Extendability of Simplicial Maps is Undecidable

Abstract

We present a short proof of the adek–Král–Matouek–Vokřínek–Wagner result from the title (in the following form due to Filakovský–Wagner–Zhechev). For any fixed even l there is no algorithm recognizing the extendability of the identity map of $S^l$to a PL map $X\rightarrow S^l$ of given 2l-dimensional simplicial complex X containing a subdivision of $S^l$ as a given subcomplex. We also exhibit a gap in the Filakovský–Wagner–Zhechev proof that embeddability of complexes is undecidable in codimension $>1$.

Appendix: Is Embeddability of Complexes Undecidable in Codimension \(>1\)?

Realizability of hypergraphs or complexes in the d-dimensional Euclidean space ${\mathbb {R}}^d$ is defined similarly to the realizability of graphs in the plane. E.g. for 2-complex one ‘draws’ a triangle for every three-element subset. There are different formalizations of the idea of realizability.

A complex (V, F) is simplicially (or linearly) embeddable in ${\mathbb {R}}^d$ if there is a set $V'$ of distinct points in ${\mathbb {R}}^d$ corresponding to V such that for any subsets $\sigma ,\tau \subset V'$ corresponding to elements of F the convex hull $\langle \sigma \rangle$ is a simplex of dimension $|\sigma |-1$, and $\langle \sigma \rangle \cap \langle \tau \rangle =\langle \sigma \cap \tau \rangle$. A complex is PL (piecewise linearly) embeddable in ${\mathbb {R}}^d$ if some its subdivision is simplicially embeddable in ${\mathbb {R}}^d$. For classical and modern results on embeddability and their discussion see e.g., surveys [10, Sect. 5], [11, 13, Sect. 3].

Theorem 3.1

(embeddability is undecidable in codimension 1)  For every fixed d, k such that $5\le d\in \{k,k+1\}$ there is no algorithm recognizing PL embeddability of k-complexes in ${\mathbb {R}}^d$.

This is deduced in [6, Thm. 1.1] from the Novikov theorem on unrecognizability of the d-sphere. Cf. [7, Rem. 3].

Conjecture 3.2

(embeddability is undecidable in codimension $>1$)  For every fixed d, k such that $8\le d\le ({3k+1})/2$ there is no algorithm recognizing PL embeddability of k-complexes in ${\mathbb {R}}^d$.

Conjecture 3.2 is stated as a theorem in [2]. The proof in [2] contains a gap described below. Their idea is to elaborate the following remark to produce the reduction (described below) to the ‘retractability is undecidable’ Theorem 1.1.

Remark 3.3

Homotopy classifications of maps $S^{2l-1}\rightarrow S^l$ and $S^{2l-1}\rightarrow S^l\vee S^l$ are related to isotopy classification of links of $S^{2l-1}\sqcup S^{2l-1}$ and of $S^{2l-1}\sqcup S^{2l-1}\sqcup S^{2l-1}$ in ${\mathbb {R}}^{3l}$ [3] (including higher-dimensional Whitehead link and Borromean rings; see [11, Sect. 3]). E.g. the generalized linking coefficients of the Whitehead link and of the Borromean rings are (the homotopy classes) of the Whitehead maps $W(1):S^{2l-1}\!\rightarrow \! S^l$ and $W_2(1):S^{2l-1}\rightarrow S^l\vee S^l$ from Theorem 1.6. Analogous results for $l=1$ do illustrate some ideas, see a description accessible to non-specialists in [12, Sect.  3.2].

We use the notation of Sect. 1. Let $a=((a^{i,j})_1,\ldots ,(a^{i,j})_m)$, $1\le i<j\le s$, and $b=(b_1,\ldots ,b_m)$ be any arrays of integers. Define the double mapping cylinder $X(a,b)=X_l(a,b)$ to be the union of ${{\,\textrm{Cyl}\,}}W_s(a)$ and ${{\,\textrm{Cyl}\,}}W_2(b)\supset Y_l$, in which $V_m^{2l-1}\subset {{\,\textrm{Cyl}\,}}W_s(a)$ is identified with $V_m^{2l-1}\subset {{\,\textrm{Cyl}\,}}W_2(b)$.

Embed $Y_{2l+1}$ standardly into $S^{3l+2}$. For l even take a small oriented $(l+1)$-disk $D\subset S^{3l+2}$ whose intersections with $Y_{2l+1}=S^{2l+1}$ is transversal and consist of exactly one point. Let ${\overline{Y}}_l=\partial D\cong S^l$ be the meridian of $Y_{2l+1}$. For l odd define analogously the meridian ${\overline{Y}}_l:=\partial D_+\cup \partial D_-\cong S^l\vee S^l$ starting with disks $D_+,D_-\subset S^{3l+2}$ intersecting at a point in $\partial D_+\cap \partial D_-$.

Conjecture 3.4

If either l is even or all $a^{i,j}_q$ are even and $l>1$ is odd, then there is a $(2l+1)$-complex $G\supset Y_l$ such that any of the following properties is equivalent to SYM for l even, and to (SKEW$'$) for l odd:

(Ex):

a PL homeomorphism $Y_l\rightarrow {\overline{Y}}_l$ extends to a PL map $X(a,b)\rightarrow S^{3l+2}-Y_{2l+1}$;

$\mathrm{(Ex}^{\prime }\mathrm{)}$:

a PL homeomorphism of $Y_l\rightarrow {\overline{Y}}_l$ extends to a PL embedding $X(a,b)\rightarrow S^{3l+2}-Y_{2l+1}$;

(Em):

$X(a,b)\cup _{Y_l}G$ embeds into $S^{3l+2}$.

All the implications except (Em) $\Rightarrow$ ()$Ex^{\prime }$ are correct results of [2]. The implication (Ex$^{\prime }$) $\Rightarrow$ (Ex) is clear. The equivalence of (Ex) and SYM/SKEW follows by Propositions 1.7 and 1.8, (a) and (b), because there is a strong deformation retraction $S^{3l+2}-Y_{2l+1}\rightarrow {\overline{Y}}_l$. The implication (Ex) $\Rightarrow$ (Ex$^{\prime }$) is implied by the following version of the Zeeman–Irwin Theorem [11, Thm. 2.9].

Lemma 3.5

For any PL map $f:X(a,b)\rightarrow S^{3l+2}-Y_{2l+1}$ there is a PL embedding $f':X(a,b)\rightarrow S^{3l+2}-Y_{2l+1}$ such that the restrictions of f and $f'$ to $Y_l\subset X(a,b)$ are homotopic.

Lemma 3.5 is essentially a restatement of [2, Thm. 10] accessible to non-specialists. See more detailed historical remark in [14, Rem. 3.9].

The idea of [2] to prove the implication (Em) $\Rightarrow$ (Ex$^{\prime }$) is to construct the complex G, and use a modification of the following lemma.

Lemma 3.6

[8, Lem. 1.4] For any integers $0\le l<k$ there is a k-complex $F_-$ containing subcomplexes $\Sigma ^k\cong S^k$ and $\Sigma ^l\cong S^l$, PL embeddable into ${\mathbb {R}}^{k+l+1}$ and such that for any PL embedding $f:F_-\rightarrow {\mathbb {R}}^{k+l+1}$ the images $f\Sigma ^k$ and $f\Sigma ^l$ are linked modulo 2.

Lemma 30 of [2] is a modification of Lemma 3.6 with ‘linked modulo 2’ replaced by ‘linked with linking coefficient $\pm 1$’. The end of proof of Lemma 30 in [2], p. 778, used the following incorrect statement: If $f:D^p\rightarrow {\mathbb {R}}^{p+q}$and $g:S^q\rightarrow {\mathbb {R}}^{p+q}$are PL embeddings such that $|f(D^p)\cap g(D^q)|=1$, then the linking coefficient of $f|_{S^{p-1}}$ and g is $\pm 1$.

Example 3.7

For any integers $p,q\ge 2$ and c there are PL embeddings $f:D^p\rightarrow {\mathbb {R}}^{p+q}$ and $g:S^q\rightarrow {\mathbb {R}}^{p+q}$ such that $|f(D^p)\cap g(S^q)|=1$ and the linking coefficient of $f|_{S^{p-1}}$ and g is c.

Proof

Take PL embeddings $f_0:S^{p-1}\rightarrow {\mathbb {R}}^{p+q-1}$ and $g_0:S^{q-1}\rightarrow {\mathbb {R}}^{p+q-1}$ whose linking coefficient is c. Take points $A,B\in {\mathbb {R}}^{p+q}-{\mathbb {R}}^{p+q-1}$ on both sides of ${\mathbb {R}}^{p+q-1}$. Then $f=f_0*A$ and $g=g_0*\{A,B\}$ are the required embeddings. $\square$

The modification [2, Lem. 30] of Lemma 3.6 is presumably incorrect, cf. [4, Thm. 1.6]. See more discussion and conjectures in [14, Sect. 3].