Embeddings of Polyhedra and Compacta in Euclidean Spacesigce.rc.unesp.br/Home/Departamentos47/matematica/topologiaalgeb... · PROBLEM: Find conditions for embeddability of k-dimensional

Embeddings of Polyhedra and Compactain Euclidean Spaces

S. Spiez (IM PAN, Warsaw)

Rio ClaroDecember 2014

S. Spiez (IM PAN, Warsaw) Embeddings of Polyhedra and Compacta in Euclidean Spaces

Menger-Nobeling Theorem

Theorem (Menger-Nobeling)

Any n-dimensional compact metric space X embeds in theEuclidean space R2n+1. Moreover, any mapping X → R2n+1 canbe approximating arbitrarily closely by embeddings.

For polyhedra this result is a consequence of the general position.


PROBLEM:

Find conditions for embeddability of k -dimensional spacesinto Euclidean spaces Rm, m ≤ 2k .

Theorem (K. Kuratowski)

A graph can be embedded in R2 if and only if it does not contain anembedded copy of one of the two graphs K5 or K3,3.


The van Kampen obstruction

In 1932, van Kampen generalized Kuratowski’s graphs ton-dimensional polyhedra which are not embeddable in R2n.

He also gave a rough description of a certain Z/2Z-equivariant2n-dimensional cohomology class ν(K) of the deleted product

K ∗ := {(x, y) ∈ K × K : x , y} = K × K \∆(K)

of an n-dimensional polyhedron K , which vanishes if and only if Kis PL-embeddable in R2n, provided n ≥ 3.

Many details in van Kampen’s paper were clarified independentlyby Shapiro and Wu.


The van Kampen obstruction ν(K) is constructed as follows.Let T be a triangulation of K . The space

T∗ :=⋃{σ × τ ∈ K × K |σ ∩ τ = ∅}

is called the simplicial deleted product (its equivariant homotopytype depends only on K ).

We choose an orientation of R2n and of n-simplices of T . For anygeneric PL-map f : K → R2n we define a cochain νf ∈ C2n(T∗;Z)by the formula

νf (σ × τ) := Σ{ε(x)|x ∈ f(σ) ∩ f(τ)}

for any disjoint oriented n-simplices σ and τ of T , whereε(x) = +1 if f(σ) and f(τ) determine at x the chosen orientation ofR2n and ε(x) = −1 otherwise.


Note that νf (σ × τ) = (−1)nνf (τ × σ). Thus νf is an element of thesubgroup

C2ns (T∗,Z) ⊂ C2n(T∗,Z)

of cochains assuming equal values on 2n-cells σ × τ and τ × σ foreven n and opposite values for odd n.

The cohomology class

ν(K) = [νf ] ∈ H2ns (K ∗,Z) ' C2n

s (T∗,Z)/B2ns (T∗,Z)

(it is independent of T and f ) is the van Kampen obstruction toembeddability of n-dimensional polyhedron K in R2n.


Theorem (E. R. van Kampen, A. Shapiro, W. T. Wu)

If an n-polyhedron P embeds in R2n then ν(K) = 0. For n ≥ 3, theconverse implication holds.

By Kuratowski’s result the converse implication holds for n = 1.

The case n = 2 was left open.


Whitney Theorem

Theorem (H. Whitney)

Every PL (smooth) n-manifold is PL (smoothly) embeddable in R2n.

The dimension 2n in the theorem is the best possible for n = 2k ,since RPn does not embed in R2n−1 for such values of n, and is notthe best possible for other n.


Conjecture

Every closed n-manifold embeds in R2n+1−α(n), wheren = 2k1 + . . . + 2kα(n) and k1 < . . . < kα(n) .

Remark. It is known that the n-manifold

M = RP2k1× . . . × RP2

kα(n)

does not embed in R2n−α(n).


Theorem (A. Haefliger)

For m ≥ 3(n+1)2 , every closed (2n −m)-connected smooth

n-manifold is smoothly embeddable in Rm.

Theorem ( R. Penrose, J. H. C. Whitehead and E. C. Zeeman,M. C. Irwin)

For m ≥ n + 3, every closed (2n −m)-connected PL n-manifold isPL embeddable in Rm.


The Whitney Obstruction

H. Whitney proved that for any orientable n-manifold M and anyimmersion f : M → Rm, the homology class

Wm−n(M) ∈ H2n−m(M;Z(m−n))

of the projection of the singular manifold

∆(f) = {(x, y) ∈ M ×M|x , y, f(x) = f(y)}

on M is independent of f .

This class is called the Whitney obstruction to embeddability of Min Rm. If M embeds in Rm, then Wm−n(M) = 0.


Theorem (A. Haefliger and M. W. Hirsh)

Let M be a closed (2n −m − 1)-connected smooth n-manifold. Form ≥ 3(n+1)

2 , M is smoothly embeddable in Rm if and only ifWm−n(M) = 0.

Theorem (C. Weber)

Let M be a closed (2n −m − 1)-connected PL n-manifold. Form ≥ n + 3, M is PL embeddable in Rm if and only if Wm−n(M) = 0.


The deleted product obstruction

Suppose that f : X → Rm is an embedding. Then the mapf∗ : X∗ → Sm−1, from the deleted product X∗ to the sphere Sm−1,defined by

(∗) f∗(x, y) =f(x) − f(y)

||f(x) − f(y)||

for each (x, y) ∈ X∗, is well defined. This map is equivariant withrespect to the involution t(x, y) = (y, x) on X∗ and the antipodalinvolution a on Sm−1.

The deleted product necessary condition for embeddability of Xinto Rm is the existence of an equivariant map X∗ → Sm−1.


The existence of an equivariant map f∗ : X∗ → Sm−1 is equivalentto the existence of a section of the bundle

p : X∗ × Sm−1/(t × a)→ X∗/t

with the fiber Sm−1 and the projection given by the formula

[(x, y), u] 7→ [(x, y)] .

If X is a polyhedron or a smooth manifold then the existence of asection of the bundle can be check by methods of obstructiontheory.

In particular the van Kampen and Whitney obstructions are the firstobstructions to the existence of a section of the bundle p.So they can be obtained from the deleted product condition in analgebraic way.


Remark. If X is a polyhedron with a triangulation T then thesimplicial deleted product T∗ is an equivariant retract of X∗.

Thus in this case the deleted product condition is equivalent to theexistence of an equivariant map T∗ → Sm−1.


The following result generalizes the theorem of van Kampen,Shapiro and Wu.

Theorem (C. Weber, cf. A. Haefliger)

Suppose K is an n-polyhedron and 2m ≥ 3(n + 1).

(W) If there exist an equivariant map F : K ∗ → Sm−1 then thereexists a PL-embedding f : K → Rm such that f∗ isequivariantly homotopic to F.

Remark. The statement (W) is true if m ≤ 2.


A continuous map f from a metric space X into Y is an ε-map,where ε > 0, if for each y ∈ Y , the diameter of f−1(y) is less than ε.

X quasi–embeds in Y if for each ε > 0 there exists an ε-mapX → Y .

Note that if a polyhedron X quasi–embeds in Rm then there existsan equivariant map X∗ → Sm−1.

Corollary (C. Weber)

If an n-polyhedron quasi–embeds in Rm and 2m ≥ 3(n + 1), then itembeds in Rm.


Problem (S. Mardesic and J. Segal)

For which pairs (m, n) of positive integers the following statementis true?

(S) An n-polyhedron embeds in Rm if it quasi–embeds in Rm.

By Weber’s result the statement (S) is true if 2m ≥ 3(n + 1);it is a generalization of an earlier Ganea’s result for m = 2n , 4.

The statement (S) is also true if m ≤ 2 or n > m.

Among the reasons for considering the above problem is that:

Its negative solution implies that the deleted product condition forthe embeddability of n-polyhedra in Rm is not sufficient.


Mardesic and Segal and L. S. Hush observed that the statements(S) for (n, n) and (n, n − 1) are false for n ≥ 4.

In fact, it is known from the work of M. Curtis and B. Mazur that forn ≥ 4 there are contractible combinatorial n-manifolds Mn suchthat π1(∂Mn) , 0 and Mn × I = In+1.The cone over (∂Mn) is an n-polyhedron which quasi–embeds inRn, but nevertheless does not embed in Rn.


The following result was established in works by J.Segal-S.S,M.H.Freedman-V.S.Krushkal- P.Teichner andJ.Segal-A.Skopenkov-S.S.

The dimension restrictions 2m ≥ 3(n + 1), in the above results ofWeber, are necessary for m ≥ max(4, n), in all cases .

By this result the only open cases were (3, 2) and (3, 3).


These two cases were covered by the following D. Goncalves andA. Skopenkov result:

For a nontrivial homology 3-ball N (or its special 2-spine N) thereexists an equivariant map N → S2, although N does not embed inR3.


PL mappings

A linear mapping of a simplicial complex K into Rm is a mappingf : |K | → Rm that is linear on each simplex.

A PL mapping of K into Rm is a linear mapping of somesubdivision K ′ of K into Rd .

A PL embedding of K into Rm is any PL mapping f : |K | → Rm thatis homeomorphism of |K | onto f(|K |).


We say that a linear mapping f of a simplicial complex K into Rm isgeneric if f |V(K) is one-to-one and the set f(V(K)) is in generalposition in Rm.

A PL mapping of K into Rm is generic if the corresponding linearmapping of some subdivision K ′ of K into Rm is generic.


Linking number

Let Sk be a PL k -shpere and S l an PL l-sphere, and letf : Sk → Rk+l+1 and g : S l → Rk+l+1 be PL-embeddings such thatf(Sk ) ∩ g(S l) = ∅. For our purposes, we may assume that f and gare mutually generic (that is, f t g : Sk t S l → Rm is generic).

The images f(Sk ) and g(S l) are unlinked if f can be extended to aPL mapping f : Bk+1 → Rk+l+1, where Bk+1 is a k-dimensional PLball with ∂Bk+1 = Sk , such that f(Bk+1) ∩ g(S l) = ∅.


By a homological linking number of f and g we understand theimage under Hk (f : Sk → Sk+l+1 \ g(S l)) of the generator ofHk (Sk ) in Hn(Sk+l+1 \ S l) ' Z; here Hk (X) denotes Hk (X ;Z).

Here, the sphere Sk+l+1 denotes the one point compactification ofRk+l+1.


Construction of the example

LemmaLet n > l ≥ 0 and m := n + l + 1. There exists an n-polyhedronQ ⊂PL R

m containing unlinked spheres Σn and Σl (of dimensions nand l, respectively) such that

(a) Σl bounds an (l + 1)-disc in Q and determines a generator ofπl(S

m \ Σn);

(b) for each embedding h : Q ↪→ Rm the homological linkingnumbers of h(Σn) and h(Σl) is odd;

Let P := (∆m+1)(n) ∪ Cone((∆m+1)(l); a), where(∆m+1)(n) ∩ Cone((∆m+1)(l); a) = (∆m+1)(l).We set Q := P \ Int Cone(σ; a), where σ is an l-face of ∆m+1.

Then Σn = ∂τ, where τ ∗ σ = ∆m+1 and Σl = ∂Cone(σ; a).


Lemma (Auxiliary Lemma)

For each n, m2 ≤ n < m, and l = m − n − 1, there exist an

n-polyhedron K ⊂PL Rm containing two disjoint wedges of spheres

Σn ∨ Σn and Σl ∨ Σl such that

(a) Σn ∨ Σn is unknotted in Rm;

(b) for each embedding K ↪→ Rm the pairs Σn, Σl and Σn,Σl arenot linked and the homological linking numbers of the pairsΣn,Σl and Σn, Σl are odd;


Lemma (Finger Move Lemma, the case m=4, n=2.)

Let K ⊂ R4 be the 2-polyhedron from Auxiliary Lemma. LetD2 ⊂ Σ2 and D2 ⊂ Σ2 be PL-disks in the interiors of some2-simplices, of some triangulation of K, adjacent to the uniquecommon point of Σn and Σn. Then there is a PL-map g : K → R2

such that

(a) g|K\Int D2 is the inclusion and g|K\Int D2 is an embedding butg(D2) ∩ g(D2) , ∅;

(b) the commutator of the inclusions Σ1 ↪→ Σ1 ∨ Σ1 andΣ1 ↪→ Σ1 ∨ Σ1 is null-homotopic in Rm \ g(Σn ∨ Σn).


Lemma (Finger Move Lemma)

Let K ⊂ Rm be the n-polyhedron from Auxiliary Lemma. LetDn ⊂ Σn and Dn ⊂ Σn be PL-disks in the interiors of somen-simplices, of some triangulation of K, adjacent to the uniquecommon point of Σn and Σn. Then there is a PL-map g : K → Rm

such that

(a) g|K\Int Dn is the inclusion and g|K\Int Dn is an embedding butg(Dn) ∩ g(Dn) , ∅;

(b) the Whitehead product of the inclusions Σl ↪→ Σl ∨ Σl andΣl ↪→ Σl ∨ Σl is null-homotopic in Rm \ g(Σn ∨ Σn).


Proposition

Let K ⊂ Rm be the n-polyhedron from Auxiliary Lemma. Letg : K → Rm be the map from Finger Move Lemma. Letr : B2l → Rm \ g(Σn ∨ Σn) be a PL-map such thatr |∂B2l : ∂B2l → Σl ∨ Σl represents the Whitehead product ofinclusions Σl ⊂ Σl ∨ Σl and Σl ⊂ Σl ∨ Σl . Let

Y = (K \ Int Dn) ∪ r(B2l) ∪ g(Dn) ⊂ Rm

andR = (K \ Int Dn) ∪ r(B2l)

⋃∂Bn=∂Dn

Bn.

Then dim R = n, R is quasi homeomorphic to Y ⊂ Rm but is nottopologically embeddable in Rm.


Proposition (The case m=4, n=2)

Let K ⊂ Rm be the n-polyhedron from Auxiliary Lemma. Letg : K → Rm be the map from Finger Move Lemma. Let g : K → R4

be the map from the above lemma. Let r : B2 → R4 \ g(Σ2 ∨ Σ2)be a PL-map such that r |∂B2 : ∂B2 → Σ1 ∨ Σ1 represents thecommutator of the inclusions Σ1 ⊂ Σ1 ∨ Σ1 and Σ1 ⊂ Σ1 ∨ Σ1. Let

R = (K \ Int D2) ∪ r(B2)⋃

∂B2=∂D2

B2.

Then dim R = 2, R quasi embeds in R4 but it is not topologicallyembeddable in R4.


Hardness of embbedings simplicial complexes in Rm

By EMBEDn→m we denote the decision problem, whose input is asimplicial complex K of at most dimension n, and where outputshould be YES or NO depending on whether K admits a PLembedding into Rm.

Theorem (Matousek-Tancer-Wagner)

EMBED(m−1)→m (and hence also EMBEDm→m) is algorithmicallyundecidable for every for every m ≥ 5.

Theorem (Matousek-Tancer-Wagner)

EMBEDn→m is NP-hard for every pair (n,m) with m ≥ 4 and(2m − 2)/3 ≤ n ≤ m.


The first theorem of Matousek-Tancer-Wagner is a consequence ofthe following celebrated result of Novikov.

Theorem (Novikov)

Fix m ≥ 5. There is effectively constructible sequence of simplicialcomplexes Ki , ı ∈ N, with the following properties:

(a) each |Ki | is a homology m-sphere;

(b) for each i, either |Ki | is a PL m-sphere, or the fundamentalgroup of |Ki | is nontrivial;

(c) There is no algorithm that decides for every given Ki which ofthe two case holds.


The second theorem of Matousek-Tancer-Wagner was proved byusing methods of:

Segal–S., Freedman–Krushkal–Teichner, andSegal–Skopenkov–S.


Embeddings of compacta X with dim(X × X) < 2 dim(X)

In early 1980’s Ancel and Sternfeld asked the following question:

Is it true that if every mapping X → R2n, where X is a compactmetric space, is a uniform limit of embeddings then dim X < n.

The answer to the above question is negative for compact metricspaces such that dim(X × X) < 2 dim X . Such compacta were firstdiscovered by Boltianski in 1940’s after Pontryagin’s construction(in 1930’s) of 2-dimensional compacta X and Y withdim(X × Y) = 3.

Theorem (Krasinkiewicz-S., Dranisnikov-Repovs-Scepin)Let X be an n-dimensional compact metric space. Then everymapping X → R2n is a uniform limit of embeddings if and only ifdim(X × X) < 2n.


TheoremLet X and Y be compacta and m = dim X + dim Y. Thendim(X × Y) < m if and only if for any two mappings X → Rn andY → Rn can be approximated by mappings with disjoint images.

This theorem was generalized by several topologist: Dranisnikov,Repovs, Scepin, Torunczyk and myself. Also a polyhedral versionof this theorem, in the metastable case, was also proved.


The following result has been obtained by Dranisnikov.

TheoremLet m be an integer and let X and Y be compacta such thatdim(X × Y) < m and dim X < m − 2. Then, any two mappingsX → Rm and Y → Rm can be approximated arbitrarily closely bymappings with disjoint images.

Problem. It is not know whether the above theorem is true ifdim X = dim Y = m − 2, except the case m = 4.


THANK YOU!



Whitehead product

Let α be a generator of the homotopy group πl(Sl).

The Whitehead product [α, α] (considered as an element ofπ2l−1(Sl ∨ Sl)) is the homotopy class of the map v, where:

v : S2l−1 = Sl−1 × B l ∪ B l × Sl−1 → Sl ∨ Sl

is the map with the fibers Sl−1 × {x} and {x} ×Sl−1 for each x ∈ Int B l

and the fiber Sl−1 × Sl−1, where B l is an l-ball and Sl−1 = ∂B l .

The case l = 1: Note that if α is a generator of π1(S1) then [α, α] isthe commutator of π1(S1 ∨ S1).


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Embeddings of Polyhedra and Compacta in Euclidean Spacesigce.rc.unesp.br/Home/Departamentos47/matematica/topologiaalgeb... · PROBLEM: Find conditions for embeddability of k-dimensional