {U}}=(U_0,\ldots ,U_k)$ and ${\mathcal {V}}=(V_0,\ldots ,V_l)$ are good open covers of X and ${\mathcal {U}}$ strongly refines ${\mathcal {V}}$ , then one can compute a simplicial map $f:K_{\mathcal {U}}\to K_{\mathcal {V}}$ sending each vertex $u_i$ for $K_{\mathcal {U}}$ to a vertex $v_j$ of $K_{\mathcal {V}}$ such that $\overline {U}_i\subseteq V_j$ . Indeed, the latter inclusion is $\Sigma ^0_1$ , so for each i one can effectively find a suitable j.
Now, from a compact presentation of X, one can compute a strong refining sequence of good open covers $({\mathcal {U}}_s)_{s\in {\mathbb {N}}}$ by Lemma 2.5. One can then compute the nerves $K_s$ of ${\mathcal {U}}_s$ as well as simplicial maps $f_s:K_{s+1}\to K_s$ . One can compute the simplicial homology groups $H_n(K_s)$ and the homomorphisms $f_{s,*}:H_n(K_{s+1};{{\mathbb {Z}}_2})\to H_n(K_s;{{\mathbb {Z}}_2})$ . Their inverse limit is $\check {H}_n(X;{{\mathbb {Z}}_2})$ , and whether it is nontrivial is $\Sigma ^0_2$ by Proposition 4.6.
Comparison between Čech homology and Čech cohomology. Lupini, Melnikov and Nies [Reference McNicholl and Stull29] and Downey and Melnikov [Reference Downey and Melnikov9] proved that the Čech cohomology groups $\check {H}^n(X)$ of a compact Polish space X can be computed from a compact presentation of X (we do not write the coefficient group, which is ${\mathbb {Z}}$ ). We will neither define nor use cohomology groups, but let us mention the important fact that for each n, the simplicial cohomology groups $H^n(K_s)$ and the induced homomorphisms $f_s^*:H^n(K_{s})\to H^{n}(K_{s+1})$ form a direct system, which is essential to show that their direct limits, the Čech cohomology groups $\check {H}^n(X)$ , are computable. As we saw, the simplicial homology groups $H_n(K_s;{{\mathbb {Z}}_2})$ form an inverse system, which makes the computation of their inverse limits, the Čech homology groups $\check {H}_n(X;{{\mathbb {Z}}_2})$ , much more difficult.
4.3 Influence of the coefficient group
Let us finally discuss the role of the coefficient group in Corollary 5.7. The same proof holds for any finite coefficient group G. Moreover, thecomputability of the Čech cohomology groups proven in [Reference McNicholl and Stull29, Reference Downey and Melnikov9] implies that Corollary 5.7 holds for the circle group $G={\mathbb {R}}/{\mathbb {Z}}$ . Indeed, there is a duality between Čech cohomology groups and Čech homology groups: $\check {H}_n(X;{\mathbb {R}}/{\mathbb {Z}})$ is the character group of $\check {H}^n(X)$ , i.e. is the group of homomorphisms from $\check {H}^n(X)$ to the circle group ${\mathbb {R}}/{\mathbb {Z}}$ ([Reference Hurewicz and Wallman22, Theorem VIII.4.G, p. 137]). Therefore, the nontriviality of $\check {H}_n(X;{\mathbb {R}}/{\mathbb {Z}})$ is equivalent to the nontriviality of $\check {H}^n(X)$ , which is $\Sigma ^0_2$ in X, as $\check {H}^n(X)$ is computable in X.
We now show that Corollary 5.7 does not hold for $G={\mathbb {Z}}$ , in which case the nontriviality of $\check {H}_n(X;{\mathbb {Z}})$ is $\Sigma ^1_1$ complete. The reason is that the groups of the inverse system are no more finite, so the existence of a path in the inverse system is similar to the existence of an infinite path in a tree.
Theorem 4.3. Let $n\geq 1$ . Given a compact presentation of a compact Polish space X, the nontriviality of $\check {H}_n(X;{\mathbb {Z}})$ is $\Sigma ^1_1$ complete.
Proof The arguments in [Reference Melnikov32, Reference McNicholl and Stull29, Reference Downey and Melnikov9] imply the computability of the inverse system of simplicial groups $(H_n(K_s;{\mathbb {Z}}),f_{s,*})$ : these groups are strongly completely decomposable (strictly speaking, their results are about cohomology groups $H^n(K_s)$ but apply equally to the homology groups $H_n(K_s;{\mathbb {Z}})$ ). Therefore, their inverse limit is nontrivial if and only if there exists a sequence $(x_s)_{s\in {\mathbb {N}}}$ with $x_s\in H_n(K_s;{\mathbb {Z}})$ , $x_{s}=f_{s,*}(x_{s+1})$ for all s and $x_s\neq 0$ for some s. This condition is $\Sigma ^1_1$ . We now show the completeness of the problem.
The set of trees $T\subseteq 2^{<\omega }$ containing an infinite path with infinitely many $1$ ’s is $\Sigma ^1_1$ complete [Reference Kechris26, Proposition 25.2]. We will reduce this problem to the nontriviliaty of $\check {H}_1(X;{\mathbb {Z}})$ , and explain the changes to be made for higherdimensional homology groups. We will build an inverse system of groups and then build an inverse limit of compact spaces whose Čech homology group isthe inverse limit of the groups. Note the similarity of this strategy with [Reference Melnikov32] which builds an inverse limit of compact spaces whose first Čech cohomology group is a given direct limit ofgroups.
We first define an inverse system of groups $(G_n,f_n)$ , i.e. a sequence $(G_n)_{n\in {\mathbb {N}}}$ of groups together with homomorphisms $f_n:G_{n+1}\to G_n$ . We will then associate to each tree T a subsystem $(L_n,f_n)$ , i.e. a sequence of subgroups $L_n\subseteq G_n$ satisfying $f_n(L_{n+1})\subseteq L_n$ .
Let $\epsilon \in 2^{<\omega }$ be the empty string, $S=\{\epsilon \}\cup \{1w:w\in 2^{<\omega }\}$ be the set of strings that do not start with $0$ and $S_n=\{w\in S:w=n\}$ . For each $w\in S$ , let $k(w)=2^{w_1}$ where $w_1$ is the number of occurrences of $1$ in w. Note that $k(\epsilon )=1$ .
For $n\in {\mathbb {N}}$ , we define
Note that $G_0={\mathbb {Z}}$ and $G_1={\mathbb {Z}}^2$ . We call each ${\mathbb {Z}}^{k(w)}$ a component of $G_n$ . If $n\geq 1$ and $w\in S_n$ , then $w_1\geq 1$ so $k(w)$ is even: we will implicitly split each component ${\mathbb {Z}}^{k(w)}$ into ${\mathbb {Z}}^{k(w)/2}\oplus {\mathbb {Z}}^{k(w)/2}$ , expressing an element of ${\mathbb {Z}}^{k(w)}$ as a pair.
We define $f_n:G_{n+1}\to G_n$ on each component of $G_{n+1}$ . For each $w\in S_n$ , one has $k(w0)=k(w)$ and $k(w1)=2k(w)$ and we let
and
An element $x\in \varprojlim (G_n,f_n)$ is described as a family $(x(w))_{w\in S}$ where $x(w)\in {\mathbb {Z}}^{k(w)}$ for each $w\in S$ , such that $x(w)=f_{w}(x(w0))+f_{w}(x(w1))$ . We define the support of x as $\mathrm {supp}(x)=\{w\in S:x(w)\neq 0\}$ . Note that $x=0$ if and only if $\mathrm {supp}(x)=S$ .
We relate the supports of nonzero elements with the infinite binary sequences containing infinitely many $1$ ’s. For an infinite sequence $q\in 2^\omega $ , let $q\upharpoonright_{n}$ be the prefix of q of length n.
Claim 1. Let $x\in \varprojlim (G_n,f_n)$ . One has $x\neq 0$ iff there exists $q\in 2^\omega $ starting with $1$ and containing infinitely many $1$ ’s, and such that $x(q\upharpoonright_{n})\neq 0$ for all sufficiently large n.
Proof Of course if such a q exists, then $x\neq 0$ . Assume that $x\neq 0$ and let $w\in S$ be such that $x(w)\neq 0$ . We build q extending w. Let $n\geq w$ and assume by induction that $v:=q\upharpoonright_{n}$ has been already defined. As $x(v)=f_{n}(x(v0))+f_{n}(x(v1))$ , one has $x(v0)\neq 0$ or $x(v1)\neq 0$ . We choose the next bit of q so that $x(q\upharpoonright_{n+1})\neq 0$ : let $q(n)=1$ if $x(v1)\neq 0$ , and $q(n)=0$ otherwise. One has $x(q\upharpoonright_{n})\neq 0$ for all $n\geq w$ by construction. Assume for a contradiction that q contains only finitely many $1$ ’s. Let v be a prefix of q extending w and such that $q=v0^\omega $ . Note that for $n\geq v$ , $k(q\upharpoonright_{n})$ is constantly $k(v)$ and $x(q\upharpoonright_{n})\in {\mathbb {Z}}^{k(v)}$ . For such n, one must have $x(q\upharpoonright_{n}1)=0$ because otherwise we would have chosen $q(n)=1$ in the construction of q. As a result, one has $x(q\upharpoonright_{n})=f_n(x(q\upharpoonright_{n}0))=f_n(x(q\upharpoonright_{n+1}))$ for all $n\geq v$ . It means that each coordinate of $x(q\upharpoonright_{v})\in {\mathbb {Z}}^{k(v)}$ is a multiple of all $2^i$ for all $i\geq 0$ , or a multiple of $3^i$ for all $i\geq 0$ , which implies that $x(q\upharpoonright_{v})=0$ . It is a contradiction because v extends w. Therefore, we have proved that q contains infinitely many $1$ ’s.
Conversely,
Claim 2. If $q\in 2^{\mathbb {N}}$ starts with $1$ and contains infinitely many $1$ ’s, then there exists $x\in \varprojlim (G_n,f_n)$ such that $\mathrm {supp}(x)$ is the set of prefixes of q.
Proof Let $(n_i)_{i\in {\mathbb {N}}}$ be the increasing enumeration of the positions n such that $q(n)=1$ , and observe that $n_0=0$ . We build $x=(x(w))_{w\in S}$ . Let $x(\epsilon )=1\in {\mathbb {Z}}$ . Let $i\in {\mathbb {N}}$ and assume by induction that $x(q\upharpoonright_{n_i})$ has been defined. The function $f_{n_i}\circ \ldots \circ f_{n_{i+1}1}:{\mathbb {Z}}^{k(q\upharpoonright_{n_{i+1}})}\to {\mathbb {Z}}^{k(q\upharpoonright_{n_i})}$ maps $(a,b)$ to $2^ja+3^jb$ , where $j=n_{i+1}n_i1$ . As $2^j$ and $3^j$ are coprime, there exists $(a,b)$ such that $2^ja+3^jb=x(q\upharpoonright_{n_i})$ , let $x(q\upharpoonright_{n_{i+1}})=(a,b)$ . For $n_i<n<n_{i+1}$ , naturally define $x(q\upharpoonright_{n})=f_n\circ \ldots f_{n_{i+1}1}(a,b)$ . Finally, let $x(w)=0$ if w is not a prefix of q. By construction, x belongs to $\varprojlim (G_n,f_n)$ and $\mathrm {supp}(x)$ is the set of prefixes of q.
Let $T\subseteq 2^{<\omega }$ be a tree. We can assume w.l.o.g. that $T\subseteq S$ , replacing T with $\{\epsilon \}\cup \{1w:x\in T\}$ if necessary. We define an inverse subsystem $(L_n,f_n)$ of $(G_n,f_n)$ . Let $T_n=\{w\in T:w=n\}$ and $L_n=\bigoplus _{w\in T_n}{\mathbb {Z}}^{k(w)}$ . As T is a tree, one has $f_n(L_{n+1})\subseteq L_n$ so $(L_n,f_n)$ is an inverse system which is a subsystem of $(G_n,f_n)$ . Its inverse limit consists of the elements of the inverse limit of $(G_n,f_n)$ whose support is contained in T.
Therefore,