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A fundamental group for digital images

Abstract

We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously established versions of a fundamental group in the digital setting. Our development gives a prominent role to subdivision of digital images. We show that our fundamental group is preserved by subdivision.

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Correspondence to Gregory Lupton.

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This work was partially supported by grants from the Simons Foundation: (#209575 to Gregory Lupton and #244393 to John Oprea).

Appendices

Appendix A. A technical result

We prove the technical result Corollary A.4 that we relied upon to establish injectivity of \((\rho _{2k+1})_*\) in the proof of Theorem 3.23. The results here should be viewed in that context. Unfortunately, the proof is rather lengthy, with many technical details. Furthermore, there is considerable notation introduced for the purpose of the proof (some of which is drawn from the following “Appendix B” below). Including all this material in the main body would interrupt the flow of ideas there, hence we have placed it here.

Suppose \(\alpha :I_M \rightarrow S(X, 2k+1)\) is a based loop in \(S(X, 2k+1)\), for \(X \subseteq \mathbb {Z}^n\) any based digital image. Denote the basepoint of X by \(\mathbf {x}_\mathbf {0} \in X\); we suppose \(\alpha \) is based at \(\overline{\mathbf {x}_\mathbf {0}} \in S(X, 2k+1)\). We use \(\alpha \) to construct a new based loop in \(S(X, 2k+1)\) as follows. For this, we use a device similar to what we called the coordinate-centring function in Lupton et al. (2019b). Namely, define a function \(C:I_{2k} \rightarrow I_{2k}\) by

$$\begin{aligned} C(r) := {\left\{ \begin{array}{ll} r+ 1 &{} 0 \le r \le k-1\\ k &{} r = k \\ r- 1 &{} k+1 \le r \le 2k.\end{array}\right. } \end{aligned}$$
(5)

Then applying C repeatedly will increase or decrease an integer in \(I_{2k}\) to k, according as the integer is less than k, or greater than k, and stabilize at k once there. For any r with \(0 \le r \le 2k\), we have \(C^p(r) = k\) for all \(p \ge k\).

Notation (leading up to Lemma A.1): For any \(i \in I_M\), suppose that \(\rho _{2k+1} \circ \alpha (i) = \mathbf {x}_\mathbf {i} = (x_1, \ldots , x_n) \in X\), so that \(\alpha (i) \in S( \mathbf {x}_\mathbf {i}, 2k+1)\). Then we have

$$\begin{aligned} \alpha (i) = \big ( (2k+1)x_1 + r_1, \ldots , (2k+1)x_n + r_n \big ), \end{aligned}$$

for some \(r_j\) with \(0 \le r_j \le 2k\), for \(j = 1, \ldots , n\). The centre of \(S( \mathbf {x}_\mathbf {i}, 2k+1)\) is

$$\begin{aligned} \overline{ \mathbf {x}_\mathbf {i}} = \big ( (2k+1)x_1 + k, \ldots , (2k+1)x_n + k \big ). \end{aligned}$$

For each \(i \in I_M\), then, define a path

$$\begin{aligned} \gamma _i:I_k \rightarrow S( \mathbf {x}_\mathbf {i}, 2k+1), \end{aligned}$$

that goes from \(\alpha (i)\) to \(\overline{\mathbf {x}_\mathbf {i}}\) then remains at \(\overline{\mathbf {x}_\mathbf {i}}\) for any remaining time in \(I_k\), as follows:

$$\begin{aligned} \gamma _i(t) := \big ( (2k+1)x_1 + C^t(r_1), \ldots , (2k+1)x_n + C^t(r_n) \big ), \end{aligned}$$
(6)

for \(0 \le t \le k\), where \(C^0(r) = r\). (Notice that the maximum difference \(|r_j - k|\) in any one coordinate of \(\alpha (i)\) and \(\overline{\mathbf {x}_\mathbf {i}}\) cannot be greater than k, so we have \(C^k(r_j) = k\) for all j.) If \(i = 0\) or \(i=M\), we have \(\alpha (0) = \alpha (M) = \overline{\mathbf {x}_\mathbf {0}}\), the centre of \(S( \mathbf {x}_\mathbf {0}, 2k+1)\). If we write the coordinates of \(\mathbf {x}_\mathbf {0}\) as \((x_1, \ldots , x_n)\), then we have

$$\begin{aligned} \alpha (0) = \alpha (M) =\big ( (2k+1)x_1 + k, \ldots , (2k+1)x_n + k \big ), \end{aligned}$$

so each \(r_j = k\), and \(\gamma _0:I_k \rightarrow S( \mathbf {x}_\mathbf {0}, 2k+1)\) and \(\gamma _M:I_k \rightarrow S( \mathbf {x}_\mathbf {0}, 2k+1)\) are both the constant path at \(\mathbf {x}_\mathbf {0}\). Then, string these paths and their reverses together to define \(\beta \). Decompose \(S(I_M, 2k+1)\) using the centres \(\overline{i} = (2k+1)i + k\) for \(i = 0, \ldots , M\), so we have

$$\begin{aligned} S(I_M, 2k+1) = [0, \overline{0}] \cup \bigcup _{i=0}^{i=M-1}\ [\overline{i}, \overline{i+1}]\ \cup [\overline{M}, (2k+1)M + 2k]. \end{aligned}$$

(This is not a disjoint union; the subintervals overlap at their endpoints.) Then define \(\beta \) as the constant path at \(\overline{\mathbf {x}_\mathbf {0}}\) on \([0, \overline{0}] \cup [\overline{M}, (2k+1)M + 2k]\). On each subinterval between centres \([\overline{i}, \overline{i+1}]\), for \(0 \le i \le M\), use (6) to define

$$\begin{aligned} \beta (\overline{i}+s) := {\left\{ \begin{array}{ll} \gamma _i(k-s) &{} 0 \le s \le k\\ \gamma _{i+1}\big (s -(k+1)\big ) &{} k+1 \le s \le 2k+1.\end{array}\right. } \end{aligned}$$
(7)

On each subinterval \([\overline{i}, \overline{i+1}]\), then, \(\beta \) restricts to the concatenation of paths \(\overline{\gamma _i}\cdot \gamma _{i+1}\), in the sense we defined concatenation of paths in Definition 3.6.

Lemma A.1

Suppose \(\alpha :I_M \rightarrow S(X, 2k+1)\) is a based loop in \(S(X, 2k+1)\), for \(X \subseteq \mathbb {Z}^n\) any based digital image. With \(\beta :S(I_M, 2k+1) \rightarrow S(X, 2k+1)\) the based loop defined as in (7) above, we have a based homotopy of based loops

$$\begin{aligned} \beta \approx \alpha \circ \rho _{2k+1} :S(I_M, 2k+1) \rightarrow S(X, 2k+1). \end{aligned}$$

Proof

To define a homotopy H over the whole of \(S(I_M, 2k+1) \times I_k\), it is sufficient to define it and check continuity on each subrectangle \([(2k+1)i, (2k+1)(i + 1)]\times I_k\), as well as the subrectangle \([(2k+1)M, (2k+1)M + 2k]\times I_k\), separately, so long as the separate definitions agree on the overlaps \(\{(2k+1)i\} \times I_k\). This is because any two points of \(S(I_M, 2k+1)\times I_k\) that are adjacent lie in one or the other of these subrectangles. We begin by defining H on \(S(i, 2k+1) \times I_k\) for each \(i \in I_M\).

For \(i=0\), we have \(S(0, 2k+1) = [0, 2k]\). Now here, we have \(\alpha \circ \rho _{2k+1}(s) = \alpha (0) = \overline{\mathbf {x}_\mathbf {0}}\) for each \(s \in [0, 2k]\). Also, above (7) we specified that \(\beta (s) = \overline{\mathbf {x}_\mathbf {0}}\) for \(s\in [0, \overline{0}]\), whilst (7) has \(\beta (\overline{0}+s) = \gamma _0(k-s)\) for \(0 \le s \le k\). But the \(\gamma _i\) are defined such that we have \(\gamma _0\) the constant path at \(\overline{\mathbf {x}_\mathbf {0}}\) also (see the comments following (6)). That is, we also have \(\beta (s) = \overline{\mathbf {x}_\mathbf {0}}\) for each \(s \in [0, 2k]\). Since \(\alpha \circ \rho _{2k+1}(s) = \beta (s) = \overline{\mathbf {x}_\mathbf {0}}\) for each \(s \in [0, 2k]\), we may define H to be the constant map \(H(s, t) = \overline{\mathbf {x}_\mathbf {0}}\) on \([0, 2k]\times I_k\). A similar state of affairs pertains at the other end of \(S(I_M, 2k+1)\): we have \(\alpha \circ \rho _{2k+1}(s) = \beta (s) = \overline{\mathbf {x}_\mathbf {0}}\) for each \(s \in [(2k+1)M, (2k+1)M+2k]\), and we define H to be the constant map \(H(s, t) = \overline{\mathbf {x}_\mathbf {0}}\) on \([(2k+1)M, (2k+1)M+2k]\times I_k\).

Now consider i for \(1 \le i \le M-1\). Here, we have \(\alpha \circ \rho _{2k+1}(s) = \alpha (i)\) for each \(s \in S(i, 2k+1)\). We write \(S(i, 2k+1) = [(2k+1)i, (2k+1)i + 2k]\) as

$$\begin{aligned} \begin{aligned} S(i, 2k+1)&= [(2k+1)i, (2k+1)i + k-1] \sqcup [(2k+1)i+k, (2k+1)i + 2k] \\&= [\overline{i-1}+k+1, \overline{i-1}+2k] \sqcup [\overline{i}, \overline{i}+k] \\&= \{ \overline{i-1}+k+1+s \mid 0 \le s \le k-1\} \sqcup \{ \overline{i}+s-k \mid k \le s \le 2k\}. \end{aligned} \end{aligned}$$

According to (7), then, we have \(\beta \) restricted to \(S(i, 2k+1) = [(2k+1)i, (2k+1)i + 2k]\) given by

$$\begin{aligned} \begin{aligned} \beta \big ( (2k+1)i + s \big )&= {\left\{ \begin{array}{ll} \beta ( \overline{i-1}+k+1+s) &{} 0 \le s \le k-1\\ \beta ( \overline{i}+s-k) &{} k \le s \le 2k\end{array}\right. }\\&= {\left\{ \begin{array}{ll} \gamma _{ (i-1)+1}\big ( (k+1+s) -(k+1)\big ) &{} 0 \le s \le k-1\\ \gamma _i\big ( k-(s-k)\big ) &{} k \le s \le 2k\end{array}\right. }\\&= {\left\{ \begin{array}{ll} \gamma _{i}(s) &{} 0 \le s \le k-1\\ \gamma _i( 2k-s) &{} k \le s \le 2k\end{array}\right. }\\&= (\gamma _i*\overline{\gamma _i})(s), \end{aligned} \end{aligned}$$

where \(\gamma _i*\overline{\gamma _i}\) denotes the short concatenation of paths as in (2) of Definition 3.6. By Lemma 3.16, we have a homotopy relative the endpoints \(H :I_{2k}\times I_k \rightarrow S(X, 2k+1)\) from \(\gamma _i*\overline{\gamma _i}\) to \(C^{\gamma _i(0)}_{2k} = C^{\alpha (i)}_{2k}\), which translates to a homotopy relative the endpoints \(H \circ (T\times \mathrm {id}_{I_k}):[(2k+1)i, (2k+1)i + 2k] \times I_k \rightarrow S(X, 2k+1)\) from the restriction of \(\beta \) to the restriction of \(\alpha \circ \rho _{2k+1}\), where \(T :[(2k+1)i, (2k+1)i + 2k] \rightarrow [0, 2k]\) is the translation \(T(s) := s-2k\).

Then, we extend H over the subrectangle \([(2k+1)i, (2k+1)(i + 1)]\times I_k\) by setting \(H\big ((2k+1)(i + 1), t\big ) = \alpha (i+1)\) for all \(t \in I_k\). Notice that this is consistent with how we would define H over the next subinterval \(S(i+1, 2k+1)\). Since H is already constant at \(\alpha (i)\) on \(\{(2k+1)i + 2k\} \times I_k\), and we have \(\alpha (i) \sim _{S(X, 2k+1)} \alpha (i+1)\) since \(\alpha \) is continuous, it follows that this extends H continuously over \([(2k+1)i, (2k+1)(i + 1)]\times I_k\). As already discussed, these homotopies piece together to give a homotopy

$$\begin{aligned} H:S(I_M, 2k+1) \times I_M \rightarrow S(X, 2k+1) \end{aligned}$$

from \(\beta \) to \(\alpha \circ \rho _{2k+1}\). Since we have \(H\big ( 0, t\big ) = \alpha (0) = \overline{x_0}\) and \(H\big ( (2k+1)M + 2k, t\big ) = \alpha (M) = \overline{x_0}\), this is a based homotopy of based loops. \(\square \)

The next step is to show \(\beta \) and the standard cover \(\widehat{\rho _{2k+1} \circ \alpha }\) (cf. Theorem B.4.1) are homotopic. We first give a careful description of \(\widehat{\rho _{2k+1} \circ \alpha }\). As observed following (7), when restricted to the subinterval \([\overline{i}, \overline{i+1}]\), we have \(\beta = \overline{\gamma _i}\cdot \gamma _{i+1}\). Our strategy is to view \(\beta \) and \(\widehat{\rho _{2k+1} \circ \alpha }\) piecewise, over the subintervals \([\overline{i}, \overline{i+1}]\) of \(S(I_M, 2k+1)\), and define a homotopy from one to the other on each of these subintervals separately.

Now the definition of \(\widehat{\rho _{2k+1} \circ \alpha }\) on each subinterval \([\overline{i}, \overline{i+1}]\) depends on both the values \(\rho _{2k+1} \circ \alpha (i) \in X\) and \(\rho _{2k+1} \circ \alpha (i+1) \in X\). We augment the notation leading up to Lemma A.1 as follows:

Notation (leading up to Lemma A.2): For any \(i \in I_M\) with \(0 \le i \le M-1\), suppose that \(\rho _{2k+1} \circ \alpha (i) = \mathbf {x}_\mathbf {i} = (x_1, \ldots , x_n) \in X\) and \(\rho _{2k+1} \circ \alpha (i+1) = \mathbf {x}_{\mathbf {i+1}} = (x'_1, \ldots , x'_n) \in X\), so that \(\alpha (i+1) \in S( \mathbf {x}_\mathbf {{i+1}}, 2k+1)\). Then we have

$$\begin{aligned} \alpha (i) = \big ( (2k+1)x_1 + r_1, \ldots , (2k+1)x_n + r_n \big ), \end{aligned}$$

and

$$\begin{aligned} \alpha (i+1) = \big ( (2k+1)x'_1 + r'_1, \ldots , (2k+1)x'_n + r'_n \big ) \end{aligned}$$

for some \(r_j\) and \(r'_j\) with \(0 \le r_j, r'_j \le 2k\) for each \(j = 1, \ldots , n\). The centres of \(S( \mathbf {x}_\mathbf {i}, 2k+1)\) and \(S( \mathbf {x}_{\mathbf {i+1}}, 2k+1)\) are

$$\begin{aligned} \overline{ \mathbf {x}_\mathbf {i}} = \big ( (2k+1)x_1 + k, \ldots , (2k+1)x_n + k \big ) \end{aligned}$$

and

$$\begin{aligned} \overline{ \mathbf {x}_{\mathbf {i+1}}} = \big ( (2k+1)x'_1 + k, \ldots , (2k+1)x'_n + k \big ) \end{aligned}$$

respectively. In the following result, and in the sequel, we denote the jth coordinate of \(\widehat{\rho _{2k+1} \circ \alpha }(t)\), for any \(t \in S(I_M, 2k+1)\), by

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(s)_j, \end{aligned}$$

for \(j = 1, \ldots , n\).

Lemma A.2

Suppose \(\alpha :I_M \rightarrow S(X, 2k+1)\) is a based loop in \(S(X, 2k+1)\), for \(X \subseteq \mathbb {Z}^n\) any based digital image, and \(\widehat{\rho _{2k+1} \circ \alpha }:S(I_M, 2k+1) \rightarrow S(X, 2k+1)\) is the standard cover of \(\rho _{2k+1} \circ \alpha :I_M \rightarrow X\) as in Theorem B.4.1.

  1. (1)

    For \(s \in [0, \overline{0}] \sqcup [\overline{M}, \overline{M} + 2k] \subseteq S(I_M, 2k+1)\), we have

    $$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(s) = \overline{\mathbf {x}_\mathbf {0}}. \end{aligned}$$
  2. (2)

    For \(i \in I_M\) with \(0 \le i \le M-1\), write \([\overline{i}, \overline{i+1}] = \{ \overline{i} + s \mid 0 \le s \le 2k+1\}\). When restricted to \([\overline{i}, \overline{i+1}]\), we may write the jth coordinate of the standard cover as

    $$\begin{aligned}&\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j \\&\qquad = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}\big ( k + k(x'_j - x_j)\big ) &{} 0 \le s \le k\\ (2k+1)x'_j + C^{s-(k+1)}\big ( k - k(x'_j - x_j)\big ) &{} k+1 \le s \le 2k+1.\end{array}\right. } \end{aligned}$$

Proof

The first item is part of the construction of the standard cover. Now consider the second item, over a subinterval \([\overline{i}, \overline{i+1}]\). From the construction of the standard cover, we have

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = (2k+1)x_j + k + s[x'_j - x_j] \text { for } 0 \le s \le 2k+1. \end{aligned}$$

For \(k+1 \le s \le 2k+1\), we may write

$$\begin{aligned} \begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j&= (2k+1)x_j + k + s[x'_j - x_j] \\&=(2k+1)x'_j - (2k+1)x'_j + k + (2k+1)x_j + sx'_j - sx_j \\&= (2k+1)x'_j + k -\big ( (2k+1) - s\big )[ x'_j -x_j].\end{aligned} \end{aligned}$$

Thus, we have

$$\begin{aligned}&\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j \\&\qquad = {\left\{ \begin{array}{ll} (2k+1)x_j + k + s(x'_j - x_j) &{} 0 \le s \le k\\ (2k+1)x'_j + k -\big ( (2k+1) - s\big )[ x'_j -x_j] &{} k+1 \le s \le 2k+1.\end{array}\right. } \end{aligned}$$

In the notation above, we have \(\rho _{2k+1} \circ \alpha (i) = \mathbf {x}_\mathbf {i}\) and \(\rho _{2k+1} \circ \alpha (i+1) = \mathbf {x}_{\mathbf {i+1}}\). Now \(\rho _{2k+1} \circ \alpha \) is continuous, and so we have \(\mathbf {x}_{\mathbf {i+1}} \sim _X \mathbf {x}_\mathbf {i}\). Therefore, \(|x'_j - x_j| \le 1\) for \(j = 1, \ldots , n\). We divide and conquer, considering \(x'_j - x_j = 0\), \(x'_j - x_j = +1\), and \(x'_j - x_j= -1\) for each j separately.

If \(x'_j - x_j =0\), then the jth coordinate of the standard cover reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = (2k+1)x_j + k \text { for } 0 \le s \le 2k+1. \end{aligned}$$
(8)

Item (2) of the enunciation (with \(x'_j = x_j\)) reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}( k) &{} 0 \le s \le k\\ (2k+1)x_j + C^{s-(k+1)}( k) &{} k+1 \le s \le 2k+1.\end{array}\right. } \end{aligned}$$
(9)

But (8) and (9) agree, since we have \(C^{k-s}( k) = k\) for each \(s = 0, \ldots , k\) and \(C^{s-(k+1)}( k) = k\) for each \(s = k+1, \ldots , 2k+1\). Notice that this case includes the case in which we have \(\rho _{2k+1} \circ \alpha (i) = \rho _{2k+1} \circ \alpha (i+1) = \mathbf {x}_\mathbf {i}\), where we have

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(s) = \overline{\mathbf {x}_\mathbf {i}} \end{aligned}$$

for each \(s \in [\overline{i}, \overline{i+1}]\).

Next, suppose that we have \(x'_j - x_j =+1\). Then the jth coordinate of the standard cover reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = (2k+1)x_j + k +s \text { for } 0 \le s \le 2k+1. \end{aligned}$$
(10)

Item (2) of the enunciation (with \(x'_j = x_j + 1\)) reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}( 2k) &{} 0 \le s \le k\\ (2k+1)(x_j + 1)+C^{s-(k+1)}( 0) &{} k+1 \le s \le 2k+1.\end{array}\right. }\nonumber \\ \end{aligned}$$
(11)

Now \(C^{k-s}(2k) = 2k - (k-s) = k+s\) for \(s=0, \dots , k\), whereas \(C^{s-(k+1)}( 0) = 0+\big ( s-(k+1) \big ) = s-(k+1)\) for \(s=k+1, \dots , 2k+1\). Hence the second term of (11) also gives \((2k+1)x_j + k+s\) for \(s=k+1, \dots , 2k+1\), and (10) agrees with (11) on \([\overline{i}, \overline{i+1}]\).

Finally, suppose we have \(x'_j - x_j =-1\). Then the jth coordinate of the standard cover reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = (2k+1)x_j + k -s \text { for } 0 \le s \le 2k+1. \end{aligned}$$
(12)

Item (2) of the enunciation (with \(x'_j = x_j - 1\)) reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}( 0) &{} 0 \le s \le k\\ (2k+1)(x_j - 1)+C^{s-(k+1)}( 2k) &{} k+1 \le s \le 2k+1.\end{array}\right. }\nonumber \\ \end{aligned}$$
(13)

Here, we have \(C^{k-s}(0) = 0 + (k-s) = k-s\) for \(s=0, \dots , k\), and \(C^{s-(k+1)}( 2k) = 2k-\big ( s-(k+1) \big ) = (2k+1) + k-s\) for \(s=k+1, \dots , 2k+1\). Here, the second term of (11) then gives \((2k+1)x_j +k-s\) for \(s=k+1, \dots , 2k+1\), and in this last case also, (12) agrees with (13) on \([\overline{i}, \overline{i+1}]\). \(\square \)

Lemma A.3

Suppose \(\alpha :I_M \rightarrow S(X, 2k+1)\) is a based loop in \(S(X, 2k+1)\), for \(X \subseteq \mathbb {Z}^n\) any based digital image. With \(\beta :S(I_M, 2k+1) \rightarrow S(X, 2k+1)\) the based loop defined as in (7) above, and \(\widehat{\rho _{2k+1} \circ \alpha }:S(I_M, 2k+1) \rightarrow S(X, 2k+1)\) the standard cover of \(\rho _{2k+1} \circ \alpha :I_M \rightarrow X\) as in Theorem B.4.1, we have a based homotopy of based loops

$$\begin{aligned} \beta \approx \widehat{\rho _{2k+1} \circ \alpha } :S(I_M, 2k+1) \rightarrow S(X, 2k+1). \end{aligned}$$

Proof

We will construct a based homotopy of based loops

$$\begin{aligned} G :S(I_M, 2k+1) \times I_k\rightarrow S(X, 2k+1) \end{aligned}$$

from \(\beta \) to \(\widehat{\rho _{2k+1} \circ \alpha }\). As discussed in the proof of Lemma A.1, we have \(\beta (s) = \overline{\mathbf {x}_\mathbf {0}}\) for (at least) \(s\in [0, \overline{0}]\) and \(s \in [(2k+1)M, (2k+1)M+2k]\). Also, the standard cover is defined as \(\widehat{\rho _{2k+1} \circ \alpha }(s) = \overline{\mathbf {x}_\mathbf {0}}\) on the same subintervals. So we set G to be the constant map \(G(s, t) = \overline{\mathbf {x}_\mathbf {0}}\) on \([0, \overline{0}]\times I_k\) and \([\overline{M}, \overline{M}+k]\times I_k\).

Refer to the notation leading up to Lemma A.1. We have

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i}) = \overline{ \rho _{2k+1} \circ \alpha (i) } = \overline{ \mathbf {x}_\mathbf {i} } \end{aligned}$$

for each centre \(\overline{i} \in S(I_M, 2k+1)\). From (7), \(\beta \) also has the property that

$$\begin{aligned} \beta (\overline{i}) = \gamma _i(k) = \overline{ \mathbf {x}_\mathbf {i} }. \end{aligned}$$

For each \(i \in I_M\) with \(0 \le i \le M-1\), we will define a homotopy

$$\begin{aligned} G:[\overline{i}, \overline{i+1}] \times I_k \rightarrow S(X, 2k+1) \end{aligned}$$

relative the endpoints, which is to say that we will have \(G(\overline{i}, t) = \beta ( \overline{i}) = \widehat{\rho _{2k+1} \circ \alpha }( \overline{i}) = \overline{ \mathbf {x}_\mathbf {i} }\) for each \(i \in I_M\) with \(0 \le i \le M-1\) and each \(t \in I_k\), from the restriction of \(\beta \) to the restriction of \(\widehat{\rho _{2k+1} \circ \alpha }\). Together with the constant homotopies on \([0, \overline{0}] \times I_k\) and \([\overline{M}, (2k+1)M + 2k]\times I_k\), these will piece together continuously to give the desired homotopy.

Refer to the notation leading up to Lemma A.2. From (6) and (7), we have the jth coordinate function of \(\beta \) described as

$$\begin{aligned} \beta (\overline{i}+s)_j = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s} (r_j) &{} 0 \le s \le k\\ (2k+1)x'_j + C^{s-(k+1)} (r'_j) &{} k+1 \le s \le 2k+1. \end{array}\right. } \end{aligned}$$
(14)

Item (2) of Lemma A.2 gives our description of the jth coordinate of the standard cover that we use. Now \(\alpha \) is continuous, so we have \(\alpha (i) \sim _{S(X, 2k+1)} \alpha (i+1)\). This means that in each coordinate we have

$$\begin{aligned} | (2k+1)x'_j + r'_j -((2k+1)x_j + r_j)| \le 1 \end{aligned}$$
(15)

for each j. Also, because \(\mathbf {x}_{\mathbf{i+1}} \sim _X \mathbf {x}_{\mathbf{i}}\), we have \(|x'_j - x_j| \le 1\) for each j.

We will work with each coordinate separately, proceeding differently according as \(x'_j - x_j = 0\) or not. We will define \(G:[\overline{i}, \overline{i+1}]\times I_{k}\rightarrow S(X, 2k+1)\) in terms of its coordinate functions, thus:

$$\begin{aligned} G(s, t) := \big ( G_1(s, t), \ldots , G_n(s, t)\big ), \end{aligned}$$

for \((s, t) \in [\overline{i}, \overline{i+1}]\times I_{k}\). In fact, we will construct the \(G_j\) so that they map

$$\begin{aligned} G_j:[\overline{i}, (2k+1)i+ 2k]\times I_{k}\rightarrow S(x_j, 2k+1) \subseteq \mathbb {Z}, \end{aligned}$$

and

$$\begin{aligned} G_j:[(2k+1)i+ 2k+1, \overline{i+1}]\times I_{k}\rightarrow S(x'_j, 2k+1) \subseteq \mathbb {Z}. \end{aligned}$$

Then the coordinate functions \(G_j\) will assemble into a homotopy G that maps

$$\begin{aligned} G:[\overline{i}, (2k+1)i+ 2k]\times I_{k}\rightarrow S(\mathbf {x}_{\mathbf {i}}, 2k+1) \subseteq S(X, 2k+1) \end{aligned}$$

and

$$\begin{aligned} G:[(2k+1)i+ 2k+1, \overline{i+1}]\times I_{k}\rightarrow S(\mathbf {x}_{\mathbf {i+1}}, 2k+1) \subseteq S(X, 2k+1). \end{aligned}$$

Thus, G will be continuous on \([\overline{i}, \overline{i+1}]\times I_{k}\) if and only if each of its coordinate functions is.

We treat the cases in which \(x'_j - x_j = \pm 1\) and \(x'_j - x_j = 0\) separately.

Case (i) \(\mathbf {x}'_\mathbf{j} - \mathbf{x}_\mathbf{j} = \pm \mathbf{1} \): Suppose that in the jth coordinates of \(\mathbf {x}_{\mathbf {i}}\) and \(\mathbf {x}_{\mathbf {i+1}}\) we have \(x'_j - x_j = +1\). Then (15) implies that we have \(| (2k+1) + r'_j -r_j| \le 1\) which, given that \(0 \le r_j, r'_j \le 2k\), implies that \(r_j = 2k\) and \(r'_j = 0\). Then the formulas for the jth coordinates of \(\beta \) and of \(\widehat{\rho _{2k+1} \circ \alpha }\) agree. Namely, because we have \(x'_j - x_j = +1\), item (2) of Lemma A.2 reduces to

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s} (2k) &{} 0 \le s \le k\\ (2k+1)x'_j + C^{s-(k+1)} (0) &{} k+1 \le s \le 2k+1. \end{array}\right. } \end{aligned}$$
(16)

But with \(r_j = 2k\) and \(r'_j = 0\), this is exactly the formula for \(\beta (\overline{i}+s)_j\) from (14). On the other hand, suppose that we have \(x'_j - x_j = -1\). Then (15) implies that we have \(| (2k+1) - (r'_j -r_j)| \le 1\) which, given that \(0 \le r_j, r'_j \le 2k\), implies that \(r_j = 0\) and \(r'_j = 2k\). In a similar way, item (2) of Lemma A.2 and (14) reduce to the same formula here, too. So, in the case in which \(x'_j - x_j = \pm 1\), we have

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = \beta (\overline{i}+s)_j, \end{aligned}$$

for each s with \(0 \le s \le 2k+1\). In this case, then, we define \(G_j(\overline{i}+s, t)\) independently of t as

$$\begin{aligned} G_j(\overline{i}+s, t) := \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s )_j = \beta (\overline{i} + s )_j \end{aligned}$$
(17)

for \(0 \le t \le k\). The value for each s is given by (16). Continuity in the jth coordinate in this case follows from the continuity of \(\beta \) (or of \(\widehat{\rho _{2k+1} \circ \alpha }\)). Indeed, suppose we have \((\overline{i} + s, t) \sim (\overline{i} + s', t')\) in \([\overline{i}, \overline{i+1}]\times I_{k}\). Then, in particular, we have \(\overline{i} + s \sim \overline{i} + s'\) in \([\overline{i}, \overline{i+1}]\). Since \(G_j(\overline{i} + s, t)\) is independent of t, we have \(|G_j(\overline{i} + s', t') - G_j(\overline{i} + s, t)| = |\beta (\overline{i} + s')_j - \beta (\overline{i} + s)_j| \le 1\), since \(\beta \) is continuous.

Case (ii) \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} =\mathbf {0}\): If \(x'_j - x_j = 0\), then item (2) of Lemma A.2 (with \(x'_j = x_j\)) gives the jth coordinate of \(\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)\) as \((2k+1)x_j + k\) for each s (recall that we have \(C^p(k) = k\) for all \(p \ge 0\)). Also, if \(x'_j - x_j = 0\), then (15) implies that we have \(|r'_j -r_j| \le 1\). Unfortunately, describing a suitable homotopy in this case involves a further divide-and-conquer, according as how \(r'_j\) and \(r_j\) compare with each other. Before embarking on the arguments, we summarize the sub-cases here:

  • Case (ii) (A)—In which we have \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {r'}_\mathbf {j} = \mathbf {r}_\mathbf {j}\);

  • Case (ii) (B)—In which we have \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {k}\le \mathbf {r'}_\mathbf {j} < \mathbf {r}_\mathbf {j}\) or \(\mathbf {r}_\mathbf {j} < \mathbf {r'}_\mathbf {j} \le \mathbf {k}\) (so that \(r'_j\) falls between k and \(r_j\));

  • Case (ii) (C)—In which we have \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {k}\le \mathbf {r}_\mathbf {j} < \mathbf {r'}_\mathbf {j}\) or \(\mathbf {r'}_\mathbf {j} < \mathbf {r}_\mathbf {j} \le \mathbf {k}\) (so that \(r_j\) falls between k and \(r'_j\))

Case (ii) (A) \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {r'}_\mathbf {j} = \mathbf {r}_\mathbf {j}\): If \(r'_j = r_j\), then we have \(C^p(r'_j) = C^p(r_j)\) for \(p \ge 0\). Then set

$$\begin{aligned} G_j(\overline{i}+s, t) := {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s} (r_j) &{} 0 \le s< k-t\\ (2k+1)x_j + C^{t} (r_j) &{} k-t \le s \le k+1+t\\ (2k+1)x_j + C^{s-(k+1)} (r_j) &{} k+1+t < s \le 2k+1.\end{array}\right. } \end{aligned}$$
(18)

When \(t=0\), this formula reduces to that for \(\beta (\overline{i}+s)\) from (14) (with \(x'_j = x_j\) and \(r'_j = r_j\)). When \(t = k\), this formula reduces to

$$\begin{aligned} G_j(\overline{i}+s, k) = (2k+1)x_j + k \end{aligned}$$

for \(0 \le s \le 2k+1\), since \(C^p(r_j) = k\) for \(p \ge k\). Namely, it reduces to the formula for \(\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)_j\) from item (2) of Lemma A.2 (with \(x'_j = x_j\)).

We now check continuity of (18); this follows an argument very similar to that used for Lemma 3.16. To check that \(G_j\) is continuous, divide \([\overline{i}, \overline{i+1}] \times I_{k}\) into two overlapping regions: Region A, consisting of those points \((\overline{i}+s, t)\) that satisfy both \(s+t \ge k\) and \(s-t \le k+1\); and region B, consisting of the points that satisfy either \(s+t \le k+1\) or \(s-t \ge k\). Now any pair of adjacent points \((\overline{i}+s, t)\sim (\overline{i}+s', t')\) in \([\overline{i}, \overline{i+1}] \times I_{k}\) satisfies \(|(s-t) - (s'-t')| \le 2\) and \(|(s+t) - (s'+t')| \le 2\), and hence either both lie in region A or both lie in region B. Suppose first that both lie in region A. From (18), for these points we have \(G_j(s, t) = C^t(r_j)\) and \(G_j(s', t') = C^{t'}(r_j)\). Since \((\overline{i}+s, t)\sim (\overline{i}+s', t')\), we have \(|t' - t| \le 1\), and hence \(|C^{t'}(r_j) - C^{t}(r_j)| \le 1\). Thus we have \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| \le 1\) when both adjacent points lie in A. On the other hand, suppose adjacent points \((\overline{i}+s, t) \sim (\overline{i}+s', t')\) both lie in region B. First suppose that \(s+t \le k+1\) and \(s'+t' \le k+1\) (the left-hand half of region B). Here, (18) specializes to

$$\begin{aligned} G_j(\overline{i}+s, t) = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}(r_j) &{} s +t \le k-1\\ (2k+1)x_j + C^{t}(r_j) &{} k \le s +t \le k+1.\end{array}\right. } \end{aligned}$$

If we have \(s+t \le k-1\) and \(s'+t' \le k-1\), then we have \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| = |C^{k-s'}(r_j) - C^{k-s}(r_j)| \le 1\), since we have \(|(k-s') - (k-s)| = |s - s'| \le 1\). If we have \(k \le s+t \le k+1\) and \(k \le s'+t' \le k+1\), then we have \(G_j(\overline{i}+s, t) = C^{t}(r_j)\) and \(G_j(\overline{i}+s', t') = C^{t'}(r_j)\), and again we have \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| \le 1\) since \(|t' - t| \le 1\). Finally, for adjacent points in the left-hand part of region B, suppose that we have (st) with \(s+t \le k-1\) and \((s', t')\) with \(k \le s'+t' \le k+1\). Now this is only possible if we also have \(s' \ge s\) which, assuming we have \((\overline{i}+s, t)\sim (\overline{i}+s', t')\), means that we have \(s' = s\) or \(s' = s+1\). Then \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| = |C^{k-s}(r_j) - C^{t'}(r_j)|\). If \(s' = s\), then the inequality \(k \le s'+t' \le k+1\) yields \(k-s \le t' \le k-s+1\), and if \(s' = s+1\), then the same inequality yields \(k-s-1 \le t' \le k-s\). In either case we have \(|t' - (k-s)| \le 1\), and thus \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| \le 1\). For the other remaining possibility, when \(k \le s+t \le k+1\) and \(s'+t' \le k-1\), we find that \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| \le 1\) follows by interchanging the roles of (st) and \((s', t')\) in the last few steps. It remains to confirm that \(G_j\) preserves adjacency for adjacent points \((\overline{i}+s, t)\sim (\overline{i}+s', t')\) that satisfy \(s-t \ge k\) and \(s'-t' \ge k\) (the right-hand half of region B). The same argument, mutatis mutandis, as we just used for the left-hand half of region B will confirm that we have \(| G_j(\overline{i}+s', t') - G_j(\overline{i}+s, t)| \le 1\) here also. We omit the details. This completes the check of continuity for \(G_j\) on \([\overline{i}, \overline{i+1}]\times I_k\) in this sub-case.

Case (ii) (B) \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {k}\le \mathbf {r'}_\mathbf {j} < \mathbf {r}_\mathbf {j}\) or \(\mathbf {r}_\mathbf {j} < \mathbf {r'}_\mathbf {j} \le \mathbf {k}\): Since \(|r'_j -r_j| \le 1\) (going back to the start of discussion for Case (ii)), we have \(C(r_j) = r'_j\) here (and hence, \(C^{p+1}(r_j) = C^p(r'_j)\) for \(p \ge 0\)). Roughly speaking, in this case our homotopy is the same as the previous case on the left half of \([\overline{i}, \overline{i+1}] \times I_k\) but, on the right half, we pause at the start for one unit of time, to allow the values \(G(k, t)_j\) to “catch-up” with those of \(G(k+1, t)_j\), as it were. This effect may be achieved as follows. We denote the homotopy in this sub-case by \(G^B_j(\overline{i}+s, t)\), for \((s, t) \in [0, 2k+1]\times I_k\), and define it in terms of the homotopy \(G_j(\overline{i}+s, t)\) from (18) of the previous sub-case. Here, because we must have \(k\le r'_j < r_j \le 2k\) or \(0 \le r_j < r'_j \le k\), it follows that we have \(C^{k-1}(r'_j) = C^k(r'_j) = k\), and thus, referring to (14) (with \(x'_j = x_j\)), we have

$$\begin{aligned} \beta (\overline{i}+s)_j = (2k+1)x_j + k = \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)_j, \end{aligned}$$

(at least) for \(s = 2k, 2k+1\) (recall that in this case, in which we have \(x'_j = x_j\), item (2) of Lemma A.2 gives the jth coordinate of \(\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)\) as \((2k+1)x_j + k\) for each s). Therefore, we may take the homotopy here to be constant for \(s = 2k\) and \(s= 2k+1\), and not just relative the endpoint \(s = 2k+1\). We define

$$\begin{aligned} G^B_j(\overline{i}+s, t) := {\left\{ \begin{array}{ll} G_j(\overline{i}+s, t) &{} 0 \le s \le k\\ G_j(\overline{i}+s+1, t) &{} k+1 \le s \le 2k\\ G_j(\overline{i}+2k+1, t) = \beta (\overline{i}+2k+1)_j&{} s = 2k+1,\end{array}\right. } \end{aligned}$$
(19)

where \(G_j(\overline{i}+s, t)\) is given by (18). Then for \(t=0\), referring to (18), this yields

$$\begin{aligned} G^B_j(\overline{i}+s, 0) = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}(r_j) &{} 0 \le s < k\\ (2k+1)x_j + C^{0}(r_j) &{} s = k\\ (2k+1)x_j + C^{(k+1)-(s+1)}(r_j) &{} k+1 \le s \le 2 k\\ (2k+1)x_j + k &{} s = 2 k+1, \end{array}\right. } \end{aligned}$$

which may be re-written as

$$\begin{aligned} G^B_j(\overline{i}+s, 0) = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s}(r_j) &{} 0 \le s \le k\\ (2k+1)x_j + C^{k-(s+1)}(r'_j) &{} k+1 \le s \le 2 k+1, \end{array}\right. } \end{aligned}$$

and thereby recognized as agreeing with \(\beta (\overline{i}+s)_j\) from (14) (with \(x'_j = x_j\)). A similar direct check confirms that we have \(G^B_j(\overline{i}+s, k) = \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s) = (2k+1)x_j + k\) for each s. Since \(G_j\) is continuous, from the previous sub-case, it follows that this \(G^B_j\) is continuous on the separate parts of its domain \([\overline{i}, \overline{i}+k] \times I_k\), \([\overline{i}+k+1, \overline{i}+2k] \times I_k\), and \(\{\overline{i}+2k+1\} \times I_k\). Where the latter two abut, the homotopy is a constant map and it is evident that \(G^B_j\) is continuous on the whole of \([\overline{i}+k+1, \overline{i}+2k+1] \times I_K\). Then, for \((s, t) \in \{\overline{i}+k, \overline{i}+k+1\} \times I_k\), (19) [and (18)] gives

$$\begin{aligned} G^B_j(\overline{i}+k, t) = G_j(\overline{i}+k, t) = (2k+1)x_j + C^{t}(r_j) \end{aligned}$$

and

$$\begin{aligned} G^B_j(\overline{i}+k+1, t) = G_j(\overline{i}+k+2, t) = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{1}(r_j) &{} t=0\\ (2k+1)x_j + C^{t}(r_j) &{} 1 \le t \le k. \end{array}\right. } \end{aligned}$$

Now, if we have \((s, t) \sim (s', t')\) in \(\{\overline{i}+k, \overline{i}+k+1\} \times I_k\), then either \(\{t, t'\} \subseteq \{0, 1\}\), or \(|t' - t| \le 1\) and \(1\le t, t' \le k\). If \(\{t, t'\} \subseteq \{0, 1\}\), then

$$\begin{aligned} \{ G^B_j(\overline{i}+s, t), G^B_j(\overline{i}+s', t') \} \subseteq \{ (2k+1)x_j + C^{1}(r_j), (2k+1)x_j + C^{0}(r_j) \}, \end{aligned}$$

and since \(|(2k+1)x_j + C^{1}(r_j)-\big ( (2k+1)x_j + C^{0}(r_j)\big )| = |C(r_j)-r_j| \le 1\), it follows that we have \(| G^B_j(\overline{i}+s, t)- G^B_j(\overline{i}+s', t')| \le 1\). But if \(1\le t, t' \le k\), then we have

$$\begin{aligned} | G^B_j(\overline{i}+s, t)- G^B_j(\overline{i}+s', t')| = | C^{t}(r_j) - C^{t'}(r_j)| \le 1, \end{aligned}$$

because we have \(|t' - t| \le 1\). It follows that \(G^B_j\) is continuous when restricted to \(\{\overline{i}+k, \overline{i}+k+1\} \times I_k\), and this is now sufficient to conclude that \(G^B_j\) is continuous over the whole of \([ \overline{i}, \overline{i+1}] \times I_k\).

Case (ii) (C) \(\mathbf {x'}_\mathbf {j} - \mathbf {x}_\mathbf {j} = \mathbf {0}\) and \(\mathbf {k}\le \mathbf {r}_\mathbf {j} < \mathbf {r'}_\mathbf {j}\) or \(\mathbf {r'}_\mathbf {j} < \mathbf {r}_\mathbf {j} \le \mathbf {k}\): This last case needs a variation on Case (ii) (A) similar to that which we just made for Case (ii) (B). Since \(|r'_j -r_j| \le 1\) (going back to the start of discussion for Case (ii)), we have \(C(r'_j) = r_j\) here (and hence, \(C^{p+1}(r'_j) = C^p(r_j)\) for \(p \ge 0\)). Here, our homotopy is the same as in Case (ii) (A) on the right half of \([\overline{i}, \overline{i+1}] \times I_k\) but, on the left half, we pause at the start for one unit of time, to allow the values \(G(k+1, t)_j\) to “catch-up” with those of \(G(k, t)_j\), as it were. We denote the homotopy in this sub-case by \(G^C_j(\overline{i}+s, t)\), for \((s, t) \in [0, 2k+1]\times I_k\), and again define it in terms of the homotopy \(G_j(\overline{i}+s, t)\) from (18). Here, because we must have \(k\le r_j < r'_j \le 2k\) or \(0 \le r'_j < r_j \le k\), it follows that we have \(C^{k-1}(r_j) = C^k(r_j) = k\), and thus, referring to (14) (with \(x'_j = x_j\)), we have

$$\begin{aligned} \beta (\overline{i}+s)_j = (2k+1)x_j + k = \widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)_j, \end{aligned}$$

(at least) for \(s = 0, 1\) (recall again that when we have \(x'_j = x_j\), item (2) of Lemma A.2 gives the jth coordinate of \(\widehat{\rho _{2k+1} \circ \alpha }(\overline{i} + s)\) as \((2k+1)x_j + k\) for each s). Therefore, we may take the homotopy here to be constant for \(s = 0\) and \(s= 1\), and not just relative the endpoint \(s = 0\). Finally, for set-up, we need to use \(r'_j\) in place of \(r_j\) in (18) (in that sub-case, \(r_j\) and \(r'_j\) were equal, but generally \(r_j\) pertains to the left half of \([\overline{i}, \overline{i+1}] \times I_k\) and \(r'_j\) to the right half, and it is on the right half that we wish to preserve the homotopy here). That is, we could equally well write the homotopy of (18) as

$$\begin{aligned} G'_j(\overline{i}+s, t) = {\left\{ \begin{array}{ll} (2k+1)x_j + C^{k-s} (r'_j) &{} 0 \le s< k-t\\ (2k+1)x_j + C^{t} (r'_j) &{} k-t \le s \le k+1+t\\ (2k+1)x_j + C^{s-(k+1)} (r'_j) &{} k+1+t < s \le 2k+1.\end{array}\right. } \end{aligned}$$
(20)

Then, with reference to (20), define here

$$\begin{aligned} G^C_j(\overline{i}+s, t) := {\left\{ \begin{array}{ll} G'_j(\overline{i}, t) = \beta (\overline{i})_j&{} s = 0\\ G'_j(\overline{i}+s-1, t) &{} 1 \le s \le k\\ G'_j(\overline{i}+s, t) &{} k+1 \le s \le 2k+1. \end{array}\right. } \end{aligned}$$
(21)

As in the previous Case (ii) (B), a direct check shows that this is a homotopy from the jth coordinate of the restriction of \(\beta \) to the jth coordinate of the restriction of \(\widehat{\rho _{2k+1} \circ \alpha }\). Continuity on \([ \overline{i}, \overline{i+1}] \times I_k\) follows from that of \(G'_j\) (which, recall, is identical with \(G_j\) in Case (ii) (A)). We omit these details.

Across Case (i) and Cases (ii) (A), (B), and (C) of the last several pages, then, we have defined coordinate homotopies \(G_j\) for each \(j = 1, \ldots , n\). As discussed before we entered Case (i) above, these coordinate homotopies assemble into a homotopy

$$\begin{aligned} G(\overline{i} + s, t) = \big ( G_1(\overline{i} + s, t), \ldots , G_n(\overline{i} + s, t)\big ) :[ \overline{i}, \overline{i+1}] \times I_k \rightarrow S(X, 2k+1), \end{aligned}$$

from \(\beta \) restricted to \([ \overline{i}, \overline{i+1}]\) to \(\widehat{\rho _{2k+1} \circ \alpha }\) restricted to \([ \overline{i}, \overline{i+1}]\). By checking each of the formulas (17), (18), (19), and (21) when \(s=0\) and \(s = 2k+1\), we see that, in each case in each coordinate, the homotopy is relative the endpoints. Indeed, we have (refer to the notation leading up to Lemma A.2)

$$\begin{aligned} G(\overline{i}, t) = \overline{\mathbf {x}_\mathbf {i}} \quad \text { and } \quad G(\overline{i}+2k+1, t) = \overline{\mathbf {x_{i+1}}}, \end{aligned}$$

for all \(t \in I_k\). Because G is continuous on each \([ \overline{i}, \overline{i+1}] \times I_k\), and is well-defined where these intervals overlap, these homotopies—as well as the constant homotopies on \([ 0, \overline{0}] \times I_k\) and \([ \overline{M}, \overline{M}+ k] \times I_k\) as we defined them at the start of this proof—assemble into a based homotopy of based loops \(G:S(I_M, k) \times I_k \rightarrow S(X, 2k+1)\) from \(\beta \) to \(\widehat{\rho _{2k+1} \circ \alpha }\). \(\square \)

Finally, we arrive at the technical result we relied upon to establish injectivity of \((\rho _{2k+1})_*\) in the proof of Theorem 3.23.

Corollary A.4

Suppose \(\alpha :I_M \rightarrow S(X, 2k+1)\) is a based loop in \(S(X, 2k+1)\), for \(X \subseteq \mathbb {Z}^n\) any based digital image. With \(\widehat{\rho _{2k+1} \circ \alpha }:S(I_M, 2k+1) \rightarrow S(X, 2k+1)\) the standard cover of \(\rho _{2k+1} \circ \alpha :I_M \rightarrow X\), we have a based homotopy of based loops

$$\begin{aligned} \widehat{\rho _{2k+1} \circ \alpha } \approx \alpha \circ \rho _{2k+1}:S(I_M, 2k+1) \rightarrow S(X, 2k+1). \end{aligned}$$

Proof

The ingredients are represented in the following diagram:

The lower-right triangle commutes tautologically; the outer rectangle commutes from the construction of the standard cover, as in Theorem B.4.1. The assertion here is that the upper-left triangle commutes up to based homotopy (generally it does not commute).

The homotopy we want follows directly from Lemmas A.1 and A.3, assuming symmetricity and transitivity of based homotopy of based loops. Since we have not given arguments for these points, we provide an explicit argument here. Lemma A.1 constructs a based homotopy of based loops

$$\begin{aligned} H:S(I_M, 2k+1) \times I_M \rightarrow S(X, 2k+1) \end{aligned}$$

from \(\beta \) to \(\alpha \circ \rho _{2k+1}\), and Lemma A.3 constructs a based homotopy of based loops

$$\begin{aligned} G :S(I_M, 2k+1) \times I_k\rightarrow S(X, 2k+1) \end{aligned}$$

from \(\beta \) to \(\widehat{\rho _{2k+1} \circ \alpha }\). Define a map \(\mathcal {H}:S(I_M, 2k+1) \times I_{2k+1} \rightarrow S(X, 2k+1)\) by

$$\begin{aligned} \mathcal {H}(s, t) := {\left\{ \begin{array}{ll} H(s, k-t) &{} 0 \le t \le k\\ G\big (s, t-(k+1)\big ) &{} k+1 \le t \le 2k+1.\end{array}\right. } \end{aligned}$$

The only possible issue with continuity of \(\mathcal {H}\) occurs where the two halves of the domain abut, namely, where \(t = k\) and \(t=k+1\). But here, we have

$$\begin{aligned} \mathcal {H}(s, k) = H(s, 0) = \beta (s) = G\big (s, 0\big ) = \mathcal {H}(s, k+1). \end{aligned}$$

Therefore, if we have \((s, t) \sim (s', t')\) in \(S(I_M, 2k+1) \times [k, k+1]\), then

$$\begin{aligned} |\mathcal {H}(s', t') - \mathcal {H}(s, t)| = |\beta (s') - \beta (s)| \le 1, \end{aligned}$$

since we have \(|s' - s| \le 1\) and \(\beta \) is continuous. Now any pair of adjacent points in \(S(I_M, 2k+1) \times I_{2k+1}\) both lie in one of the sub-rectangles \(S(I_M, 2k+1) \times [0, k]\), \(S(I_M, 2k+1) \times [k+1, 2k+1]\), or \(S(I_M, 2k+1) \times [k, k+1]\), and it follows that \(\mathcal {H}\) is continuous on the whole of \(S(I_M, 2k+1) \times I_{2k+1}\). A direct check confirms that \(\mathcal {H}(s, 0) = \alpha \circ \rho _{2k+1}(s)\) and \(\mathcal {H}(s, 2k+1) = (\widehat{\rho _{2k+1} \circ \alpha })(s)\), and also that \(\mathcal {H}(0, t) = \overline{\mathbf {x}_\mathbf {0}} = \mathcal {H}((2k+1)M+2k, t)\), so that \(\mathcal {H}\) is the desired based homotopy of based loops. \(\square \)

Appendix B. Based maps and homotopies

Here we present some basic material on maps and homotopies in the based setting. As we pointed out in the Introduction, whereas Lupton et al. (2019a, 2019b) are concerned with un-based maps and homotopies, we need based versions of all definitions and results. Whilst some of the items given here extend or amplify those reviewed in Sect. 2, they are nonetheless useful in the development of ideas in the main body. We refer to this appendix from numerous points in the main body, as well as from the material in “Appendix A”. But including all this material in the main body would slow the progression of ideas there. So we have elected to collect it here, rather than disperse it throughout the main body.

Also in this appendix, and for the convenience of the reader, we give the statements of two results from Lupton et al. (2019b) that are used in some of our main results.

Products and homotopy

We record a number of elementary observations that are used, implicitly or explicitly, in the main body.

Lemma B.1.1

For based digital images \((X, x_0)\) and \((Y, y_0)\), the projections onto either factor \(p_1:X \times Y \rightarrow X\) and \(p_2:X \times Y \rightarrow Y\) are based maps. Suppose given based maps of digital images \(f:(A, a_0) \rightarrow (X, x_0)\) and \(g:(A, a_0) \rightarrow (Y, y_0)\). Then there is a unique based map, which we write \((f, g):(A, a_0) \rightarrow \big (X \times Y, (x_0, y_0)\big )\) that satisfies \(p_1\circ (f, g) = f\) and \(p_2\circ (f, g) = g\).

Proof

The first assertion follows immediately from the definitions. The map (fg) is defined as \((f, g)(a) := \big ( f(a), g(a) \big )\). It is immediate from the definitions that this map is continuous and based. This is evidently the unique map with the suitable coordinate functions. \(\square \)

Because of the rectangular nature of the digital setting, it is often convenient to consider the product of maps, as follows.

Definition B.1.2

Given functions of digital images \(f_i :X_i \rightarrow Y_i\) for \(i = 1, \ldots , n\), we define their product function

$$\begin{aligned} f_1 \times \cdots \times f_n :X_1 \times \cdots \times X_n \rightarrow Y_1 \times \cdots \times Y_n \end{aligned}$$

as \((f_1 \times \cdots \times f_n) (x_1, \ldots , x_n) := \big (f_1(x_1), \ldots , f_n(x_n) \big )\).

Lemma B.1.3

Given based maps of based digital images \(f_i :X_i \rightarrow Y_i\) for \(i = 1, \ldots , n\), their product \(f_1 \times \cdots \times f_n\) is a (continuous) based map.

Proof

This follows directly from the definitions. \(\square \)

We defined based homotopy in Definition 2.4.

Lemma B.1.4

Suppose based maps \(f, f':X \rightarrow Y\) of based digital images \(X \subseteq \mathbb {Z}^m\) and \(Y \subseteq \mathbb {Z}^n\) are based-homotopic, and also based maps \(g, g':Y \rightarrow Z\) of based digital images with \(Z \subseteq \mathbb {Z}^p\) are based-homotopic. Then \(g\circ f, g'\circ f':X \rightarrow Z\) are based-homotopic.

Proof

Suppose \(H :X \times I_N \rightarrow Y\) is a based homotopy from f to \(f'\) and \(G :Y \times I_M \rightarrow Z\) is a based homotopy from g to \(g'\). Define \(P :X \times I_{N+M} \rightarrow Z\) by

$$\begin{aligned} P(x, t) := {\left\{ \begin{array}{ll} g\circ H(x, t) &{} 0 \le t \le N\\ G\circ (f' \times T)(x, t) &{} N \le t \le N+M,\end{array}\right. } \end{aligned}$$

where \(T :[N, N+M] \rightarrow I_M\) is the (evidently continuous) translation defined by \(T(t) := t - N\). We check that P defines a continuous map on \(X \times I_{N+M}\). For this, \(f' \times T\) is continuous by Lemma B.1.3. As compositions of continuous maps, the two parts of P are continuous on \(X \times I_N\) and \(X \times [N, N+M]\). They agree on their overlap \(X \times \{N\}\), as is easily checked. Furthermore, they piece together to give a continuous map. This is because any two adjacent points \((x, t) \sim (x', t')\) of \(X \times I_{N+M}\) must have \(|t - t'| \le 1\), and thus either both are in \(X \times I_N\) or both are in \(X \times [N, N+M]\). From the continuity of the two parts, we have \(P(x, t) \sim _Z P(x', t')\), so P is continuous on \(X \times I_{N+M}\). Now P is a homotopy from \(g\circ f\) to \(g'\circ f'\), as is easily checked. Furthermore, we have

$$\begin{aligned} P(x_0, t) = {\left\{ \begin{array}{ll} g\circ H(x_0, t) = g(y_0) = z_0 &{} 0 \le t \le N\\ G\big (f'(x_0), T(t)\big ) = G\big (y_0, T(t)\big ) = z_0&{} N \le t \le N+M,\end{array}\right. } \end{aligned}$$

since H and G are both based homotopies. Hence P is a based homotopy from \(g\circ f\) to \(g'\circ f'\). \(\square \)

We defined based homotopy of based loops in Definition 3.2.

Lemma B.1.5

Suppose based loops \(\alpha , \beta :I_L \rightarrow Y\) in a based digital image \(Y \subseteq \mathbb {Z}^n\) are based homotopic as based loops, and based maps \(g, g':Y \rightarrow Z\) of based digital images with \(Z \subseteq \mathbb {Z}^p\) are based-homotopic. Then the based loops \(g\circ \alpha , g'\circ \beta :I_L \rightarrow Z\) are based homotopic as based loops in Z. Furthermore, considering \(\alpha \circ \rho _k, \beta \circ \rho _k:S(I_L, k) \rightarrow Y\) as based loops \(I_{kL+k-1} \rightarrow Y\) in Y, they are based homotopic as based loops in Y.

Proof

These are basically two special cases of Lemma B.1.4; we just need to be sure that both ends of the loop are preserved through the homotopy. For the first point, suppose \(H :I_L \times I_N \rightarrow Y\) is a based homotopy of based loops from \(\alpha \) to \(\beta \) and \(G :Y \times I_M \rightarrow Z\) is a based homotopy from g to \(g'\). Define \(P :I_L \times I_{N+M} \rightarrow Z\) as in the proof of Lemma B.1.4 by

$$\begin{aligned} P(s, t) := {\left\{ \begin{array}{ll} g\circ H(s, t) &{} 0 \le t \le N\\ G\circ (\beta \times T)(x, t) &{} N \le t \le N+M,\end{array}\right. } \end{aligned}$$

Then P is a (continuous) homotopy from \(g\circ \alpha \) to \(g'\circ \beta \) that satisfies \(P(0, t) = z_0\) for \(t \in I_{N+M}\), by Lemma B.1.4. In addition, here we have

$$\begin{aligned} P(L, t) = {\left\{ \begin{array}{ll} g\circ H(L, t) = g(y_0) = z_0 &{} 0 \le t \le N\\ G\big (\beta (L), T(t)\big ) = G\big (y_0, T(t)\big ) = z_0&{} N \le t \le N+M,\end{array}\right. }, \end{aligned}$$

since \(\beta \) is a based loop in Y and H is a based homotopy of based loops. Thus, P is a based homotopy of based loops.

For the second point, recall that \(\rho _k:S(I_L, k) \rightarrow I_L\) satisfies \(\rho _k(0) = 0\) and \(\rho _k(kL+k-1) = L\). A special case of the argument of Lemma B.1.4, in which H is redundant, may be used here. Namely, define \(P' :S(I_L, k) \times I_{N} \rightarrow Y\) by

$$\begin{aligned} P'(s, t) := G\circ (\rho _k \times \mathrm {id}_{I_N})(s, t), \end{aligned}$$

where \(G :I_L \times I_N \rightarrow Y\) is a based homotopy of based loops from \(\alpha \) to \(\beta \). Continuity follows directly from Lemma B.1.3 here. A direct check confirms that \(P'\) is a based homotopy of based loops from \(\alpha \circ \rho _k\) to \(\beta \circ \rho _k\). \(\square \)

Definition B.1.6

(Based Homotopy Equivalence) Let \(f :X \rightarrow Y\) be a based map of based digital images. If there is a based map \(g :Y \rightarrow X\) such that \(g\circ f \approx \text {id}_X\) and \(f \circ g \approx \text {id}_Y\) (with both homotopies based), then f is a based homotopy equivalence, and X and Y are said to be based homotopy equivalent, or to have the same based homotopy type.

Based homotopy equivalence is an equivalence relation on based digital images. This may be shown with an argument identical to that used to show the topological counterpart of this fact. We leave the proof as an exercise.

Subdivision

Subdivision behaves well with respect to products. For any digital images \(X \subseteq \mathbb {Z}^m\) and \(Y \subseteq \mathbb {Z}^n\) and any \(k \ge 2\) we may identify

$$\begin{aligned} S(X \times Y, k) \cong S(X, k) \times S(Y, k) \end{aligned}$$

and, furthermore, the standard projection \(\rho _k:S(X \times Y, k) \rightarrow X \times Y\) may be identified with the product of the standard projections on X and Y, thus:

$$\begin{aligned} \rho _k = \rho _k \times \rho _k:S(X, k) \times S(Y, k) \rightarrow X \times Y. \end{aligned}$$

The projection \(\rho _k :S(X, k) \rightarrow X\) may be factored in various ways. For example, if \(k = pq\), then we may write

$$\begin{aligned} \rho _k = \rho _p\circ \rho _q :S(X, k) \rightarrow S(X, p) \rightarrow X. \end{aligned}$$

A different sort of “partial projection” that may also be used to factor \(\rho _k\) is as follows.

Definition B.2.1

For any \(x \in \mathbb {Z}\) and any \(k \ge 2\), recall that the subdivision S(xk) may be described as \(S(x, k) = [kx, kx + k-1]\). Then, for \(k \ge 3\), define a function

$$\begin{aligned} \rho ^c_{k} :S(x, k) \rightarrow S(x, k-1) \end{aligned}$$

as

$$\begin{aligned} \rho ^c_{k}(kx + j) := {\left\{ \begin{array}{ll} (k-1)x + j &{} 0 \le j \le \lfloor k/2 \rfloor -1\\ (k-1)x + j-1 &{} \lfloor k/2 \rfloor \le j \le k-1.\end{array}\right. } \end{aligned}$$

Next, for any \(x = (x_1, \ldots , x_n) \in \mathbb {Z}^n\), with the identifications

$$\begin{aligned} S(x, k) = S(x_1, k) \times \cdots \times S(x_n, k) \end{aligned}$$

and

$$\begin{aligned} S(x, k-1) = S(x_1, k-1) \times \cdots \times S(x_n, k-1), \end{aligned}$$

in which each \(S(x_i, k)\) and \(S(x_i, k-1)\) are viewed as 1D, define \(\rho ^c_{k} :S(x, k) \rightarrow S(x, k-1)\) as the product of functions

$$\begin{aligned} \rho ^c_{k}\times \cdots \times \rho ^c_{k} :S(x_1, k) \times \cdots \times S(x_n, k) \rightarrow S(x_1, k-1) \times \cdots \times S(x_n, k-1). \end{aligned}$$

Finally, for any digital image \(X \subseteq \mathbb {Z}^n\), define

$$\begin{aligned} \rho ^c_{k} :S(X, k) \rightarrow S(X, k-1) \end{aligned}$$

by viewing each subdivision as a (disjoint) union

$$\begin{aligned} S(X, k) = \coprod _{x \in X}\ S(x, k) \qquad \text {and} \qquad S(X, k-1) = \coprod _{x \in X}\ S(x, k-1) \end{aligned}$$

and assembling a global \(\rho ^c_{k}\) on S(Xk) from the individual \(\rho ^c_{k} :S(x, k) \rightarrow S(x, k-1)\) as just defined. Lupton et al. (2019b), we show that, for each \(k\ge 3\), the function \(\rho ^c_{k} :S(X, k) \rightarrow S(X, k-1)\) is continuous.

These partial projections are often useful, because they allow us to factor the projection \(\rho _k:S(X, k) \rightarrow X\) as

$$\begin{aligned} \rho _k = \rho _{k-1}\circ \rho ^c_k:S(X, k) \rightarrow S(X, k-1) \rightarrow X, \end{aligned}$$

with \(\rho _{k_1}:S(X, k-1) \rightarrow X\) the standard projection and \(\rho ^c_k:S(X, k) \rightarrow S(X, k-1)\) the map from Definition B.2.1. Indeed, we make crucial use of these partial projections in several results in the main body.

Subdivision-based homotopy

Recall from Definition 2.8 our conventions on basepoints vis-à-vis subdivision. Also, recall our notational convention that when \(k=1\), \(S(X, k) = S(X, 1) = X\) and \(\rho _1:S(X, 1) \rightarrow X\) is just the identity map.

We defined subdivision-based homotopy of maps in Definition 2.9.

Proposition B.3.1

Suppose we have based digital images \(X \subseteq \mathbb {Z}^m\) and \(Y \subseteq \mathbb {Z}^n\). Consider the set

$$\begin{aligned} \mathcal {S} := \left\{ f:S(X, k) \rightarrow Y \mid f \text { is a based map and } k \ge 1\right\} \end{aligned}$$

of all based maps from any subdivision of X to Y. Subdivision-based homotopy is an equivalence relation on the set \(\mathcal {S}\).

Proof

The proofs of reflexivity and symmetry are more-or-less tautological from Definition 2.9. We omit their details. For transitivity, we argue as follows. Suppose we have based maps \(f :S(X, k) \rightarrow Y\), \(g :S(X, l) \rightarrow Y\), and \(h :S(X, m) \rightarrow Y\), with f and g subdivision-based homotopic, and g and h subdivision-based homotopic. We wish to show that f and h are subdivision-based homotopic.

For f and g , we have \(k', l'\) with \(kk' = ll'\) and a based homotopy \(H :S(X, kk')\times I_N = S(X, ll')\times I_N \rightarrow Y\) from \(f\circ \rho _{k'}\) to \(g\circ \rho _{l'}\). Likewise, for g and h , we have \(l'', m'\) with \(ll'' = mm'\) and a based homotopy \(G :S(X, ll'')\times I_M = S(X, mm')\times I_M \rightarrow Y\) from \(g\circ \rho _{l''}\) to \(h\circ \rho _{m'}\). From the special case of Lemma B.1.4 in which the first homotopy is redundant,

$$\begin{aligned} P := H\circ (\rho _{l''} \times \mathrm {id}_{I_N}):S(X, ll'l'')\times I_N = S(X, kk'l'')\times I_N \rightarrow Y \end{aligned}$$

is a based homotopy of based maps from \(f\circ \rho _{k'}\circ \rho _{l''}\) to \(g\circ \rho _{l'}\circ \rho _{l''} = g\circ \rho _{l'l''}\). Also,

$$\begin{aligned} P' := G\circ (\rho _{l''} \times \mathrm {id}_{I_M}):S(X, ll''l')\times I_M = S(X, mm'l')\times I_M \rightarrow Y \end{aligned}$$

is a based homotopy of based maps from \(g\circ \rho _{l''}\circ \rho _{l'} = g\circ \rho _{l''l'}\) to \(h\circ \rho _{m'}\circ \rho _{l'}\). Then we piece these homotopies together into \(P'':S(X, kk'l'')\times I_{N+M} = S(X, mm'l')\times I_{N+M}\rightarrow Y\) defined by

$$\begin{aligned} P''(x, t) := {\left\{ \begin{array}{ll} P(x, t) &{} 0 \le t \le N\\ P'(x, t-N) &{} N \le t \le N+M.\end{array}\right. } \end{aligned}$$

The two parts of this homotopy agree on their overlap (when \(t = N\)) and assemble into a continuous whole by the same argument as was used in the proof of Lemma B.1.4. It is straightforward to check that \(P''\) is a based homotopy of based maps from \(f\circ \rho _{k'l''}\) to \(h\circ \rho _{l'm'}\) The result follows. \(\square \)

We defined subdivision-based homotopy of based loops in Definition 3.4.

Proposition B.3.2

For a based digital image \(Y \subseteq \mathbb {Z}^n\), consider the set

$$\begin{aligned} \mathcal {S} := \left\{ \alpha :I_N \rightarrow Y \mid \alpha \text { is a based loop of length } N \ge 1\right\} \end{aligned}$$

of all based loops in Y of any length. Subdivision-based homotopy of based loops is an equivalence relation on the set \(\mathcal {S}\).

Proof

This is basically a special case of Proposition B.3.1; we just need to be sure that both ends of a loop are preserved through homotopies. The proofs of reflexivity and symmetry here are more-or-less tautological from Definition 3.4. We omit their details, and focus on transitivity. Suppose we have based loops \(\alpha :I_L \rightarrow Y\), \(\beta :I_M \rightarrow Y\), and \(\gamma :I_N \rightarrow Y\), with \(\alpha \) and \(\beta \) subdivision-based homotopic as based loops, and \(\beta \) and \(\gamma \) subdivision-based homotopic as based loops. We wish to show that \(\alpha \) and \(\gamma \) are subdivision-based homotopic as based loops.

For \(\alpha \) and \(\beta \) , we have lm with \(l(L+1) = m(M+1)\) and a based homotopy of based loops \(H :I_{lL + l-1}\times I_S = I_{mM + m-1}\times I_S \rightarrow Y\) from \(\alpha \circ \rho _l\) to \(\beta \circ \rho _m\). Likewise, for \(\beta \) and \(\gamma \), we have \(\mu , \nu \) with \(\mu (M+1) = \nu (N+1)\) and a based homotopy of based loops \(G :I_{\mu M + \mu -1}\times I_T = I_{\nu N + \nu -1}\times I_T \rightarrow Y\) from \(\beta \circ \rho _{\mu }\) to \(\gamma \circ \rho _{\nu }\).

From the (proof of the) second point of Lemma B.1.5,

$$\begin{aligned} P := H\circ (\rho _{\mu } \times \mathrm {id}_{I_S}):S(I_L, l\mu )\times I_S = S(M, m\mu )\times I_S \rightarrow Y \end{aligned}$$

is a based homotopy of based loops from \(\alpha \circ \rho _l\circ \rho _{\mu }\) to \(\beta \circ \rho _m\circ \rho _{\mu } = \beta \circ \rho _{m\mu }\). Also,

$$\begin{aligned} P' := G\circ (\rho _{m} \times \mathrm {id}_{I_T}):S(I_M, \mu m)\times I_T = S(I_N, \nu m)\times I_T \rightarrow Y \end{aligned}$$

is a based homotopy of based loops from \(\beta \circ \rho _{m\mu } = \beta \circ \rho _{\mu }\circ \rho _{m}\) to \(\gamma \circ \rho _{\nu }\circ \rho _{m}\). Then we piece these homotopies together into \(P'':S(I_L, l\mu )\times I_{S+T} = S(I_N, \nu m)\times I_{S+T}\rightarrow Y\) defined by

$$\begin{aligned} P''(s, t) := {\left\{ \begin{array}{ll} P(s, t) &{} 0 \le t \le S\\ P'(s, t-S) &{} S \le s \le S+T.\end{array}\right. } \end{aligned}$$

The two parts of this homotopy agree on their overlap (when \(t = S\)) and assemble into a continuous whole by the same argument as was used in the proof of Lemma B.1.4. It is straightforward to check that \(P''\) is a homotopy from \(\alpha \circ \rho _{l\mu }\) to \(\gamma \circ \rho _{m\nu }\), and also that \(P''(0, t) = P''(l\mu L + l \mu -1, t) = P''(m\nu N + m \nu -1, t) = y_0\), so that \(P''\) is a based homotopy of based loops. The result follows. \(\square \)

Next we define a relation between digital images that is less rigid than the relation of homotopy equivalence (per Definition B.1.6) and to which we refer from several places in the Introduction and the main body.

Definition B.3.3

(Subdivision-Based Homotopy Equivalence) For based digital images \(X \subseteq \mathbb {Z}^m\) and \(Y\subseteq \mathbb {Z}^n\) and some \(k, l \ge 1\), suppose that we have the following data:

  1. (a)

    Based maps \(f:S(X, k) \rightarrow Y\) and \(g:S(Y, l) \rightarrow X\);

  2. (b)

    Based maps (coverings) F and G that make the following diagrams commute

    We say that F is a cover of f and G is a cover of g.

  3. (c)

    \(f\circ G:S(Y, kl) \rightarrow Y\) subdivision-based homotopic to \(\mathrm {id}_Y:Y \rightarrow Y\) and \(g\circ F:S(X, kl) \rightarrow X\) subdivision-based homotopic to \(\mathrm {id}_X:X \rightarrow X\).

Then we say that X and Y are subdivision-based homotopy equivalent.

Remark B.3.4

With \(k = l = 1\) and \(F=f\), \(G=g\) and the subdivision-based homotopies of (c) ordinary based homotopies, the notion of subdivision-based homotopy equivalence reduces to that of based homotopy equivalence.

We complete this basic material on subdivison-based homotopy with a discussion of based-contractible and subdivision-based contractible digital images. A primary goal here is to explain assertions we made above Corollary 3.19 and in Definition 3.28, as well as to provide a little background for Examples 3.20 and 3.29.

First, revert to the notion of a based-contractible digital image as defined above Corollary 3.19. It is easy to see the equivalence of the two versions of the definition asserted there. For suppose we have a based homotopy

$$\begin{aligned} \mathrm {id}_X \approx C_{x_0}:X \rightarrow X \end{aligned}$$

from the identity map of \(X \subseteq \mathbb {Z}^n\) to the constant map of X at \(x_0\), the basepoint of X. Then this serves as a based homotopy from \(\mathrm {id}_X\) to the composition \(X \rightarrow \{ x_0\} \rightarrow X\). On the other hand, the composition \(\{ x_0\} \rightarrow X \rightarrow \{ x_0\}\) equals the identity map of \(\{ x_0\}\). That is, we have X and \(\{ x_0\}\) based-homotopy equivalent. Conversely, a based-homotopy equivalence between X and \(\{ y_0\}\) entails a based homotopy from the composition \(X \rightarrow \{y_0\} \rightarrow X\), namely the constant map \(C_{x_0}:X \rightarrow X\), and the identity \(\mathrm {id}_X:X \rightarrow X\).

Now we would like to repeat this discussion and, as we asserted in Definition 3.28, identify X being subdivision-based homotopy equivalent to a point (as defined by Definition B.3.3 above) with the existence, for some K, of a based homotopy

$$\begin{aligned} \rho _K \approx C_{x_0}:S(X, K) \rightarrow X \end{aligned}$$

from some standard projection to the constant map at \(x_0\) (this is the remark we made in the proof of Corollary 3.30). But adding subdivisions into the mix complicates things, due to the elaborate combinations of subdivisions involved. At one point in our discussion we will want to contract a cubical lattice of the form \(S(y_0, k)\), with \(y_0\) a single point in some \(\mathbb {Z}^m\), to its basepoint via a based contracting homotopy. Whilst our intuition suggests this is straightforward, there are some subtleties and so we proceed carefully.

In Example 3.20 we indicated how to contract an interval to its initial point. Suppose, instead, we wish to contract to the terminal point. In the topological (continuous) setting, a contracting homotopy from \(\mathrm {id}_I :I \rightarrow I\) to the constant map \(C_{ \{1\} }:I \rightarrow \{ 1 \} \subseteq I\), where I denotes the unit interval, is given by \(H :I \times I \rightarrow I\) with

$$\begin{aligned} H(s, t) := {\left\{ \begin{array}{ll} s+t &{} 0 \le s \le 1-t\\ 1 &{} 1 - t \le s \le 1. \end{array}\right. } \end{aligned}$$

But consider mimicking this in the digital setting, e.g. as a two-step contracting homotopy from \(\mathrm {id}_{I_2} :I_2 \rightarrow I_2\) to the constant map \(C_{ \{2\} }:I_2 \rightarrow \{ 2 \} \subseteq I_2\). We might be led to try \(H :I_2 \times I_2 \rightarrow I\) with

$$\begin{aligned} H(s, t) := {\left\{ \begin{array}{ll} s+t &{} 0 \le s \le 2-t\\ 2 &{} 2 - t \le s \le 2. \end{array}\right. } \end{aligned}$$

This H is not a continuous map, however, since we would have \(H(0, 0) = 0\), \(H(1, 1) = 2\), with \((0, 0) \sim (1, 1)\) in \(I_2 \times I_2\) but 0 and 2 not adjacent in \(I_2\). The issue here ultimately derives from our choice of adjacencies in the product of two digital images and indicates that we must take some care over the steps in a homotopy if it is to satisfy the requirements of continuity.

So consider contracting an interval \(I_N\) to any point \(m \in \{ 0, 1, \ldots , N\}\) via a homotopy that fixes m. Namely, a homotopy \(H :I_N \times I_M \rightarrow I_N\) for some M with \(H(s, 0) = s\), \(H(s, M) = m\) and \(H(m, t) = m\) for \(s \in I_N\) and \(t \in I_M\). For this, take \(M = N\) and separate \(I_N \times I_N\) into \(I_m \times I_N \cup [m, N] \times I_N\). On the “left-side” \(I_m \times I_N\) (or \(s \le m\)) define

$$\begin{aligned} H(s, t) := {\left\{ \begin{array}{ll} t &{} 0 \le s \le t\\ s &{} t \le s \le m. \end{array}\right. } \end{aligned}$$
(22)

On the “right-side” \([m, N] \times I_N\) (or \(m \le s \le N\)) define

$$\begin{aligned} H(s, t) := {\left\{ \begin{array}{ll} s &{} m \le s \le N - t\\ N-t &{} N- t \le s \le N. \end{array}\right. } \end{aligned}$$
(23)

Lemma B.3.5

(Based contraction of an interval) The map \(H :I_N \times I_N \rightarrow I_N\) defined by the formulas (22) and (23) above is a homotopy from \(\mathrm {id} :I_N \rightarrow I_N\) to \(C_m :I_N \rightarrow I_N\), the constant map at \(m \in I_N\), that fixes m, i.e., that satisfies \(H(m, t) = m\) for each \(t \in I_N\).

Proof

These formulas agree when \(s = m\) and give \(H(m, t) = m\), thus are based at \(m \in I_N\). Since they agree on \(\{s\} \times I_N\), it is sufficient for continuity on all of \(I_N \times I_N\) that H be continuous on each “side.” On the left side, we confirm that \(H(s, t) \sim H(s', t')\) when \((s, t) \sim (s', t')\). If \(s \le t\) and \(s' \le t'\), then we have \(H(s, t) = t \sim t' = H(s', t')\). If \(t \le s\) and \(t' \le s'\) (with \(s, s' \le m\)) we have \(H(s, t) = s \sim s' = H(s', t')\). Now suppose we have \(s \le t\) with \(H(s, t) = t\) and \(t' \le s'\) with \(H(s', t') = s'\). Because s and \(s'\) are adjacent, we have \(s' - 1 \le s\) and thus \(s' - 1 \le t\) or \(-1 \le t - s'\). Because t and \(t'\) are adjacent, we have \(t \le t' + 1\) and thus \(t \le s' + 1\) or \(t - s' \le 1\). Combining these last two, we have \(| t - s'| \le 1\) and hence \(H(s, t) = t \sim s' = H(s', t')\). Similar reasoning shows that we have \(H(s, t) = s \sim t' = H(s', t')\) also in the case in which \(t \le s\) and \(s' \le t'\). So the given formula for H is continuous on the left-side \(I_m \times I_N\). A similar argument (we omit the details) shows that H is also continuous when restricted to the right side \([m, N] \times I_N\). It follows that H defines a (continuous) homotopy that satisfies \(H(s, 0) = s\), \(H(s, N) = m\) and \(H(m, t) = m\). Note that, restricted to the left-side, the homotopy stabilizes at \(H(s, t) = m\) for \(t \ge m\) and on the right side stabilizes at \(H(s, t) = m\) for \(t \ge N-m\), but we allow for \(t \le N\) because an N-step homotopy would be needed to contract to either endpoint with these formulas. \(\square \)

Now consider contracting a cube to any of its points. First, suppose we have the cube \((I_N)^m \subseteq \mathbb {Z}^m\) with basepoint \(\mathbf {b}_\mathbf {0} = (b_1, b_2, \ldots , b_m)\). Define a map

$$\begin{aligned} \mathcal {H} :(I_N)^m \times I_N \rightarrow (I_N)^m \end{aligned}$$

by \(\mathcal {H}(x_1, \ldots , x_m, t) = \left( H_1(x_1, t), H_2(x_2, t), \ldots , H_m(x_m, t) \right) \) where, in the i’th coordinate, \(H_i\) denotes the homotopy from Lemma B.3.5 that contracts the interval \(I_N\) to the i’th coordinate \(b_i\) of the basepoint \(\mathbf {b}_\mathbf {0}\).

Lemma B.3.6

(Based contraction of a cube) The map \(\mathcal {H} :(I_N)^m \times I_N \rightarrow (I_N)^m\) defined by the formula above is a homotopy from \(\mathrm {id} :(I_N)^m \rightarrow (I_N)^m\) to \(C_{\mathbf {b}_\mathbf {0}} :(I_N)^m \rightarrow (I_N)^m\), the constant map at \(\mathbf {b}_\mathbf {0} \in (I_N)^m\), that fixes \(\mathbf {b}_\mathbf {0}\), i.e., that satisfies \(H(\mathbf {b}_\mathbf {0}, t) = \mathbf {b}_\mathbf {0}\) for each \(t \in I_N\).

Proof

The only issue is one of continuity of \(\mathcal {H}\). To confirm this, let \(\Delta :I_N \rightarrow (I_N)^m\) denote the m-fold diagonal map defined by \(\Delta (t) = (t, t, \ldots , t)\) for \(t \in I_N\). Also, let \(P :(I_N)^m \times (I_N)^m \rightarrow (I_N \times I_N)^m\) denote the map that permutes (or shuffles) coordinates defined by

$$\begin{aligned} P\left( (x_1, x_2, \ldots , x_m), (t_1, t_2, \ldots , t_m)\right) = \left( (x_1, t_1), (x_2, t_2) \ldots , (x_m, t_m)\right) . \end{aligned}$$

Both \(\Delta \) and P are easily seen to be continuous. Then we may write \(\mathcal {H}\) as the composition

$$\begin{aligned} \mathcal {H} = (H_1 \times \cdots \times H_m) \circ P \circ (\mathrm {id}_{(I_N)^m}\times \Delta ) \end{aligned}$$

mapping \((I_N)^m \times I_N \rightarrow (I_N)^m \times (I_N)^m \rightarrow (I_N \times I_N)^m \rightarrow (I_N)^m\). As observed in Lemma B.1.3, the products of continuous maps \(H_1 \times \cdots \times H_m \) and \(\mathrm {id}_{(I_N)^m}\times \Delta \) are again continuous. It follows that \(\mathcal {H}\) is continuous. \(\square \)

Now we come to the main point of this discussion.

Lemma B.3.7

Let X be a based digital image \(X\subseteq \mathbb {Z}^m\) with basepoint (any) \(\mathbf {x}_\mathbf {0} \in X\). The following are equivalent:

  1. (a)

    X is subdivision-based homotopy equivalent to a point, namely to some singleton point \(\mathbf {y}_\mathbf {0} \in \mathbb {Z}^n\);

  2. (b)

    for some \(K \ge 1\) we have a based homotopy

    $$\begin{aligned} \rho _K \approx C_{\mathbf {x}_\mathbf {0}}:S(X, K) \rightarrow X \end{aligned}$$

    from the standard projection to the constant map at \(\mathbf {x}_\mathbf {0}\).

Proof

\((a) \Rightarrow (b)\): With \(Y = \{ \mathbf {y}_\mathbf {0} \} \in \mathbb {Z}^n\), the data from Definition B.3.3 include maps

for some k and l with \(g \circ F\) subdivision-based homotopic to the identity \(\mathrm {id} :X \rightarrow X\). That is, for some m, we have based-homotopic maps

$$\begin{aligned} g\circ F \circ \rho _m \approx \rho _{K} :S(X, K) \rightarrow X \end{aligned}$$

where \(K = klm\). Now \(S( \{ \mathbf {y}_\mathbf {0} \} , l) \) is based-contractible, in the ordinary sense as defined above Corollary 3.19. This is because \(S( \{ \mathbf {y}_\mathbf {0} \} , l)\) is some cube in \(\mathbb {Z}^n\). Although Lemma B.3.6 applies directly to a standard cube of the form \((I_l)^n\), we may easily extend it to the present situation. Namely, suppose that \(T :\mathbb {Z}^n \rightarrow \mathbb {Z}^n\) is the translation that translates the standard cube \((I_l)^n\) to \(S( \{ \mathbf {y}_\mathbf {0} \} , l)\). Specifically, if we have \(\mathbf {y}_\mathbf {0} = (y_1, y_2, \ldots , y_n)\), then T will be the translation given by \(T(x_1, \ldots , x_n) = (x_1 + l y_1, \ldots , x_n+ l y_n)\). The translation T and its inverse \(T^{-1}\) are clearly continuous. If \(\overline{\mathbf {y}_\mathbf {0}} \in S( \{ \mathbf {y}_\mathbf {0} \} , l)\) is the basepoint (either following our convention in Definition 2.8 or chosen in some other way), then let \(\mathbf {b}_\mathbf {0} = T^{-1}\left( \overline{\mathbf {y}_\mathbf {0}} \right) \) be the basepoint in \((I_l)^n\). Then the composition

in which \(\mathcal {H}\) denotes the based homotopy from Lemma B.3.6 that ends at the constant map \(C_{\mathbf {b}_\mathbf {0}}\), gives a based homotopy from \(\mathrm {id} :S( \{ \mathbf {y}_\mathbf {0} \} , l) \rightarrow S( \{ \mathbf {y}_\mathbf {0} \} , l)\) to \(C_{\overline{\mathbf {y}_\mathbf {0}}}\). Then we have based homotopies

$$\begin{aligned} \rho _{K}\approx & {} g\circ F \circ \rho _m \approx g\circ \mathrm {id}_{ S( \{ \mathbf {y}_\mathbf {0} \} , l) } \circ F \circ \rho _m \\\approx & {} g\circ C_{\overline{\mathbf {y}_\mathbf {0}}} \circ F \circ \rho _m = C_{\mathbf {x}_\mathbf {0} } :S(X, K) \rightarrow X. \end{aligned}$$

\((b) \Rightarrow (a)\): From the given K and based homotopy, we assemble the various data of Definition B.3.3. Take \(Y = \{ \mathbf {x}_\mathbf {0} \}\), \(k = 1\) and \(l = K\) (the given K). Then \(f :X \rightarrow Y = \{ \mathbf {x}_\mathbf {0} \}\) is the constant map, and \(g :S(Y, K) = S( \{ \mathbf {x}_\mathbf {0} \}, K) \rightarrow X\) is the constant map at \(\mathbf {x}_\mathbf {0} \in X\). For F, a cover of f, we may take \(F :S(X, K) \rightarrow S(Y, K) = S( \{ \mathbf {x}_\mathbf {0} \}, K)\) to be the constant map at \(\overline{\mathbf {x}_\mathbf {0}} \in S( \{ \mathbf {x}_\mathbf {0} \}, K)\) (however this basepoint is chosen). Then take \(G = g\) since we have \(\rho _k = \mathrm {id}\). The commutative diagrams of item (b) of Definition B.3.3 now look like the following:

with the horizontal maps constant maps. Then the composition \(g \circ F :S(X, K) \rightarrow X\) is the constant map at \(\mathbf {x}_\mathbf {0}\), which we are given is based-homotopic to the standard projection \(\rho _K\). That is, the composition is subdivision-based homotopic to the identity map of X. On the other hand, the composition \(f \circ G = f \circ g:S( \{ \mathbf {x}_\mathbf {0} \}, K) \rightarrow \{ \mathbf {x}_\mathbf {0} \}\) equals the constant map \(\rho _K\) so is tautologically subdivision-based homotopic to the identity map of \(\{ \mathbf {x}_\mathbf {0} \}\). The terms of Definition B.3.3 have all been met, so (b) follows. \(\square \)

Results about subdivision of maps from Lupton et al. (2019b)

For the convenience of the reader, we state two results from Lupton et al. (2019b) that we use in the proof of Theorem 3.23 and its annexe “Appendix A”. We refer to Lupton et al. (2019b) for all details of the proofs, and simply comment on some of the ingredients used.

Theorem B.4.1

[Th.4.1 and Cor.6.2.(C) of Lupton et al. (2019b)] Suppose we are given \(\alpha :I_N \rightarrow Y\), a path of length N in any digital image \(Y \subseteq \mathbb {Z}^n\). For any odd \(2k+1 \ge 3\), there is a canonical map of subdivisions

$$\begin{aligned} {\widehat{\alpha }}:S(I_N, 2k+1) = I_{N(2k+1)+2k} \rightarrow S(Y, 2k+1), \end{aligned}$$

or standard cover of \(\alpha \), that covers the given path, in the sense that the following diagram commutes:

For Y a based digital image and \(\alpha \) a based loop, with \(\alpha (0) = \alpha (N) = y_0\), then the standard cover \({\widehat{\alpha }}\) is a based loop: we have \({\widehat{\alpha }}(0) = {\widehat{\alpha }}\big ( (2k+1)N + 2k \big ) = \overline{y_0}\).

Proof (Gloss on the proof of Theorem 4.1 of Lupton et al. (2019b)) Refer to Definition 2.8 for our conventions and notation on basepoints and centres. For instance, for \(i \in I_N \subseteq \mathbb {Z}\) we write \(\overline{i} = (2k+1)i+k \in S(i, 2k+1)\). Then the standard cover may be described as

$$\begin{aligned} {\widehat{\alpha }}(j) :={\left\{ \begin{array}{ll} \ \ \overline{\alpha (0)} &{} 0 \le j < k \\ \\ {\widehat{\alpha }}(\overline{i} + t) = \overline{\alpha (i)} + t\big [ \alpha (i+1) - \alpha (i) \big ] &{} 0 \le i \le N, 0 \le t \le 2k\\ \\ \overline{\alpha (N)} &{} \overline{N} \le j \le \overline{N} + 2k \\ \end{array}\right. } \end{aligned}$$
(24)

In the middle line, we use coordinate-wise (vector) addition and scalar multiplication. If \(\alpha (i) = (y_1, \ldots , y_n)\) and \(\alpha (i+1) = (y'_1, \ldots , y'_n)\), then the coordinates of the middle line of (24) may be specified as

$$\begin{aligned} {\widehat{\alpha }}(\overline{i} + t) = \big ( (2k+1)y_1 +k + t[ y'_1 - y_1 ], \ldots , (2k+1)y_n + k + t[ y'_n - y_n ]\big )\qquad \end{aligned}$$
(25)

for \(0 \le t \le 2k\). A basic feature of the way in which we construct \({\widehat{\alpha }}\) is that it satisfies \({\widehat{\alpha }}\big (\overline{i}\big ) = \overline{ \alpha (i)}\). In particular, if \(\alpha \) is a based loop, then we have \({\widehat{\alpha }}\big (\overline{0}\big ) = {\widehat{\alpha }}\big (\overline{N}\big ) = \overline{ y_0}\). But also, in the construction, we define \({\widehat{\alpha }}(j) = \overline{ \alpha (0)}\) each j with \(0 \le j \le \overline{0} = k\), and \({\widehat{\alpha }}(j) = \overline{ \alpha (N)}\) for each j with \((2k+1)N + k = \overline{N} \le j \le (2k+1)N +2k\). With 0 the basepoint of \(S(I_N, 2k+1) = I_{(2k+1)N + 2k}\) and \(\overline{ y_0}\) the basepoint of \(S(Y, 2k+1)\), this justifies the last part of the theorem. \(\square \)

The following result from Lupton et al. (2019b) functions as a homotopy lifting result for based paths or loops.

Theorem B.4.2

[Th.5.7 and Cor.6.2.(C) of Lupton et al. (2019b)] Suppose we are given a map \(H :I_M \times I_N \rightarrow Y\) with \(Y \subseteq \mathbb {Z}^n\) any digital image. For any \(k \ge 1\), there is a canonical choice of map \({\widehat{H}}:S(I_M, 2k+1) \times S(I_N, 2k+1) \rightarrow S(Y, 2k+1)\) that makes the following diagram commute:

Furthermore, if H is a based homotopy of based loops from \(\alpha :I_M \rightarrow Y\) to \(\beta :I_M \rightarrow Y\), then \({\widehat{H}}:I_{(2k+1)M+ 2k} \times I_{(2k+1)N+ 2k} \rightarrow S(Y, 2k+1)\) is a based homotopy of based loops from \({\widehat{\alpha }}:I_{(2k+1)M+ 2k} \rightarrow S(Y, 2k+1)\) to \({\widehat{\beta }} :I_{(2k+1)M+ 2k} \rightarrow S(Y, 2k+1)\), the standard covers of \(\alpha \) and \(\beta \) as in Theorem B.4.1.

Proof (Gloss on the proof of Theorem 5.7 of Lupton et al. (2019b)) In the proof of (Lupton et al. 2019b, Th.5.7), we define the covering homotopy \({\widehat{H}}\) around the edges of \(S(I_M, 2k+1) \times S(I_N, 2k+1)\) using the standard covers of the paths in Y gotten by restricting H to the edges of \(I_M \times I_N\). In particular, this construction yields \({\widehat{H}}(s, 0) = {\widehat{\alpha }}(s)\) and \({\widehat{H}}(s, (2k+1)N+ 2k) = {\widehat{\beta }}(s)\) for \(s \in I_{(2k+1)M+ 2k}\). Also, along the vertical edges, since H is a based homotopy of based loops, we have \(H(0, t) = y_0 = H((2k+1)M+ 2k, t)\) for \(t \in I_{(2k+1)N+ 2k}\). Now for the constant map \(C_N :I_N \rightarrow Y\), the standard cover is again a constant map (at \(\overline{y_0} \in S(Y, 2k+1)\)): we have \(\widehat{C_N} = C_{(2k+1)N+ 2k}:I_{(2k+1)N+ 2k} \rightarrow S(Y, 2k+1)\). Hence, again from the construction of the covering homotopy \({\widehat{H}}\), we have

$$\begin{aligned} {\widehat{H}}(0, t) = \overline{y_0} = {\widehat{H}}((2k+1)M+ 2k, t) \end{aligned}$$

for \(t \in I_{(2k+1)N+ 2k}\). Hence \({\widehat{H}}\) is indeed a based homotopy of based loops. \(\square \)

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Lupton, G., Oprea, J. & Scoville, N.A. A fundamental group for digital images. J Appl. and Comput. Topology 5, 249–311 (2021). https://doi.org/10.1007/s41468-021-00067-1

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Keywords

  • Digital image
  • Digital topology
  • Subdivision
  • Digital fundamental group
  • Digital simple closed curve
  • Winding number

Mathematics Subject Classification

  • Primary 55Q99
  • Secondary 68U10
  • 68R99