We improve the bound on Kúhnel’s problem to determine the smallest
n
such that the
k
-skeleton of an
n
-simplex
Δ
n
(
k
)
does not embed into a compact PL 2
k
-manifold
M
by showing that if
Δ
n
(
k
)
...embeds into
M
, then
n
≤
(
2
k
+
1
)
+
(
k
+
1
)
β
k
(
M
;
Z
2
)
. As a consequence we obtain improved Radon and Helly type results for set systems in such manifolds. Our main tool is a new description of an obstruction for embeddability of a
k
-complex
K
into a compact PL 2
k
-manifold
M
via the intersection form on
M
. In our approach we need that for every map
f
:
K
→
M
the restriction to the
(
k
-
1
)
-skeleton of
K
is nullhomotopic. In particular, this condition is satisfied in interesting cases if
K
is
(
k
-
1
)
-connected, for example a
k
-skeleton of
n
-simplex, or if
M
is
(
k
-
1
)
-connected. In addition, if
M
is
(
k
-
1
)
-connected and
k
≥
3
, the obstruction is complete, meaning that a
k
-complex
K
embeds into
M
if and only if the obstruction vanishes. For trivial intersection forms, our obstruction coincides with the standard van Kampen obstruction. However, if the form is non-trivial, the obstruction is not linear but rather ’quadratic’ in a sense that it vanishes if and only if certain system of quadratic diophantine equations is solvable. This may potentially be useful in attacking algorithmic decidability of embeddability of
k
-complexes into PL 2
k
-manifolds.
Whenever a finitely generated group G acts properly discontinuously by isometries on a metric space X, there is an induced uniform embedding (a Lipschitz and uniformly proper map) rho: G right arrow ...X given by mapping G to an orbit. We study when there is a difference between a finitely generated group G acting properly on a contractible n-manifold and uniformly embedding into a contractible n-manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible 3-manifold but do not act on a contractible 3-manifold. We show that k-fold products of certain examples do not act on contractible 3k-manifolds. Mathematics Subject Classification (2020). Primary: 20F65, 20F69, 57S30, 57M07; Secondary: 20F67, 57M60. Keywords. Van Kampen obstruction, Wu invariant, uniformly proper dimension, action dimension.
A map φ:K→R2 of a graph K is approximable by embeddings, if for each ε>0 there is an ε-close to φ embedding f:K→R2. Analogous notions were studied in computer science under the names of cluster ...planarity and weak simplicity. This short survey is intended not only for specialists in the area, but also for mathematicians from other areas.
We present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen (or Hanani–Tutte) obstruction for approximability by embeddings and discuss its completeness. We discuss analogous problems of moving graphs in the plane apart (cf. S. Spież and H. Toruńczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional generalizations, including completeness of the van Kampen obstruction and its algorithmic corollary (D. Repovš and A. Skopenkov, 1998).
We obtain a criterion for approximability of PL maps
S
1→
R
2
by embeddings, analogous to the one proved by Minc for PL maps
I→
R
2
.
Theorem.
Let
ϕ
:S
1→
R
2
be a PL map, which is simplicial for ...some triangulation of
S
1 with
k vertices. The map
ϕ is approximable by embeddings if and only if for each
i=0,…,
k the
ith derivative
ϕ
(
i)
(
defined by Minc)
neither contains transversal self-intersections nor is the standard winding of degree, ∉{−1,0,1}.
We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of PL maps
I→
R
2
(Corollary 1.4). We also generalize these criteria to simplicial maps
T→S
1⊂
R
2
, where
T is a graph without vertices of degree >3 (Theorem 1.5).
We prove the following theorem:
Suppose that
m ⩾
3(n + 1)
2
and that
ƒ : K →
R
m
is a PL map of an
n-dimensional
finite polyhedron
K.
Then ƒ
is approximable by embeddings if and only if there exists ...an equivariant homotopical extension
Φ :
K
̃
→ S
m−1
of the map
\
̃
tf :
K
̃
ƒ → S
m−1
,
defined by
f
̃
(x,y) =
(ƒ(x) − ƒ(y))
(∥ƒ(x) − ƒ(y)∥)
,
where
K
̃
ƒ = {(x, y) ε K × K ¦ ƒ(x) ≠ ƒ(y)}
. Our result is a controlled version of the classical deleted product criterion of embeddability of
n-dimensional polyhedra in
R
m
. The proof requires additional (compared with the classical result) general position arguments, for which the restriction
m ⩾
3(n + l)
2
is again necessary. We also introduce the van Kampen obstruction for approximability by embeddings.
We have collected several open problems on graphs which arise in geometric topology, in particular in the following areas:
1.
(1) basic embeddability of compacta into the plane
R
2;
2.
(2) ...approximability of maps by embeddings;
3.
(3) uncountable collections of continua in
R
2 and their span; and
4.
(4) representations of closed PL manifolds by colored graphs. These problems should be of interest to both topologists and combinatorists.
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