aa r X i v : . [ m a t h . A T ] A ug EXTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE
A. SKOPENKOV
Abstract.
We present a short proof of the ˇCadek-Krˇc´al-Matouˇsek-Vokˇr´ınek-Wagner resultfrom the title (in the following form due to Filakovsk´y-Wagner-Zhechev).
For any fixed integer l > there is no algorithm recognizing the extendability of the identitymap of S l ∨ S l to a PL map X → S l ∨ S l of given l -dimensional simplicial complex X containinga subdivision of S l ∨ S l as a given subcomplex. We also exhibit a gap in the Filakovsk´y-Wagner-Zhechev proof that embeddability of com-plexes is undecidable in codimension > Contents
1. Extendability of simplicial maps is undecidable 12. Appendix: an alternative proof for l even 63. Appendix: embeddability of complexes is undecidable 8References 111. Extendability of simplicial maps is undecidable
We present a short proof of a recent topological undecidability results for hypergraphs (com-plexes) [CKM+]. A k -hypergraph (more precisely, k -dimensional, or ( k + 1)-uniform, hypergraph) ( V, F ) isa finite set V together with a collection F ⊂ (cid:0) Vk +1 (cid:1) of ( k + 1)-element subsets of V . Elements of V and of F are called vertices and faces . For instance, a complete k -hypergraph on n vertices(or the k -skeleton of the ( n − k + 1)-element subsets of an n -element set. For n = k + 2 we denote this hypergraph by S k .In topology, topological combinatorics and algorithmic topology it is more traditional (be-cause sometimes more convenient) to work not with hypergraphs but with complexes (we shallnot use longer name ‘abstract finite simplicial complexes’). The following results are stated forcomplexes, although some of them are correct for hypergraphs.A complex ( V, F ) is a finite set V together with a collection F ⊂ V of subsets of V . suchthat if a subset σ is in the collection, then each subset of σ is in the collection, In an equivalentgeometric language, a complex is a collection of closed faces (=subsimplices) of some simplex. A k -complex is a complex containing at most ( k + 1)-element subsets, i.e. at most k -dimensionalsimplices. A simplicial map f : ( V, F ) → ( V ′ , F ′ ) between complexes is a map f : V → V ′ (not necessarily injective) such that f ( σ ) ∈ F ′ for each σ ∈ F .The subdivision of an edge operation is shown in fig. 1 left (excercise: represent the subdivisionof a face operation is shown in fig. 1 right as composition of several subdivisions of an edge and I would like to thank M. ˇCadek, R. Karasev, E. Kogan, B. Poonen, L. Vokˇr´ınek and U. Wagner for helpfuldiscussions.Moscow Institute of Physics and Technology, and Independent University of Moscow. Email: [email protected] . https://users.mccme.ru/skopenko/ . Supported by the Russian Foundation for Ba-sic Research Grant No. 19-01-00169. This paper is based on the courses [HT]. In these courses topological concepts are exposed in the wayinteresting and accessible to non-specialists, in particular, to computer science students. Besides this note, thecourses are based on some sections of [Sk20, Sk].
Figure 1.
Subdivision of an edgeinverse operations). A subdivision of a complex K is any complex which can be obtained from K by several subdivisions of an edge. The body (or geometric realization) | K | of a complex K is the union of simplices of K . are defined. A piecewise-linear (PL) map K → K ′ betweencomplexes a simplicial map between certain their subdivisions. A simplicial or PL map betweencomplexes induces a map between their bodies, which is called simplicial or PL , respectively. Below we often abbreviate | K | to K ; no confusion should arise.The wedge K ∨ . . . ∨ K m of complexes K = ( V , F ) , . . . , K m = ( V m , F m ) with disjointvertices is the complex whose vertex set is obtained by choosing one vertex from each V j and identifying chosen vertices, and whose edge set is obtained from F ⊔ . . . ⊔ F m by suchidentification. The choice of vertices is important in general, but is immaterial in the examplesbelow. Let K ∨ K be the wedge of two copies of K . Theorem 1.1 (retractability is undecidable) . For any fixed integer l > there is no algorithmrecognizing the extendability of the identity map of S l ∨ S l to a PL map X → S l ∨ S l of given l -complex X containing a subdivision of S l ∨ S l as a given subcomplex. This was easily deduced in [FWZ] from the following theorem.Let A m = S l − ∨ . . . ∨ S l − m be the wedge of m copies of S l − . Theorem 1.2 (extendability is undecidable) . For some fixed integer m and any fixed l > there is no algorithm recognizing extendability of given simplicial map A m → S l ∨ S l to a PLmap X → S l ∨ S l of given l -complex X containing a subdivision of A m as a given subcomplex. This is a ‘concrete’ version of [CKM+, Theorem 1.1.a].Remarks and examples below are formally not used later.
Remark 1.3. (a)
Relation to earlier known results.
For l > S → S l ∨ S l extends to D . The analogues of the above two theorems for S l ∨ S l replaced by a complex Y without this property (called simply-connectedness ) were well-known by mid 20th century. Seemore in [CKM+, § Why this text might be interesting.
Below I present a shorter exposition of the proofsof the two undecidability theorems. The exposition is shorter because I structure the proofby explicitly stating the Brower-Hopf-Whitehead Theorem 1.6 below (see also Remark 1.5),and Propositions 1.7, 1.9 on the equivalence of extendability / retractability to solvability ofcertain system of Diophantine equations. These results are essentially known before [CKM+]and are essentially deduced in [CKM+] from other known results. As far as I know, theywere never stated (not even in [CKM+]) in the explicit form below, which is convenient tothe ‘undecidability’ applications. Also I present main definitions in a more economic way The related but different notion of a continuous map between bodies of complexes is not required to stateand prove the results of this text. In theorems below the existence of a continuous extension is equivalent tothe existence of a PL extension (by the PL Approximation Theorem).
XTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE 3 accessible to non-specialists (in particular, to computer scientists). In particular, I make allthe construction for complexes and so co not use cell complexes and simplicial sets. All thisallows to omit theory not required for the statements and the proofs (e.g. compare the Brower-Hopf-Whitehead Theorem 1.6 to [CKM+, § l even separately, and to cover a minor gap in the proof of [CKM+, Proposition 5.2], seedetails in Remark 1.10. The recovery does not require new ideas but works by leaving out someunnecessary ideas. Figure 2.
Borromean triangles (Valknut) and quadrilaterals (icosahedron)(c) The Brower-Hopf-Whitehead Theorem 1.6 would relate homotopy classification of maps S l − → S l ∨ . . . ∨ S l , l >
1, and quadratic functions on integers. Thus it would allow a reductionof the topological undecidability results to Lemma 1.4. Observe that such a classification isrelated to Borromean rings S l − ⊔ S l − ⊔ S l − ⊂ R l . Analogous results for l = 1 do illustratesome ideas, see a description accessible to non-specialists in [Sk20, § Lemma 1.4 (proved in [CKM+, Lemma 2.1]) . For some (fixed) integers m, s there is noalgorithm which for given arrays a = ( a qi,j ) , ≤ i < j ≤ s , ≤ q ≤ m and b = ( b , . . . , b m ) ofintegers decides whether(SKEW) there are integers x , . . . , x s , y , . . . , y s such that X ≤ i A particular case of the Brower-Hopf-Whitehead Theorem 1.6. For anysimplicial map ϕ : P → Q between subdivisions of S one can effectively construct an integerdeg ϕ (the degree of ϕ ) such that the following holds.(i) For any integer k there exists an effectively constructible PL map b k : S → S of degree k . (ii) If deg ϕ = deg ψ for maps ϕ, ψ : S → S , then ϕ ≃ ψ .(iii) For any integer a there exists an effectively constructible PL map W ( a ) : S → S ∨ S such that • W ( a ) ≃ W ( a ′ ) only when a = a ′ ; • κ ◦ W ( a ) ≃ W ( a ( x y − x y )), where κ = ( b x ∨ b x ) ∨ ( b y ∨ b y ) : S ∨ S → S ∨ S .(See the leftmost triangle of the diagram before Proposition 1.9 involving A = S , Y = V = S ∨ S , α = W ( a ) and β = W ( a ( x y − x y )).)(b) Sketch of a proof of (a). Define deg ϕ by to be the sum of signs of a finite number ofpoints from ϕ − y , where y ∈ S is a ‘random’ (i.e. regular ) value of ϕ . (More precisely, take y outside the image of any edge of P .)For details and proof of (i,ii) see e.g. [Ma03] or [Sk20, § A. SKOPENKOV Decompose S = ∂ ( D × D ) = S × D ∪ S × S D × S . Define the Whitehead map w : S → S ∨ S as the ‘union’ of the compositions S × D → D c → S i → S ∨ ∗ and D × S → D c → S i → ∗ ∨ S . Here pr j — is the projection onto the j -the factor, c — contraction of the boundary to a point,and i , i — ‘identical’ maps. Define W ( a ) in the same way as w except that i = b a . It is easy to modify this ‘topological’definition to obtain an effectively constructible PL map W ( a ).Denote by lk the linking coefficient of two collections of closed oriented broken lines in 3-space.See e.g. [ST80], [Sk, § 4] and [Sk20u, § § ψ : S → S ∨ S define H ∨ ( ψ ) := lk( ψ − y , ψ − y ), where y ∈ S ∨ ∗ , y ∈ ∗ ∨ S are ‘random’ (i.e. regular ) values of ψ . (More precisely, take subdivisions of S and of S ∨ S for which ψ is simplicial. Then take y , y outside the image of any edge of the subdivision of S .)This is a well-defined homotopy invariant of ψ ( Whitehead invariant ).Clearly, H ∨ ( W ( a )) = a . Hence W ( a ) ≃ W ( a ′ ) only when a = a ′ .Using the definition of the degree and simple properties of linking coefficients, we see that H ∨ ( κ ◦ W ( a )) = a ( x y − x y ).A map ψ : S → S ∨ S is Borromean if its composition with each of the contractions S ∨ S → S ∨ ∗ and S ∨ S → ∗ ∨ S is homotopic to a constant map. Clearly, W ( a ) and κ ◦ W ( a ) are Borromean.The Hilton Theorem on homotopy classification of maps S → S ∨ S implies that if H ∨ ( ϕ ) = H ∨ ( ψ ) for Borromean maps ϕ, ψ : S → S ∨ S , then ϕ ≃ ψ (this corollary was presumablyproved earlier by Whitehead). This implies the relation of (iii). (cid:3) Theorem 1.6 (Brower; Hopf-Whitehead) . For any integer l and simplicial map ϕ : P → Q between subdivisions of S l one can effectively construct an integer deg ϕ (the degree of ϕ ) suchthat the following holds.(i) For any integer k there exists an effectively constructible PL map b k : S l → S l of degree k .(ii) If deg ϕ = deg ψ for maps ϕ, ψ : S l → S l , then ϕ ≃ ψ .(iii) Let V s := S l ∨ . . . ∨ S ls be the wedge of s copies of S l .For an array x = ( x , . . . , x s ) of integers let b x : V s → S l be the map whose restriction to the j -th sphere is b x j .For any array a = ( a ij ) , ≤ i < j ≤ s , of integers there exists an effectively constructiblePL map W s ( a ) : S l − → V s such that for any l > • W s ( a ) ≃ W s ( a ′ ) only when a = a ′ ; • ( b x ∨ b y ) ◦ W s ( a ) ≃ W ( R x,y ( a )) , where R x,y ( a ) := P ≤ i For any l > the property (SKEW) is equivalent to(LD) there is a PL map κ : V s → S l ∨ S l such that κ ◦ W s ( a ) ≃ W ( b ) . Thus ‘left divisibility is undecidable’, see (LD). See the leftmost triangle of the diagrambefore Proposition 1.9 involving A = A m , Y = S l ∨ S l , V = V s , α = W s ( a ) and β = W ( b ). Observe that S × S ∼ = D / ∼ , where x ∼ y ⇔ (cid:0) x, y ∈ S and w ( x ) = w ( y ) (cid:1) . XTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE 5 Proposition 1.9 easily follows from the Brower-Hopf-Whitehead Theorem 1.6, see the detailsbelow. Proof that ( SKEW ) ⇒ ( LD ) . Take an integer solution ( x, y ) = ( x , . . . , x s , y , . . . , y s ). Let κ := b x ∨ b y . Then κ ◦ W s ( a q ) ≃ W ( R x,y ( a q )) = W ( b q ) for each q . Thus κ ◦ W s ( a ) ≃ W ( b ). (cid:3) Proof that ( LD ) ⇒ ( SKEW ) . Take the PL map κ : V s → S l ∨ S l . Denote by p j : S l ∨ S l → S lj the contraction of the (3 − j )-th sphere. Let x j := deg( p ◦ κ | S lj ) and y j := deg( p ◦ κ | S lj ). Then κ ≃ b x ∨ b y . Take any q . Then W ( R x,y ( a q )) ≃ ( b x ∨ b y ) ◦ W s ( a q ) ≃ κ ◦ W s ( a q ) ≃ W ( b q ). Hence R x,y ( a q ) = b q . (cid:3) Let us present a construction which relates homotopy and extendability. For a map f : P → Q between subsets P ⊂ R p and Q ⊂ R q define the mapping cylinder Cyl f to be theunion of 0 × Q × ⊂ R p × R q × R = R p + q +1 and segments joining points ( u, , ∈ R p + q +1 to(0 , f ( u ) , ∈ R p + q +1 , for all u ∈ P . See [CKM+, Figure in p. 14].We identify P with P × × Q with 0 × Q × Example 1.8. (a) For the 2-winding b S → S (i.e. for the quotient map S → R P ) themapping cylinder Cyl b is the M¨obius band (i.e. the complement to a 2-disk in R P ).(b) For the Hopf map η : S → S (i.e. for the quotient map S → C P ) the mappingcylinder Cyl η is the complement to a 4-ball in C P (i.e. the ‘complexified’ M¨obius band).(c) Cyl w is the complement to a 4-ball in S × S . For a simplicial map g : P → Q between complexes denote by | g | : | P | → | Q | the corre-sponding PL map between their bodies. Then Cyl | g | is the body of certain complex • whose vertices are the vertices of P and the vertices of Q ; • whose simplices are the simplices of P , the simplices of Q and another simplices that arenot hard to define. A α (cid:15) (cid:15) ⊂ / / β " " ❉❉❉❉❉❉❉❉❉ Cyl α ⊂ / / (cid:15) (cid:15) ✤✤✤ Cyl( α, β ) y y r r r r r r V κ / / ❴❴❴❴ Y Y ⊂ O O . Proposition 1.9. For any complexes A, V, Y and PL maps α : A → V , β : A → Y thefollowing properties are equivalent:(1) there is a map κ : V → Y such that β ≃ κ ◦ α ;(2) the map β extends to a PL map Cyl α → Y ;(3) the identity map Y → Y extends to a PL map Cyl( α, β ) → Y , where the ‘double mappingcylinder’ Cyl( α, β ) is the union of Cyl α and Cyl β ⊃ Y with A ⊂ Cyl α identified with A ⊂ Cyl β .Sketch of a proof. (3) ⇒ (1) (or (2) ⇒ (1)). Let κ be the restriction to V ⊂ Cyl α of givenextension.(1) ⇒ (2). Define the map ret α : Cyl α → Q by mapping to α ( u ) the segment containing( u, , 0) from the definition of Cyl α . By the Borsuk Homotopy Extension Theorem extend-ability is equivalent to homotopy extendability. So we can take the required extension to be κ ◦ ret α .(2) ⇒ (3). Define the required extension to be ret β on Cyl β and to be the given extensionon Cyl α . (cid:3) Observe that (LD) is (1) for the Y, A, V, α and β given after Proposition 1.7. Recall that themap W s ( a ) is effectively constructed. Hence • the ‘retractability is undecidable’ Theorem 1.1 follows by Lemma 1.4 together with theequivalence of ( SKEW ) and (3) for the above Y, A, V, α and β . A. SKOPENKOV • the ‘extendability is undecidable’ Theorem 1.2 follows by Lemma 1.4 together with theequivalence of ( SKEW ) and (2) for the above Y, A, V, α and β : take X = Cyl W s ( a ) and asimplicial subdivision of W ( b ) : A m → S ∨ S . Remark 1.10. Proposition 5.2 of [CKM+] asserts the equivalence of ( SKEW ) and (2) ofProposition 1.9 for Y, A, V, α and β given after Proposition 1.7. The proof of Proposition 5.2was not formally presented in [CKM+], it is written that the proposition follows from the textbefore. That text requires multiplication by 2 in the group π k − ( S k ∨ S k ) (at one place denotedby π k − ( S d ∨ S d )). Since this group can have elements of order 2, such a multiplication of anequation does not produce an equivalent equation. Thus the ‘if’ part of Proposition 5.2 is notproved in [CKM+]. This gap is easy to recover; e.g. it is recovered here. Appendix: an alternative proof for l even Theorem 2.1 (retractability is undecidable) . For any fixed even l there is no algorithm recog-nizing the extendability of the identity map of S l to a PL map X → S l of given l -complex X containing a subdivision of S l as a given subcomplex. Theorem 2.2 (extendability is undecidable) . For some fixed integer m and any fixed even l there is no algorithm recognizing extendability of given simplicial map A m → S l to a PL map X → S l of given l -complex X containing a subdivision of A m as a given subcomplex. Lemma 2.3 (proved in [CKM+, Lemma 2.1]) . For some (fixed) integers m, s there is noalgorithm which for given arrays a = ( a qi,j ) , ≤ i < j ≤ s , ≤ q ≤ m and b = ( b , . . . , b m ) ofintegers decides whether(SYM) there are integers x , . . . , x s such that X ≤ i There is a misprint in the statement of [CKM+, Proposition 5.2] ; namely, the assumption that all coefficients a ( q ) ij in [CKM+, (Q-SKEW)] = ( SKEW ) should be even is missing. With this assumption in place, I am notaware of any gap. Remark 1.10 is incorrect in assuming that [CKM+] uses multiplication by 2 in the homotopygroup π k − ( S k ∨ S k ) ; instead, the system of equations [CKM+, (Q-SKEW)] = ( SKEW ) with values in Z getsmultiplied. In my opinion, this does not show that Remark 1.10 is not proper. Indeed, Remark 1.10 concerns only thetext before [CKM+, Proposition 5.2], not any other non-existent text, cf. [Sk20d, Remark 3.d]. Also, in orderto justify ‘We get the following:’ before [CKM+, Proposition 5.2] one needs to prove that multiplication by 2of ‘the system of s equations in π k − ( S k ∨ S k )’ produces an equivalent system; thus Remark 1.10 is correctin assuming that the text before [CKM+, Proposition 5.2] uses multiplication by 2 in the homotopy group π k − ( S k ∨ S k ) (although the part of the argument using this multiplication is omitted). XTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE 7 Sketch of a proof of the particular case. Use the text from Remark 1.5.b up to the definition of W ( a ).For a PL map ψ : S → S define H ( ψ ) := lk( ψ − y , ψ − y ), where y , y ∈ S are distinct‘random’ (i.e. regular ) values of ψ . (More precisely, take subdivisions of S and of S such that ψ is simplicial. Then take y , y outside the image of any edge of the subdivision of S .)This is a well-defined homotopy invariant of ψ ( Hopf invariant ).Clearly, HW ( a ) = 2 a . Hence W ( a ) ≃ W ( a ′ ) only when a = a ′ .Using the definition of the degree and simple properties of linking coefficients, we see that H (( b x ∨ b x ) ◦ W ( a )) = 2 ax x .The Freudenthal-Pontryagin Theorem on homotopy classification of maps S → S statesthat if H ( ϕ ) = H ( ψ ) for maps ϕ, ψ : S → S , then ϕ ≃ ψ . This implies the relation of(iii). (cid:3) Theorem 2.5 (Brower; Hopf-Whitehead) . For any integer l and simplicial map ϕ : P → Q between subdivisions of S l one can effectively construct an integer deg ϕ (the degree of ϕ ) suchthat the following holds.(i) For any integer k there exists an effectively constructible PL map b k : S l → S l of degree k .(ii) If deg ϕ = deg ψ for maps ϕ, ψ : S l → S l , then ϕ ≃ ψ .(iii) Let V s := S l ∨ . . . ∨ S ls be the wedge of s copies of S l .For an array x = ( x , . . . , x s ) of integers let b x : V s → S l be the map whose restriction to the j -th sphere is b x j .For any integer b and array a = ( a ij ) , ≤ i < j ≤ s , of integers there exists an effectivelyconstructible PL map W s ( a ) : S l − → V s such that for the map W ( b ) : S l − → S l defined by W ( b ) := ( b ∨ b ◦ W ( b ) and any even l we have • W ( b ) ≃ W ( b ′ ) only when b = b ′ ; • b x ◦ W s ( a ) ≃ W ( Q x ( a )) , where Q x ( a ) := P ≤ i For any even l the property (SYM) is equivalent to(LD’) there is a PL map κ : V s → S l such that κ ◦ W s ( a ) ≃ W ( b ) . Thus ‘left divisibility is undecidable’, see (LD’). See the leftmost triangle of the diagrambefore Proposition 1.9 involving Y = S , A = A m , V = V s , α = W s ( a ), β = W ( b ) and κ = b x .The proposition easily follows from the Brower-Hopf-Whitehead theorem, see the details below. Proof that ( SY M ) ⇒ ( LD ′ ) . Take an integer solution x = ( x , . . . , x s ). Let κ := b x . Then κ ◦ W s ( a q ) ≃ W ( Q x ( a q )) = W ( b q ) for each q . Thus κ ◦ W s ( a ) ≃ W ( b ). Proof that ( LD ′ ) ⇒ ( SY M ) . Take the PL map κ : V s → S l . Let x j := deg( κ | S lj ). Then κ ≃ b x . Take any q . Then W ( Q x ( a q )) ≃ b x ◦ W s ( a q ) ≃ κ ◦ W s ( a q ) ≃ W ( b q ). Hence Q x ( a q ) = b q .Observe that (LD’) is (1) for Y, A, V, α and β described after Proposition 2.6. Recall thatthe map W s ( a ) is effectively constructed. Hence • the ‘retractability is undecidable’ Theorem 2.1 follows for l even by Lemma 2.3 togetherwith the equivalence of ( SY M ) and (3) for the above Y, A, V, α and β . • the ‘extendability is undecidable’ Theorem 2.2 follows for l even by Lemma 2.3 togetherwith the equivalence of ( SY M ) and (2) for the above Y, A, V, α and β : take X = Cyl W s ( a )and a simplicial subdivision of W ( b ) : A m → S . A. SKOPENKOV Appendix: embeddability of complexes is undecidable Realizability of hypergraphs or complexes in the d -dimensional Euclidean space R d is definedsimilarly to the realizability of graphs in the plane. E.g. for k = 2 one ‘draws’ a triangle forevery three-element subset. There are different formalizations of the idea of realizability.A complex ( V, F ) is simplicialy (or linearly) embeddable in R d if there is a set V ′ of distinctpoints in R d corresponding to V such that for any subsets σ, τ ⊂ V ′ corresponding to elementsof F the convex hull conv σ is a non-degenerate simplex and conv σ ∩ conv τ = conv( σ ∩ τ ).A ‘small shift’ (or ‘general position’) argument shows that every graph is simplicialy em-beddable in R . A straightforward generalization shows that every k -complex is simpliciallyembeddable in R k +1 . Let us define piecewise-linear (PL) embeddability of complexes which is the analogue of theplanarity of graphs. Complexes are PL homeomorphic if one can be obtained from theother by several subdivisions of an edge and inverse operations (i.e. if they have a commonsubdivision). A complex is PL embeddable in R d if some homeomorphic complex is simplicialyembeddable in R d . For classical results on embeddability and their discussion see e.g. surveys[Sk06], [Sk18, § § Theorem 3.1 (embeddability is undecidable in codimension 1) . For every fixed d, k such that ≤ d ∈ { k, k + 1 } there is no algorithm recognizing PL embeddability of k -complexes in R d . This is deduced in [MTW11, Theorem 1.1] from the Novikov theorem on unrecognizabilityof the d -sphere. Cf. [NW97, Remark 3]. Conjecture 3.2 (embeddability is undecidable in codimension > . For every fixed d, k suchthat ≤ d ≤ k +12 there is no algorithm recognizing PL embeddability of k -complexes in R d . This is stated as a theorem in [FWZ]. The proof in [FWZ] contains a gap described in § g : K → R d of a complex K is called an almost embedding if gα ∩ gβ = ∅ for anytwo disjoint simplices α, β ⊂ K . Conjecture 3.3 (almost embeddability is undecidable) . For every fixed d, k such that(a) ≤ d ∈ { k, k + 1 } ; (b) ≤ d ≤ k +12 there is no algorithm recognizing PL embeddability of k -complexes in R d . Conjecture 3.2 easily follows from its ‘extreme’ case 2 d = 3 k + 1 = 6 l + 4 [FWZ, Corollaries4 and 6]. The extreme case is implied by the equivalence ( SKEW ) ⇔ ( Em ) of the followingConjecture 3.4. We use the notation of § 1. Let X ( a, b ) := Cyl( W s ( a ) , W ( b )). Assume that S l +1 ∨ S l +1 isstandardly embedded into S l +2 . Take a small oriented ( l + 1)-disks D + , D − ⊂ S l +2 • intersecting at a point in ∂D + ∪ ∂D − ; • whose intersections with S l +1 ∨ S l +1 are transversal and consist of exactly one point D + ∩ ( S l +1 ∨ S l +1 ) ∈ S l +1 ∨ ∗ and D − ∩ ( S l +1 ∨ S l +1 ) ∈ ∗ ∨ S l +1 .Define the meridian Σ l ∨ Σ l of S l +1 ∨ S l +1 in S l +2 to be ∂D + ∪ ∂D − . Conjecture 3.4. For any integer l there is a (2 l + 1) -complex G ⊃ S l ∨ S l such that any ofthe following properties is equivalent to (SKEW): The related but different notions of being topologically homeomorphic and topologically embeddable are notrequired to state and prove the results of this text. Embeddability (simplicial, PL or topological) of a complexin R d is alternatively defined as the existence of an injective (simplicial, PL or continuous) map of its body in R d . The extreme case is also implied by the equivalence between ( SKEW 1) of Conjecture 3.11.a and theanalogue of ( Em 2) from Conjecture 3.14 for ‘almost embedding’ replaced by ‘embedding’. The extreme casefor l even is also implied by the equivalence between ( SY M 1) of Conjecture 3.11.b and the analogue of ( Em XTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE 9 (Ex) a PL homeomorphism of S l ∨ S l → Σ l ∨ Σ l of S l +1 ∨ S l +1 in S l +2 extends to a PLmap X ( a, b ) → S l +2 − ( S l +1 ∨ S l +1 ) .(Ex’) a PL homeomorphism of S l ∨ S l → Σ l ∨ Σ l extends to a PL embedding X ( a, b ) → S l +2 − ( S l +1 ∨ S l +1 ) .(Em) X ( a, b ) ∪ S l ∨ S l G embeds into S l +2 . All the implications except ( Em ) ⇒ ( Ex ′ ) are correct results of [FWZ].The implication ( Ex ′ ) ⇒ ( Ex ) is clear.The equivalence of ( Ex ) and (3) (and thus to undecidable (SKEW)) for Y = S ∨ S , A = A m , V = V s , α = W s ( a ) and β = W ( b ) follows because there is a strong deformation retraction S l +2 − ( S l +1 ∨ S l +1 ) → Σ l ∨ Σ l .The implication ( Ex ) ⇒ ( Ex ′ ) is implied by the following version of the Zeeman-IrwinTheorem [Sk06, Theorem 2.9]. Lemma 3.5. For any map f : X ( a, b ) → S l +2 − ( S l +1 ∨ S l +1 ) there is a PL embedding f ′ : X ( a, b ) → S l +2 − ( S l +1 ∨ S l +1 ) such that the restrictions of f and f ′ to S l ∨ S l ⊂ X ( a, b ) are homotopic. Remark 3.6. (a) Lemma 3.5 is essentially a restatement of [FWZ, Theorem 10] accessibleto non-specialists. Analogous lemma for X ( a, b ) replaced by 2 l -dimensional ( l − § § 2, theparagraph before remark 1]).(b) Proposition 34 of [FWZ] is a detailed general position argument for the following state-ment: If Z is a subcomplex of a complex X and Z < d , then any map of X to a PL d -manifold is homotopic to a PL map the closure of whose self-intersection set misses Z . (Thisshould be known, at least in folklore, but I do not immediately see a reference. )(c) Lemma 41 of [FWZ] is a version of the following theorem: Any map of S n × I to an (2 n + 3 − m ) -connected m -manifold Q is homotopic to a PL embedding (this is a particular caseof [Hu69, Theorem 8.3]). The novelty of [FWZ, Lemma 41] is the property S ( g ) ⊂ S ( g ). Thisproperty is not checked in [FWZ, proof Lemma 41] but does follow from C ∩ g (Cl( A × [0 , − σ )) = g ( e I ); the latter holds because of the ‘metastable dimension restriction’ 2(3 l + 2) ≥ l + 1).(d) In the proof of [FWZ, Lemma 42] the property S ( g ) ⊂ S ( g ) is not checked. This propertyensures that we can make new improvements without destroying the older ones. Cf. [Sk98,line 5 after the display formula in p. 2468]. This property presumably holds because of the‘metastable dimension restriction’ 2(3 l + 2) ≥ l + 1).The idea of [FWZ] to prove the implication ( Em ) ⇒ ( Ex ′ ) is to construct the complex G ,and use a modification of the following Lemma 3.7. Lemma 3.7 (Segal-Spie˙z; [SS92, Lemma 1.4], [ST17, § . For any integers ≤ l < k there is a k -complex F − containing subcomplexes Σ k ∼ = S k and Σ l ∼ = S l , PLembeddable into R k + l +1 and such that for any PL almost embedding f : F − → R k + l +1 theimages f Σ k and f Σ l are linked modulo 2. Lemma 30 of [FWZ] is a modification of Lemma 3.7 with ‘linked modulo 2’ replaced by‘linked with linking coefficient ± If f : D p → R p + q and g : S q → R p + q are PL embeddingssuch that | f ( D p ) ∩ g ( D q ) | = 1 , then the linking coefficient of f | S p − and g is ± . Example 3.8. For any integers p, q ≥ and c there are PL embeddings f : D p → R p + q and g : S q → R p + q such that | f ( D p ) ∩ g ( S q ) | = 1 and the linking coefficient of f | S p − and g is c . Proof. Take PL embeddings f : S p − → R p + q − and g : S q − → R p + q − whose linking coeffi-cient is c . Take points A, B ∈ R p + q − R p + q − on both sides of R p + q − . Then f = f ∗ A and g = g ∗ { A, B } are the required embeddings. (cid:3) The modification [FWZ, Lemma 30] of Lemma 3.7 is presumably incorrect: Theorem 3.9 ([KS20]) . For any integers < l < k and z there is a PL almost embedding f : F − → R k + l +1 such that lk f = 2 z + 1 . Conjecture 3.10. The complex F/ ∼ is the image of a map from [KS20, Lemma 2.3.a] , or isdefined in [FWZ] (we have F/ ∼⊃ F − ).(a) For any l ≥ and integer z there is a PL embedding f : F/ ∼→ R l +2 such that lk f = 2 z + 1 .(b) Same as (a) with ‘embedding’ replaced by ‘almost embedding’. (This can perhaps be provedusing Theorem 3.9 and the idea of Example 3.8.) Conjecture 3.3 easily follows from its ‘extreme’ case 2 d = 3 k + 1 = 6 l + 4 analogouslyto [FWZ, Corollaries 4 and 6]. The extreme case for l even is implied by the equivalence( SY M ⇔ ( Em 1) of the following Conjectures 3.11.b and Proposition 3.13. The extreme casefor any l is implied by the equivalence ( SKEW ⇔ ( Em 2) of the following Conjectures 3.11.aand 3.14. Conjecture 3.11. (a) For some fixed integers m, s there is no algorithm which for given arrays a = ( a qi,j ) , ≤ i < j ≤ s , ≤ q ≤ m and b = ( b , . . . , b m ) of integers decides whether(SKEW1) there are integers x , . . . , x s , y , . . . , y s , z such that X ≤ i B. Moroz conjectured and E. Kogan sketched a proof that Conjecture 3.11.a isequivalent to: (*) for some fixed positive integers m, s there is no algorithm which for a given system of m Diophantine equations in s variables decides whether the system has a solution in rationalnumbers with odd denominators. Since m equations are equivalent to 1 equation (sum of squares) and since work of J. Robinsoncharacterizes the rational numbers with odd denominators among all rational numbers in aDiophantine way, (*) is in turn is equivalent to: (**) for some fixed positive integer s there is no algorithm which for a given polynomialequation with integer coefficients in s variables decides whether the system has a solution inrational numbers. The statement (**) is an open problem.An odd (almost) embedding is a PL (almost) embedding f : S l → S l +2 − S l +1 such that f ( S l ) is linked modulo 2 with S l +1 . Proposition 3.13. For any even l there is a (2 l + 1) -complex G ⊃ S l such that any of thefollowing properties is equivalent to (SYM1):(Ex1) some odd almost embedding extends to a PL map of X ( a, b ) .(Ex’1) some odd almost embedding extends to a PL embedding of X ( a, b ) .(Em1) X ( a, b ) ∪ S l G embeds into S l +2 . XTENDABILITY OF SIMPLICIAL MAPS IS UNDECIDABLE 11 All the implications except ( Em ⇒ ( Ex ′ 1) (and their analogues for ‘almost embedding’replaced by ‘embedding’) are proved analogously to the corresponding correct implications ofConjecture 3.4. The implication ( Em ⇒ ( Ex ′ 1) (and its analogue) follows by Theorem 3.9(by the conjecture in [KS20, Remark 1.7.b]) analogously to [FWZ].An odd (almost) embedding is a PL (almost) embedding f : S l ∨ S l → S l +2 − S l +11 ∨ S l +12 such that the mod 2 linking coefficient of f ( S li ) and S l +1 j equals to the Kronecker delta δ ij . Conjecture 3.14. For any l > there is a (2 l + 1) -complex G ⊃ S l ∨ S l such that any of thefollowing properties is equivalent to (SKEW1):(Ex2) some odd almost embedding extends to a PL map of X ( a, b ) .(Ex’2) some odd almost embedding extends to a PL embedding of X ( a, b ) .(Em2) X ( a, b ) ∪ S l ∨ S l G embeds into S l +2 . All the implications except ( Em ⇒ ( Ex ′ 2) (and their analogues for ‘almost embedding’replaced by ‘embedding’) are proved analogously to the corresponding correct implications ofConjecture 3.4. The implication ( Em ⇒ ( Ex ′ 2) (and its analogue) would follow by a ‘wedge’analogue of Theorem 3.9 (and of the conjecture in [KS20, Remark 1.7.b]) analogously to [FWZ]. References [CKM+] M. ˇCadek, M. Krˇc´al. J. Matouˇsek, L. Vokˇr´ınek, U. Wagner. Extendability of continuous maps isundecidable, Discr. and Comp. Geom. 51 (2014) 24–66. arXiv:1302.2370.[FWZ] M. Filakovsk´y, U. Wagner, S. Zhechev. Embeddability of simplicial complexes is un-decidable. 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