$\check{H}_{n}(X;G)$ and $\check{H}^{n}(X;G)$ respectively. However, these
different theories are equivalent for ANRs.

######  Theorem 7.22 (Theorem 1 in [Mar58]).

If $X$ is an ANR, then for all $G$ and $n$,

$\displaystyle H_{n}(X;G)$ $\displaystyle\cong\check{H}_{n}(X;G),$
$\displaystyle H^{n}(X;G)$ $\displaystyle\cong\check{H}^{n}(X;G).$

The result applies to compact manifolds with and without boundary, which are
ANRs (Corollary A.9 in [Hat02]).

##### Partial characterization of $\mathsf{E}_{n}$

For compact spaces of dimension at most $n+1$, $\mathsf{E}_{n}$ can be
characterized in terms of Čech homology and cohomology. For a pair $(X,A)$,
let $i:A\to X$ be the inclusion map. Like any continuous map, it induces
natural homomorphisms on the homology and cohomology groups, written
$i_{*}:\check{H}_{n}(A;G)\to\check{H}_{n}(X;G)$ and
$i^{*}:\check{H}^{n}(X;G)\to\check{H}^{n}(A;G)$. It uses the abelian groups
$(\mathbb{Z},+)$ and $(\mathbb{R}/\mathbb{Z},+)$.

######  Theorem 7.23 (Corollaries VIII.2 and VIII.3 (p. 149) in [HW41]).

Let $(X,A)$ be a compact pair with $\dim(X)\leq n+1$. One has

$\displaystyle(X,A)\in\mathsf{E}_{n}$ $\displaystyle\iff
i_{*}:\check{H}_{n}(A;\mathbb{R}/\mathbb{Z})\to\check{H}_{n}(X;\mathbb{R}/\mathbb{Z})\text{
is not injective}$ $\displaystyle\iff
i^{*}:\check{H}^{n}(X;\mathbb{Z})\to\check{H}^{n}(A;\mathbb{Z})\text{ is not
surjective}.$

The equivalence does not hold in general. For instance, Hopf’s fibration is a
continuous function $f:\SS_{3}\to\SS_{2}$ which is not null-homotopic, hence
does not extend to $\mathbb{B}_{4}$, implying that
$(\mathbb{B}_{4},\SS_{3})\in\mathsf{E}_{2}$. However, the second homology and
cohomology groups of both $\mathbb{B}_{4}$ and $\SS_{3}$ are trivial, so the
only homomorphism between them is bijective.

##### Partial characterization of $\mathsf{H}_{n}$

The invariant $\mathsf{H}_{n}$ also admits a characterization in terms of Čech
homology and cohomology, for compact spaces of dimension at most $n$.

######  Theorem 7.24 (Corollary 4 (p. 150) in [HW41]).

Let $X$ be a compact space with $\dim(X)\leq n$. One has

$\displaystyle X\in\mathsf{H}_{n}$
$\displaystyle\iff\check{H}_{n}(X;\mathbb{R}/\mathbb{Z})\neq 0$
$\displaystyle\iff\check{H}^{n}(X;\mathbb{Z})\neq 0.$

Again, the equivalence does not hold in general. Hopf’s fibration implies that
$\SS_{3}\in\mathsf{H}_{2}$, but
$\check{H}_{2}(\SS_{3};\mathbb{R}/\mathbb{Z})\cong\check{H}^{2}(\SS_{3};\mathbb{Z})\cong
0$.

Lupini, Melnikov and Nies [LMN23], and Melnikov and Downey [DM23] with a
different proof, proved that if $X\subseteq Q$ is given in the Vietoris
topology $\tau_{\mathcal{V}}$, then its Čech cohomology groups
$\check{H}^{n}(X;\mathbb{Z})$ can be computably presented (relative to $X$ and
uniformly): one can enumerate a list of generators and the words, i.e. finite
combinations of generators and their inverses, that are equal to the $0$
element of the group. Their result implies that the non-triviality of a group
$\check{H}^{n}(X;\mathbb{Z})$ is a $\Sigma^{0}_{2}$ invariant in the topology
$\tau_{\mathcal{V}}$: the group is non-trivial iff there exists a word which
does not equal $0$, which is a $\Sigma^{0}_{2}$ formula. It seems that their
proof even shows that it is a $\Sigma^{0}_{2}$ invariant in the topology
$\tau_{up\mathcal{V}}$. However Proposition 5.19 implies that being
$\Sigma^{0}_{2}$ in $\tau_{\mathcal{V}}$ is enough for the purpose of proving
strong computable type. More precisely,

######  Corollary 7.25.

Let $X$ be a compact space. If $X$ is minimal such that
$\check{H}^{n}(X;\mathbb{Z})\ncong 0$, then $X$ has strong computable type.

###### Proof.

The invariant
$\mathsf{I}_{n}=\\{X\in\mathcal{K}(Q):\check{H}^{n}(X;\mathbb{Z})\ncong 0\\}$
is $\Sigma^{0}_{2}$ in the topology $\tau_{\mathcal{V}}$ by [LMN23] and
[DM23]. Therefore, $\text{$\uparrow$}\mathsf{I}_{n}$ is $\Sigma^{0}_{2}$ in
the topology $\tau_{up\mathcal{V}}$ and has the same minimal elements as
$\mathsf{I}_{n}$ (Proposition 5.19), so we can apply Theorem 4.8. ∎

We also mention a stronger connection between maps to the sphere and
cohomology. Theorem 7.24 is actually a particular case of Hopf’s
classification theorem, which states that for a compact space of dimension at
most $n$, there is a one-to-one correspondence between homotopy classes of
maps to the $n$-sphere and elements of the $n$th Čech cohomology group. It can
be found as Theorem 2.2 (p. 17) in [Ale47], reformulated in [Hu48].

######  Theorem 7.26 (Hopf classification theorem).

Let $X$ be a compact space of dimension $\leq n$. There is a bijection between
$[X;\SS_{n}]$ and $\check{H}^{n}(K;\mathbb{Z})$.

##  8 Examples

In this section, we illustrate our framework by revisiting previous results
stating that a pair has computable type, and showing that it can be explained
by their minimality w.r.t. some suitable $\Sigma^{0}_{2}$ invariant. The
advantages of this approach are multiple. A single $\Sigma^{0}_{2}$ invariant
can be used for many different sets or pairs, therefore capturing in a unified
way the computability-theoretic part of the argument, leaving a purely
topological analysis to each specific set or pair, which can often leverage
the well-developed field of topology. It makes the study of (strong)
computable type more explicitly related to topology.

In particular, most of the arguments in this section are purely topological.

###  8.1 Examples of $\mathsf{E}_{n}$-minimal pairs

In this section we show how $\mathsf{E}_{n}$-minimality covers many of the
known examples of pairs having computable type. We also give a new example to
illustrate its scope.

####  8.1.1 Chainable continuum between two points

We revisit a result due to Iljazović in [Ilj09]. Let us first recall the
definitions. A continuum is a connected compact metric space.

######  Definition 8.1.

Let $X$ be a metric space. A finite sequence $\mathcal{C}$ of nonempty open
subsets $C_{0},\ldots,C_{m}$ of $X$ is said to be a chain if $C_{i}\cap
C_{j}\neq\emptyset\Leftrightarrow|i-j|\leq 1$, $\forall
i,j\in\\{0,\ldots,m\\}$. A chain $\mathcal{C}=\\{C_{0},\ldots,C_{m}\\}$ covers
$X$ if $X\subseteq\bigcup_{i}C_{i}$; for $\epsilon>0$, it is an
$\epsilon$-chain if $\max_{i}(\mathrm{diam}(C_{i}))<\epsilon$.

For $a,b\in X$, $X$ is chainable from $a$ to $b$ if for every $\epsilon>0$,
there exists an $\epsilon$-chain $\mathcal{C}=\\{C_{0},\ldots,C_{m}\\}$ in $X$
which covers $X$ and such that $a\in C_{0}$ and $b\in C_{m}$.

Iljazović proved in [Ilj09] that if $X$ is chainable from $a$ to $b$, then the
pair $(X,\\{a,b\\})$ has computable type. This result and its proof can be
reformulated as the $\mathsf{E}_{0}$-minimality of the pair.

######  Proposition 8.2.

If $X$ is compact connected and is chainable from $a$ to $b$, then the pair
$(X,\\{a,b\\})$ is $E_{0}$-minimal.

###### Proof.

Take $\SS_{0}=\\{a,b\\}$ and $f=\mathrm{id}_{\SS_{0}}:\SS_{0}\to\SS_{0}$. As
$X$ is connected, the function $f$ has no continuous extension
$g:X\rightarrow\SS_{0}$, because $g^{-1}(a)$ and $g^{-1}(b)$ would disconnect
$X$. Hence, $(X,\\{a,b\\})$ is in $E_{0}$.

Let $x\in X\setminus\\{a,b\\}$ and $\epsilon>0$ be such that
$B(x,\epsilon)\subseteq X\setminus\\{a,b\\}$. Let $f:\\{a,b\\}\to\SS_{0}$. The
points $a,b$ must be disconnected in $X\setminus B(x,\epsilon)$. Indeed, let
$\mathcal{C}=\\{C_{0},\ldots,C_{m}\\}$ be an $\epsilon$-chain covering $X$
with $a\in C_{0}$ and $b\in C_{m}$. Let $k$ be such that $x\in C_{k}$. As
$\mathrm{diam}(C_{k})<\epsilon$, $a\notin C_{k},b\notin C_{k}$ and
$C_{k}\subseteq B(x,\epsilon)$. Therefore $X\setminus B(x,\epsilon)$ is
covered by the two disjoint open sets $U_{a}:=\bigcup_{i<k}C_{i}$ and
$U_{b}:=\bigcup_{i>k}C_{i}$, containing $a$ and $b$ respectively.

The disconnection can be turned into a continuous extension $g:X\setminus
B(x,\epsilon)\to\SS_{0}$, sending $U_{a}$ to $f(a)$ and $U_{b}$ to $f(b)$.
Therefore, $(X\setminus B(x,\epsilon),\\{a,b\\})\notin\mathsf{E}_{0}$. It
implies that $(X,\\{a,b\\})$ is $\mathsf{E}_{0}$-minimal because any proper
subset of $X$ is contained in such a $X\setminus B(x,\epsilon)$. ∎

We will cover other examples from [Ilj09] in Section 8.3, using another
$\Sigma^{0}_{2}$ invariant.

####  8.1.2 Pseudo $n$-cubes

In [HIP20], the authors introduce the notion of a pseudo-$n$-cube, which is a
pair $(X,A)$ which can somehow be approximated by $n$-dimensional cubes and
their boundaries. They prove that pseudo-$n$-cubes have computable type.

We show how the result can be proved using the invariant $\mathsf{E}_{n-1}$,
by showing how the $\mathsf{E}_{n-1}$-minimality of the $n$-dimensional cube
implies the $\mathsf{E}_{n-1}$-minimality of pseudo-$n$-cubes.

We first recall the definition of pseudo-$n$-cubes from [HIP20]. Fix some
$n\in\mathbb{N}^{*}$ and let $S^{0}_{1},\ldots,S^{0}_{n}$,
$S^{1}_{1},\ldots,S^{1}_{n}$ be the faces of the $n$-cube $I^{n}$ defined by

$\displaystyle S^{0}_{i}$ $\displaystyle=\\{(x_{1},\ldots,x_{n})\in
I^{n}:x_{i}=0\\},$ $\displaystyle S^{1}_{i}$
$\displaystyle=\\{(x_{1},\ldots,x_{n})\in I^{n}:x_{i}=1\\},$

where $i\in\\{1,\ldots,n\\}$. Let $S=\bigcup_{i}S^{0}_{i}\cup S^{1}_{i}$ be
the boundary of $I^{n}$.

The Hausdorff distance between two non-empty compact subsets $A,B$ of a metric
space $(X,d)$ is

$\text{d}_{H}(A,B)=\max\left(\max_{x\in A}\min_{y\in B}d(x,y),\max_{y\in
B}\min_{x\in A}d(x,y)\right).$

######  Definition 8.3.

A compact pair $(X,A)$ in $\mathbb{R}^{n}$ is called a pseudo-$n$-cube with
respect to some finite sequence of compact subsets $A^{0}_{i}$ and
$A^{1}_{i}$, $1\leq i\leq n$, if $A=\bigcup_{i}A^{0}_{i}\cup A^{1}_{i}$,
$A^{0}_{i}\cap A^{1}_{i}=\emptyset$ and for each $\epsilon>0$ there exists a
continuous injection $f:I^{n}\rightarrow\mathbb{R}^{n}$ such that

  1. 1.

$f(I^{n})\subseteq X$,

  2. 2.

$f(\mathrm{int}_{\mathbb{R}^{n}}(I^{n}))\cap A=\emptyset$,

  3. 3.

$X\setminus
f(\mathrm{int}_{\mathbb{R}^{n}}(I^{n}))\subseteq\mathcal{N}_{\epsilon}(f(S))$,

  4. 4.

$\text{d}_{H}(f(S^{0}_{i}),A^{0}_{i})<\epsilon$ and
$\text{d}_{H}(f(S^{1}_{i}),A^{1}_{i})<\epsilon$.

We now show how the proof that pseudo-$n$-cubes have computable type can be
simplified by showing that they are $\mathsf{E}_{n-1}$-minimal.

######  Proposition 8.4.

Every pseudo-$n$-cube is $\mathsf{E}_{n-1}$-minimal.

###### Proof.

A pseudo-cube is approximated by cubes. We use Propositions 7.5 and 7.2 to
derive $\mathsf{E}_{n-1}$-minimality of pseudo-$n$-cubes from
$\mathsf{E}_{n-1}$-minimality of $n$-cubes (Proposition 7.12).

We recall that the pair $(I^{n},S)\cong(\mathbb{B}_{n},\SS_{n-1})$ is in
$\mathsf{E}_{n-1}$ because the identity on $S$ has no continuous extension to
$I^{n}$. A similar function $h:A\to S$ can be defined that has no extension to
$X$. The function $h$ satisfies $h(A^{0}_{i})\subseteq S^{0}_{i}$ and
$h(A^{1}_{i})\subseteq S^{1}_{i}$ and is built as follows. On an intersection
$A^{b_{1}}_{1}\cap\ldots\cap A^{b_{n}}_{n}$ (where $b_{i}\in\\{0,1\\}$), $h$
sends every point to the vertex $(b_{1},\ldots,b_{n})$ of $I^{n}$. We then
progressively extend $h$, sending intersections of $n-1$ sets to segments,
intersections of $n-2$ sets to squares, then cubes, etc. and finally sending
single sets $A^{0}_{i}$ and $A^{1}_{i}$ to the $n-1$-cubes which are the faces
$S^{0}_{i}$ and $S^{1}_{i}$ of $I^{n}$. All the extensions can be done using
the Tietze extension theorem.

Once $h:A\to S$ is defined, it has a continuous extension $\tilde{h}$ to a
neighborhood of $A$ because $S$ is an ANR. For sufficiently small $\epsilon$
and a cube $(I^{n},S)$ embedded in $X$ as in the definition, $\tilde{h}$ is
defined on $S$ and sends $S^{b}_{i}$ close to $S^{b}_{i}$, in particular
disjoint from $S^{1-b}_{i}$. Thus, $\tilde{h}|_{S}:S\to S$ is homotopic to
$\mathrm{id}_{S}$. In other words, the functions $\mathrm{id}_{S}$ on the
copies of $(I^{n},S)$ are asymptotically homotopic to $h:A\to S$. As
$\mathrm{id}_{S}$ has no continuous extension to $I^{n}$, $h$ has no
continuous extension to $X$ by Proposition 7.5. As a result,
$(X,A)\in\mathsf{E}_{n-1}$.

We now prove minimality. By definition of pseudo-$n$-cubes, for each
$\epsilon$ there exists a copy $(X_{i},A_{i})$ of the $n$-cube in $X$
satisfying the conditions of Proposition 7.2:

  * •

$X_{i}\setminus A_{i}$ is open in $X$ (indeed, it is the image by a continuous
injection map of an open subset of $\mathbb{R}^{n}$ in
$X\subseteq\mathbb{R}^{n}$, so it is open by invariance of domain) and
contained in $X\setminus A$,

  * •

$X\setminus(X_{i}\setminus A_{i})\subseteq\mathcal{N}_{\epsilon}(A)$.

If $U\subseteq X\setminus A$ is an open set intersecting $X$, then the same
conditions are satisfied by the pairs $(X_{i}\setminus U,A_{i}\setminus U)$
and $(X\setminus U,A)$. For sufficiently large $i$, $U$ intersects $X_{i}$ so
$(X_{i}\setminus U,A_{i}\setminus U)\notin\mathsf{E}_{n-1}$ by
$\mathsf{E}_{n-1}$-minimality of $(X_{i},A_{i})$. We can apply Proposition
7.2, which implies that $(X\setminus U,A)\notin\mathsf{E}_{n-1}$. Therefore,
$(X,A)$ is $\mathsf{E}_{n-1}$-minimal (we again use Corollary 7.11 which can
be applied, as $A\subseteq\mathbb{R}^{n}$ has empty interior, therefore has
dimension at most $n-1$ by Theorem 3.7.1 in [vM01]). ∎

####  8.1.3 A new example

In this section we illustrate how $\mathsf{E}_{n}$-minimality can be used to
obtain new examples of spaces having computable type property, with little
effort. We present an example, but many variations are possible.

######  Definition 8.5.

The topologist’s sine curve $T$, or Warsaw sine curve, is defined by

$T=\left\\{\left(x,\sin\tfrac{1}{x}\right):0<x\leq
1\right\\}\cup\left\\{(0,y):-1\leq y\leq 1\right\\}.$

We define the Warsaw saucer $S$ by

$S=\left\\{(x,y,z)\in\mathbb{R}^{3}:0<x^{2}+y^{2}\leq 1\text{ and
}z=\sin\tfrac{1}{\sqrt{x^{2}+y^{2}}}\right\\}\cup\\{(0,0,z):-1\leq z\leq
1\\}.$

We define its boundary $\partial S$ as its bounding circle

$\partial
S=\left\\{(x,y,\sin(1))\in\mathbb{R}^{3}:x^{2}+y^{2}=1\right\\}\cong\SS_{1}.$

It is obtained from the closed topologist’s sine curve, making it rotate
around the vertical segment, see Figure 4.

(a) The Warsaw saucer $S$ (b) A half-cut of $S$

Figure 4:

The set $S$ is compact and connected but neither locally connected nor path-
connected. It is neither a manifold, nor a simplicial complex, nor a
pseudo-$n$-cube so none of the previous results applies to $(S,\partial S)$.
However, it can easily be proved to have strong computable type, because it is
closely related to the pair $(\mathbb{B}_{2},\SS_{1})$ and is therefore
$\mathsf{E}_{1}$-minimal.

######  Proposition 8.6.

The pair $(S,\partial S)$ is $\mathsf{E}_{1}$-minimal and hence has strong
computable type.

###### Proof.

We show the $\mathsf{E}_{1}$-minimality of $(\mathbb{B}_{2},\SS_{1})$ implies
the $\mathsf{E}_{1}$-minimality of $(S,\partial S)$.

The surface $S$ can be approximated by copies of the disk $\mathbb{B}_{2}$
that share the same boundary $\SS_{1}$. Indeed, let

$X_{i}=\left\\{(x,y,z):\frac{1}{i\pi}\leq\sqrt{x^{2}+y^{2}}\leq 1\text{ and
}z=\sin\tfrac{1}{\sqrt{x^{2}+y^{2}}}\right\\}\cup\left\\{(x,y,0):\sqrt{x^{2}+y^{2}}\leq\frac{1}{i\pi}\right\\}.$

The sets $X_{i}$ converge to $S$, and they all have the same boundary
$\SS_{1}$. We know that
$(X_{i},\SS_{1})=(\mathbb{B}_{2},\SS_{1})\in\mathsf{E}_{1}$, moreover the same
function, which is the identity $\mathrm{id}:\SS_{1}\to\SS_{1}$, has no
continuous extension to $X_{i}$. Therefore, this function has no continuous
extension to $S$ by Corollary 7.6.

We now prove $\mathsf{E}_{1}$-minimality. The projection $\pi$ on the
horizontal plane sends $S$ to $\mathbb{B}_{2}$ and is the identity on
$\partial S=\SS_{1}$. If $X$ is a proper compact subset of $S$ containing
$\partial S$, then $\pi(X)$ is a proper compact subset of $\mathbb{B}_{2}$. As
$(\mathbb{B}_{2},\SS_{1})$ is $\mathsf{E}_{1}$-minimal,
$(\pi(X),\SS_{1})\notin\mathsf{E}_{1}$ so $(X,\partial S)\notin\mathsf{E}_{1}$
by Proposition 7.1 (more directly, every $f:\SS_{1}\to\SS_{1}$ has a
continuous extension $F:\pi(X)\to\SS_{1}$, inducing a continuous extension
$F^{\prime}=F\circ\pi:X\to\partial S$ of $f$ to $X$). As a result,
$(S,\partial S)$ is $\mathsf{E}_{1}$-minimal. ∎

###  8.2 Compact $n$-manifolds are $\mathsf{H}_{n}$-minimal

It was proved in [Ilj13] and [IS18] that compact manifolds with or without
boundary have computable type, as pairs and single spaces respectively. In
this section, we show how this result can be derived from classical results in
algebraic topology implying that $n$-dimensional manifolds are
$H_{n}$-minimal, and applying Theorem 4.8. We first recall the definition of a
manifold.

######  Definition 8.7.

A manifold of dimension $n$, or more concisely an $n$-manifold, is a Hausdorff
space $M$ in which each point has an open neighborhood homeomorphic to
$\mathbb{R}^{n}$.

An $n$-manifold with boundary is a Hausdorff space $M$ in which each point has
an open neighborhood homeomorphic either to $\mathbb{R}^{n}$ or to the half-
space $\mathbb{R}_{+}^{n}=\\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{n}\geq
0\\}$ by a homeomorphism which maps $x$ to a point $(x_{1},\ldots,x_{n-1},0)$.
The set of points which have an open neighborhood homeomorphic to the half-
space is called the boundary of $M$ and is denoted by $\partial M$.

The next result seems to be folklore but is not stated in any standard
textbook on algebraic topology. However, it can be derived from classical
results about homology of manifolds.

######  Theorem 8.8.

Every connected compact $n$-manifold $M$ is $\mathsf{H}_{n}$-minimal. Every
connected compact $n$-manifold with boundary $(M,\partial M)$ is
$\mathsf{H}_{n}$-minimal.

We give here the proof for manifolds without boundary, and put the proof for
manifolds with boundary in the appendix (it is similar but requires more
technical results about pairs).

###### Proof.

The result is a consequence of Theorem 7.24, which reduces $\mathsf{H}_{n}$ to
homology groups, and of classical results about homology groups of manifolds.

Let us use the abelian group $G=\mathbb{R}/\mathbb{Z}$. As it contains an
element of order $2$, namely $1/2$, Corollary 7.12 in [BEGH93] implies that if
$M$ is a connected $n$-manifold, then $H_{n}(M;G)\neq 0$ if and only if $M$ is
compact. Therefore, if $M$ is a compact connected $n$-manifold and $x\in M$,
then

$\displaystyle H_{n}(M;G)$ $\displaystyle\ncong 0,$ $\displaystyle
H_{n}(M\setminus\\{x\\};G)$ $\displaystyle\cong 0,$

because $M\setminus\\{x\\}$ is a non-compact connected manifold.

If $x\in M$ and $B$ is an open Euclidean ball around $x$, then
$M\setminus\\{x\\}$ _deformation retracts_ to $M\setminus B$, which means that
there is a retraction $r:M\setminus\\{x\\}\to M\setminus B$ such that if
$i:M\setminus B\to M\setminus\\{x\\}$ is the inclusion map, then $i\circ r$ is
homotopic to $\mathrm{id}_{M\setminus\\{x\\}}$. It is a classical result that
deformation retractions preserve homology groups, so $H_{n}(M\setminus B)\cong
H_{n}(M\setminus\\{x\\})\cong 0$.

Therefore, Theorem 7.24 implies that $M\in\mathsf{H}_{n}$ and $M\setminus
B\notin\mathsf{H}_{n}$. It implies that $M$ is $\mathsf{H}_{n}$-minimal by
Lemma 7.17: if $X$ is a proper subset of $M$, then $X\subseteq M\setminus B$
for some $B$, so $X\notin\mathsf{H}_{n}$. ∎

Theorem 8.8 implies the result in [Ilj13] that connected compact manifold with
or without boundary have (strong) computable type. However the proof is
inherently different and has new implications. For instance, it implies that
$\mathsf{SCT}_{M,\partial M}\leq_{W}\mathsf{C}_{\mathbb{N}}$ by Theorem 5.17.
Note that a disconnected compact manifold also has strong computable type
because it is a finite union of connected compact manifolds, but it is not
$\mathsf{H}_{n}$-minimal because every connected component is in
$\mathsf{H}_{n}$.

Moreover, Theorem 8.8 immediately implies that other pairs have strong
computable type, as follows.

######  Corollary 8.9.

If $M$ is a compact connected $n$-manifold with possibly empty boundary
$\partial M$, then the pair

$\mathrm{Cone}(M,\partial M)=(\mathrm{Cone}(M),M\cup\mathrm{Cone}(\partial
M))$

is $\mathsf{E}_{n}$-minimal hence has strong computable type.

###### Proof.

All the assumptions of Proposition 7.18 are met: $\partial M$ has empty
interior in $M$, $\dim(M)=n$ and $\dim(\partial M)=n-1$. As $(M,\partial M)$
is $\mathsf{H}_{n}$-minimal by Theorem 8.8, $\mathrm{Cone}(M,\partial M)$ is
$\mathsf{E}_{n}$-minimal. ∎

The cone of a manifold $M$ is a manifold only when $M$ is a sphere of a ball,
so this result is indeed new.

###  8.3 Circularly chainable continuum which are not chainable

In [Ilj09], it is proved that every compact metrizable space which is
circularly chainable but not chainable has computable type. The simplest
examples of such sets are given by the circle and the Warsaw circle. We
briefly show how the proof of this result can be reformulated in our framework
by finding a suitable $\Sigma^{0}_{2}$ invariant.

We already saw the notion of chain in Definition 8.1. We recall other related
notions, taken from [Ilj09]. Let $(X,d)$ be a metric space. A finite sequence
$C=(C_{0},\ldots,C_{n})$ of non-empty open subsets of $X$ is called:

  * •

A chain if for all $i,j\in\\{0,\ldots,n\\}$, $C_{i}\cap
C_{j}\neq\emptyset\iff|i-j|\leq 1$,

  * •

A circular chain if for all $i,j\in\\{0,\ldots,n\\}$, $C_{i}\cap
C_{j}\neq\emptyset\iff(|i-j|\leq 1$ or $\\{i,j\\}=\\{0,n\\})$.

  * •

A quasi-chain if for all $i,j\in\\{0,\ldots,n\\}$, $|i-j|>1\implies C_{i}\cap
C_{j}=\emptyset$.

Its mesh is defined by $\mathrm{mesh}(C)=\max_{0\leq i\leq
n}(\mathrm{diam}(C_{i}))$. It covers a set $S\subseteq X$ if
$S\subseteq\bigcup_{i}C_{i}$.

A set $S\subseteq X$ is chainable (resp. circularly chainable, quasi-
chainable) if for every $\epsilon>0$ there exists a chain (resp. circular
chain, quasi-chain) of mesh $<\epsilon$ covering $S$.

If $S$ is connected, then $S$ is chainable if and only if $S$ is quasi-
chainable (Lemma 30 in [Ilj09]).

######  Proposition 8.10.

Not being quasi-chainable is a $\Sigma_{2}^{0}$ invariant in
$\tau_{up\mathcal{V}}$. If a compact space is circularly chainable but not
chainable, then it is minimal satisfying this invariant and hence it has
strong computable type.

###### Proof.

If $U\subseteq Q$ is a finite union of rational open balls, then we write
$\overline{U}$ for the corresponding union of closed balls. By a standard
compactness argument, a compact set $S\subseteq Q$ is quasi-chainable if for
every rational $\epsilon>0$ there exists a quasi-chain
$C=(C_{0},\ldots,C_{n})$ of mesh $<\epsilon$ covering $S$, with the following
additional properties: $\overline{C_{i}}\cap\overline{C_{j}}=\emptyset$ for
$|i-j|>1$, and $\mathrm{diam}(\overline{C_{i}})<\epsilon$. Finite unions of
rational closed balls are effectively compact, so these additional properties
are c.e., and whether $C$ covers $S$ is effectively open in the upper Vietoris
topology. Therefore, being quasi-chainable is a $\Pi^{0}_{2}$ invariant.

If $S$ is not chainable then $S$ is not quasi-chainable. If in addition $S$ is
circularly chainable, then every proper compact subset of $S$ is quasi-
chainable. Indeed, let $T\subsetneq S$ be compact, $x\in S\setminus T$ and
$\epsilon<d_{Q}(x,T)$, and let $C=(C_{0},\ldots,C_{n})$ be a circular chain of
mesh $<\epsilon$ covering $S$. We can assume that $x\in C_{n}$, applying a
circular permutation if needed. As $\mathrm{diam}(C_{n})<\epsilon<d_{Q}(x,T)$,
the set $C_{n}$ is disjoint from $T$. As a result, $(C_{0},\ldots,C_{n-1})$ is
a quasi-chain covering $T$. ∎

It can be shown that if a compact space is in $\mathsf{H}_{1}$, then it is not
quasi-chainable. We do not know whether the converse implication holds, in
particular whether spaces that are circularly chainable but not chainable are
$\mathsf{H}_{1}$-minimal.

###  8.4 A space that properly contains copies of itself

Theorem 4.8 states in particular that if a compact space $X$ is minimal for
some $\Sigma^{0}_{2}$ topological invariant then $X$ has strong computable
type. We show here that the converse implication does not hold. We actually
prove more: we build a compact space $X$ that has strong computable type and
contains a proper subset $X^{\prime}$ that is homeomorphic to $X$. It implies
that $X$ cannot be minimal for _any_ topological invariant, because any
invariant satisfied by $X$ is also satisfied by $X^{\prime}$. Moreover, the
space $X$ is to our knowledge the first example of an infinite-dimensional
space having strong computable type.

The space $X$ is built as follows. Let $f:\SS_{1}\to Q$ be a so-called “space-
filling curve”, i.e. a surjective continuous function (such a function from
$[0,1]$ to $Q$ exists by the Hahn-Mazurkiewicz theorem, see Theorem 6.3.14 in
[Eng89], and can be extended to the circle by joining the two endpoints). Let
$X=Q\cup_{f}\mathbb{B}_{2}$ be the adjunction space, which is the quotient of
the disjoint union $Q\sqcup\mathbb{B}_{2}$ by the equivalence relation
generated by $x\sim f(x)$ for $x\in\SS_{1}=\partial\mathbb{B}_{2}$.

Note that $X$ contains a proper subset which is homeomorphic to $X$: indeed,
$X$ properly contains $Q$, which contains a copy of $X$.

######  Proposition 8.11.

If $f:\SS_{1}\to Q$ is surjective continuous, then the space
$Q\cup_{f}\mathbb{B}_{2}$ has strong computable type.

###### Proof.

Although $X$ cannot be $\mathsf{H}_{2}$-minimal, we show that there is a
continuous function $q:X\to\SS_{2}$ which is not null-homotopic (i.e.
$X\in\mathsf{H}_{2}$) and such that for any compact subset $Y\subsetneq X$,
the restriction $q|_{Y}:Y\to\SS_{2}$ is null-homotopic.

The condition that $f$ is surjective implies that $Q$ has empty interior in
$X$: every $y\in Q$ is the image by $f$ of some $x\in\partial\mathbb{B}_{2}$,
which is a limit of a sequence
$x_{n}\in\mathbb{B}_{2}\setminus\partial\mathbb{B}_{2}$; therefore in $X$, $y$
is the limit of $x_{n}\in X\setminus Q$.

The quotient space $X/Q$ is homeomorphic to $\mathbb{B}_{2}/\partial\SS_{2}$,
which is homeomorphic to $\SS_{2}$, so we can type the quotient map $q:X\to
X/Q$ as $q:X\to\SS_{2}$. We claim that $q$ is not null-homotopic.

Proposition 0.17 in [Hat02] states that if a pair $(X,A)$ satisfies the
homotopy extension property and $A$ is contractible, then the quotient map
$q:X\to X/A$ is a homotopy equivalence. The pair $(X,Q)$ satisfies the
homotopy extension property because $X$ is obtained by attaching a disk to $Q$
along its boundary (see the proof of Proposition 0.16 in [Hat02]). The space
$Q$ is contractible, so the quotient map $q:X\to X/Q\cong\SS_{2}$ is a
homotopy equivalence, i.e. there exists a function $p:\SS_{2}\to X$ such that
$p\circ q$ is homotopic to $\mathrm{id}_{X}$ and $q\circ p$ is homotopic to
$\mathrm{id}_{\SS_{2}}$. It implies that $q$ is not null-homotopic: otherwise
the function $\mathrm{id}_{\SS_{2}}$, which is homotopic to $q\circ p$, would
be null-homotopic as well, i.e. $\SS_{2}$ would be contractible.

We now show that if $Y$ is a compact proper subset of $X$, then the
restriction $q|_{Y}:Y\to\SS_{2}$ is null-homotopic. As $Q$ has empty interior
in $X$, the interior of $\mathbb{B}_{2}$ is dense in $X$ so it contains a
point $x\in X\setminus Y$. The space $X\setminus\\{x\\}$ is contractible,
because it deformation retracts to $Q$ which is contractible. Therefore, the
restriction of $q$ to $X\setminus\\{x\\}$ is null-homotopic, so its
restriction to $Y\subseteq X\setminus\\{x\\}$ is null-homotopic, by
restriction of the homotopy.

As a result, for a compact set $Y\subseteq X$, one has $Y\neq X$ if and only
if the restriction of $q$ to $Y$ is null-homotopic.

We now prove that $X$ has computable type. Let $X_{0}\subseteq Q$ be a
semicomputable copy of $X$, and $\phi:X_{0}\to X$ a homeomorphism. The
previous discussion implies that for an open set $U\subseteq Q$, $U$
intersects $X_{0}$ if and only if the restriction of $q\circ\phi$ to
$X_{0}\setminus U$ is null-homotopic.

We now use the properties of the numberings $\nu_{[X;Y]}$ from Theorem 6.7.
Let $i_{0}$ be an index of a constant computable function
$f_{i_{0}}:Q\to\SS_{2}$. Note that $i_{0}\in\mathrm{dom}(\nu_{[Z;\SS_{2}]})$
for all $Z\subseteq Q$. Let $i_{1}$ be such that
$\nu_{[X_{0};\SS_{2}]}(i_{1})$ is the homotopy class of $q\circ\phi$. The
definition of $\nu_{[X;Y]}$ implies that for any open set $U\subseteq Q$,
$i_{1}\in\mathrm{dom}(\nu_{[X_{0}\setminus U;\SS_{2}]})$ and
$\nu_{[X_{0}\setminus U;\SS_{2}]}(i_{1})$ is the homotopy class of the
restriction of $q$ to $X_{0}\setminus U$.

We saw that a basic open set $U\subseteq Q$ intersects $X_{0}$ if and only if
$q|_{X_{0}\setminus U}$ is null-homotopic, which is equivalent to
$\nu_{[X_{0}\setminus U;\SS_{2}]}(i_{1})=\nu_{[X_{0}\setminus
U;\SS_{2}]}(i_{0})$. As $X_{0}$ is semicomputable, so is $X_{0}\setminus U$,
therefore this predicate is c.e. and $X_{0}$ is computable, which shows that
$X_{0}$ has computable type.

The argument holds relative to any oracle, i.e. if $X_{0}$ is semicomputable
relative to an oracle $O$, then $X_{0}$ is computable relative to $O$.
Therefore, $X$ has strong computable type. ∎

##  9 Open questions

Our study leaves several questions open.

We have revisited several results from the literature on computable type,
identifying a $\Sigma^{0}_{2}$ invariant for which the space or the pair is
minimal. We do not know whether the results in [AH22] on finite simplicial
complexes can be revisited in a similar way.

######  Question 1.

If a finite simplicial pair $(X,A)$ has strong computable type, is it minimal
satisfying some $\Sigma^{0}_{2}$ invariant?

The example from Section 8.4 is a space that has strong computable type and
properly contains copies of itself. The argument strongly relies on the fact
that it contains the Hilbert cube, which is infinite-dimensional.

######  Question 2.

Is there a finite-dimensional space that has strong computable type and
properly contains copies of itself?

Theorem 5.17 assumes that the pair is minimal for some $\Sigma^{0}_{2}$
invariant. Can this assumption be dropped?

######  Question 3.

Is it always true that if $(X,A)$ has strong computable type, then
$\mathsf{SCT}_{(X,A)}\leq^{t}_{W}\mathsf{C}_{\mathbb{N}}$?

######  Question 4.

Is there a compact space having (strong) computable type and infinitely many
connected components?

We also mention a surprisingly difficult question, already raised in [ČI21b].

######  Question 5.

If two pairs $(X_{1},A_{1})$ and $(X_{2},A_{2})$ have (strong) computable
type, does $(X_{1},A_{1})\times(X_{2},A_{2})=(X_{1}\times X_{2},X_{1}\times
A_{2}\cup A_{1}\times X_{2})$ have (strong) computable type?

##### Acknowledgements.

We thank Emmanuel Jeandel for interesting discussions on the topic, as well as
the anonymous referees for their careful reading and helpful comments.

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##  Appendix A Compact $n$-manifolds are $\mathsf{H}_{n}$-minimal

###  A.1 Background

We gather some results from Bredon [BEGH93] and Hatcher [Hat02]. We do not
recall the definitions of homology groups, but recall results from algebraic
topology and then show how they imply Theorem 8.8.

If $X$ is a topological space, $A\subseteq X$ is a subset and $G$ is an
abelian group, then for each $n\in\mathbb{N}$ one can define the $n$th
homology group $H_{n}(X,A;G)$, which is an abelian group. When $A=\emptyset$,
we write $H_{n}(X;G)$ for $H_{n}(X,A;G)$. When $G=\mathbb{Z}$, one denotes
$H_{n}(X,A;\mathbb{Z})$ by $H_{n}(X,A)$.

Homology of manifolds is very well understood. We will mainly use the fact
that homology can detect compactness of manifolds, as follows.

######  Theorem A.1 (Corollary VI.7.12 (p. 346) in [BEGH93]).

Let $M$ be a connected $n$-manifold and assume that $G$ contains an element of
order $2$. One has

$H_{n}(M)\ncong 0\iff M\text{ is compact.}$

The next result is a generalization to pairs.

######  Theorem A.2 (Corollary VI.7.11 (p. 346) in [BEGH93]).

Let $M$ be an $n$-manifold and assume that $G$ contains an element of order
$2$. If $A\subseteq M$ is closed and connected, then

$H_{n}(M,M\setminus A;G)\ncong 0\iff A\text{ is compact.}$

Note that Theorem A.1 is just a particular case of Theorem A.2, with $A=M$.

###  A.2 Proof of Theorem 8.8

These results imply that if $(M,\partial M)$ is a compact connected
$n$-manifold with (possibly empty) boundary, then removing a point makes the
$n$th homology group trivial, because compactness is lost. The argument is
very classical but we did not find this particular statement in any reference
textbook, so we include a proof. For $x\in M$, we write
$M_{x}=M\setminus\\{x\\}$.

######  Theorem A.3.

Let $(M,\partial M)$ be a compact connected $n$-manifold with (possibly empty)
boundary. If $G$ contains an element of order $2$, then $H_{n}(M,\partial
M;G)\ncong 0$. If $x\in M\setminus\partial M$ then $H_{n}(M_{x},\partial
M;G)\cong 0$.

###### Proof.

If $\partial M$ is empty, then it is a direct consequence of Theorem A.1,
because $M_{x}$ is a non-compact connected $n$-manifold. Let us now consider
the case when $\partial M$ is non-empty.

In $M$, $\partial M$ has a _collar neighborhood_ , i.e. an open neighborhood
that is homeomorphic to $\partial M\times[0,1)$, in which $\partial M$ is
identified with $\partial M\times\\{0\\}$ (Proposition 3.42 in [Hat02]).
Moreover we can assume that this collar neighborhood does not contain $x$
(otherwise take the subset that is sent to $\partial M\times[0,\epsilon)$ by
the homeomorphism, for sufficiently small $\epsilon>0$). For simplicity of
notation, we identify the collar neighborhood with $\partial M\times[0,1)$. As
the collar neighborhood deformation retracts to $\partial M$, one has for any
abelian group $G$,

$\displaystyle H_{n}(M_{x},\partial M;G)$ $\displaystyle\cong
H_{n}(M_{x},\partial M\times[0,1);G)$ $\displaystyle\cong
H_{n}(M_{x}\setminus\partial M,\partial M\times(0,1);G)$

where the last isomorphism exists by excision (it is a classical derivation
that can be found for instance in [BEGH93], at the beginning of section VI.9).
Similarly, $H_{n}(M,\partial M;G)\cong H_{n}(M\setminus\partial M,\partial
M\times(0,1);G)$. We apply Theorem A.2 to the manifolds $N=M\setminus\partial
M$ and $N_{x}=M_{x}\setminus\partial M$ and their subsets
$A=N\setminus(\partial M\times(0,1))$ and $A_{x}=N_{x}\setminus(\partial
M\times(0,1))$. $A$ is compact but $A_{x}$ is not because $x$ has been
removed, so by Theorem A.2,

$\displaystyle H_{n}(M\setminus\partial M,\partial M\times(0,1);G)$
$\displaystyle=H_{n}(N,N\setminus A)\ncong 0.$ $\displaystyle
H_{n}(M_{x}\setminus\partial M,\partial M\times(0,1);G)$
$\displaystyle=H_{n}(N_{x},N_{x}\setminus A_{x})\cong 0.$

As a result, $H_{n}(M,\partial M;G)\ncong 0$ and $H_{n}(M_{x},\partial
M;G)\cong 0$. ∎

We can now prove that compact connected $n$-manifolds with boundary are
$\mathsf{H}_{n}$-minimal, using the same argument as for manifolds without
boundary.

If $x\in M\setminus\partial M$ and $B\subseteq M\setminus\partial M$ is an
open Euclidean ball around $x$, then $M_{x}$ _deformation retracts_ to
$M\setminus B$, which means that there is a retraction $r:M_{x}\to M\setminus
B$ such that if $i:M\setminus B\to M_{x}$ is the inclusion map, then $r\circ
i$ is homotopic to $\mathrm{id}_{M_{x}}$, and the homotopy is constant on
$M\setminus B$. It is a classical result that deformation retractions preserve
homology groups (Proposition 2.19 in [Hat02]), so $H_{n}(M\setminus B,\partial
M;G)\cong H_{n}(M_{x},\partial M;G)\cong 0$.

Theorem 7.24 relates $\mathsf{H}_{n}$ with homology groups, but is only for
single sets. However, pairs can be reduced to single sets as follows. One the
one hand, relative homology groups of pairs satisfying the homotopy extension
property are isomorphic to homology groups of their quotients, so:

$\displaystyle H_{n}(M/\partial M;G)$ $\displaystyle\cong H_{n}(M,\partial
M;G)\ncong 0,$ $\displaystyle H_{n}(M_{x}/\partial M;G)$ $\displaystyle\cong
H_{n}(M_{x},\partial M;G)\cong 0.$

On the other hand, a pair $(X,A)$ is in $\mathsf{H}_{n}$ iff $X/A$ is in
$\mathsf{H}_{n}$ (Proposition 7.20).

Therefore, Theorem 7.24 implies that $(M,\partial M)\in\mathsf{H}_{n}$ and
$(M\setminus B,\partial M)\notin\mathsf{H}_{n}$. It implies that $(M,\partial
M)$ is $\mathsf{H}_{n}$-minimal by Lemma 7.17: if $(X,A)$ is a proper subspair
of $M$, then $(X,A)\subseteq(M\setminus B,\partial M)$ for some $B\subseteq
M\setminus\partial M$, so $(X,A)\notin\mathsf{H}_{n}$.