On fundamental group of Riemannian manifolds with ommited fractal subsets
We show that if $K$ is a closed and bounded subset of a Riemannian manifold $M$ of dimension $m > 3$, and the fractal dimension of $K$ is less than $m - 3$, then the fundamental groups of $M$ and $M - K$ are isomorphic.
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| Дата: | 2011 |
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Institute of Mathematics, NAS of Ukraine
2011
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508735986728960 |
|---|---|
| author | Mirzaie, R. Мірзай, Р. |
| author_facet | Mirzaie, R. Мірзай, Р. |
| author_sort | Mirzaie, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:35:28Z |
| description | We show that if $K$ is a closed and bounded subset of a Riemannian manifold $M$ of dimension $m > 3$, and the fractal dimension of $K$ is less than $m - 3$, then the fundamental groups of $M$ and $M - K$ are isomorphic. |
| first_indexed | 2026-03-24T02:29:56Z |
| format | Article |
| fulltext |
UDC 512.5
R. Mirzaie (I. Kh. Int. Univ. (IKIU) Qazvin, Iran)
ON FUNDAMENTAL GROUP OF RIEMANNIAN MANIFOLDS
WITH OMITTED FRACTAL SUBSETS
ПРО ФУНДАМЕНТАЛЬНУ ГРУПУ РIМАНОВИХ
МНОГОВИДIВ З ПРОПУЩЕНИМИ ФРАКТАЛЬНИМИ
ПIДМНОЖИНАМИ
We show that if K is a closed and bounded subset of a Riemannian manifold M of dimension m > 3, and the
fractal dimension of K is less than m− 3, then the fundamental groups of M and M −K are isomorphic.
Показано, що якщо K — замкнена й обмежена пiдмножина рiманового многовиду M розмiрностi m > 3,
а фрактальна розмiрнiсть K менша за m− 3, то фундаментальнi групи M i M −K є iзоморфними.
1. Introduction. If K is a subset of a connected topological space M, it is interesting
(but usually hard) to study, relations between fundamental groups of M and M − K.
When the difference of the fractal dimensions (box dimension or Hausdorff dimension)
of K and M is big enough, we expect that the fundamental groups of M and M −K be
isomorphic. It is proved in [1] that if M = Rm or M = Sm, m ≥ 2 and F is a compact
subset of M and the Hausdorff dimension of F is strictly less than m − k − 1, then
M − F is k-connected (i.e., its homotopy groups πi vanish for i ≤ k). Consequently
if dimH(F ) < m − 2 then Rn − F and Sn − F are simply connected. In this paper,
we consider a more general case when M is a Riemannian manifold then we prove the
following theorem.
Theorem 1.1. Let Mm be a Riemannian manifold of dimension m > 3, and K
be a bounded and closed subset of M such that dimB(K) < m − 3. Then π1(M) is
isomorphic to π1(M −K).
Before giving the proof of the theorem, we mention some preliminaries. Let A be a
subset of a metric space (M,d). We denote by dimA the topological dimension of A.
Let ε be a positive number and put
Bε(A) =
{
x ∈M : d(x, a) < ε for some a ∈ A
}
.
If A is bounded then the upper box dimension of A is defined by
dimBA = lim sup
δ→0
log(mδA)
− log δ
,
where, mδA is the maximum number of disjoint balls of radius δ, with centers contained
in A. The lower box dimension dimB(A) is defined in similar way. Another definition
for dimension, which is widely used in fractal geometry is Hausdorff dimension (see
[2]). We use the upper box dimension in our theorem. But a similar result is true for
lower box dimension and also for Hausdorff dimension.
Remark 1.1. (a) If A is a submanifold of a Riemannian manifold M, then
dimB(A) = dim(A).
(b) If (M,d) and (N, d′) are metric spaces and f : M → N is a map such that for
some positive number c > 0, d′(f(x), f(y)) ≤ cd(x, y) (f is Lipschitz), then
c© R. MIRZAIE, 2011
854 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON FUNDAMENTAL GROUP OF RIEMANNIAN MANIFOLDS WITH OMITTED . . . 855
dimB(f(A)) ≤ dimB(A).
(c) If A1 and A2 are bounded subsets of M, then
dimB(A1 ×A2) ≤ dimB(A1) + dimB(A2).
Remark 1.2. In the followings, for each positive number r, we denote by Sn−1(r)
the sphere of radius r and center at the origin of Rn. Let D be a closed (n− 1)-disc in
Rn and let a be a point outside of D. The set C =
{
ta+ (1− t)d : d ∈ D, 0 ≤ t ≤ 1
}
is called a cone with vertex a, over D. The following map is called a radial projection
f : C → D : f(ta+ (1− t)d) = d.
If x1, x2 ∈ C and x1 → a, x2 → a then |x2−x1| → 0. Thus f is not Lipschitz (because
|f(x1) − f(x2)| is bounded). But, if W is an open neighborhood of a in Rn, the map
f : (C −W )→ D is a Lipschitz map.
2. Proof of Theorem 1.1.
Step 1. Let 0 < r2 < r1, A(r1, r2) =
{
x ∈ Rn : r2 ≤ |x| ≤ r1
}
, n > 2, and let
K be a closed subset of A(r1, r2), such that dimB(K) < n− 1. Then there are points
a1 ∈ Sn−1(r1) and a2 ∈ Sn−1(r2) such that the line segment a2a1, joining two points
a1 and a2, does not intersect K.
Proof. Since dimB(K) < n− 1, then Sn−1(r1)−K 6= ∅. Let a1 ∈ Sn−1(r1)−K
and let o be the origin of Rn. Denote by oa1 the line segment joining o to a1. Put
b = oa1 ∩ Sn−1(r2) and let c be the mid point of ob and consider the (n − 1)-disc D,
with the center at c and boundary on Sn−1(r2), which is perpendicular to ob at the point
c. Since K is closed, there is an open neighborhood W of a1, such that K ∩W = ∅.
Let C be the cone over D with the vertex a1, and consider the radial projection map
f : (C −W )→ D. f is a Lipschitz map. Thus
dimB(f(K ∩ (C −W ))) ≤ dimB(K ∩ (C −W )) < n− 1.
Thus, f(K ∩ (C −W )) does not cover D. If d ∈ (D− f((C −W )∩K)) then the line
segment a1d does not intersect K. If a2 = a1d∩Sn−1(r2), then a1a2 is the desired line
segment.
Step 2. If K ⊂ Rn, n > 2, and dimB(K) < n − 1, then there is a path σ :
[0, 1]→ Rn such that σ(0) = o and for each t ∈ (0, 1], σ(t) /∈ K.
Proof. Consider the spheres Sn−1
(
1
m
)
, m ∈ N. Since dimB(K) < n − 1, then
for each r > 0, Sn−1(r) − K 6= ∅. Let a1 ∈ (Sn−1(1) − K). By Step 1, there is
point a2 ∈ Sn−1
(
1
2
)
, such that a1a2 ∩ K = ∅. Let σ1 :
[
1
2
, 1
]
→ Rn be a path
from a2 to a1 along the line segment a2a1. Now, by induction, we can find the points
am ∈ Sn−1
(
1
m
)
, m > 1, and the paths σm−1 :
[
1
m
,
1
m− 1
]
→ Rn, along the line
segments amam−1, such that am−1am ∩K = ∅. The following path is the desired path
σ : [0, 1]→ Rn, σ(0) = 0, and σ(t) = σm(t) if t ∈
[
1
m
,
1
m− 1
]
, m > 1.
Let α, β : I = [0, 1]→ M be two continuous paths in M with the same end-points.
We recall that a continuous map F : [0, 1] × [0, 1] → M with the following properties,
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
856 R. MIRZAIE
is called a homotopy equivalence between α and β
F (s, 0) = α(s), F (s, 1) = β(s), s ∈ I,
F (0, t) = α(0) = β(0), F (1, t) = α(1) = β(1), t ∈ I.
Step 3. Let E be a closed and bounded subset of Rn, n > 3, such that dimB(E) <
< n − 3. Let α, β : I → (Rn − E) be two loops at the point x0 ∈ (Rn − E) and
F : I × I → Rn be a differentiable homotopy equivalence between α and β (in Rn).
If ε > 0 then there is a homotopy equivalence G : I × I → (Rn − E) (homotopy
equivalence in (Rn − E)) between α and β such that
max
{
|F (s, t)−G(s, t)| : (s, t) ∈ I × I
}
< ε.
Proof. Put N = F (I × I) and let
φ : N ×Rn → Rn, φ(x, y) = y − x.
Consider the following metric on N ×Rn:
d((x1, y1), (x2, y2)) = |x1 − x2|+ |y1 − y2|.
Put K = φ(N × E). φ is a Lipschitz map, so
dimB(K) = dimBφ(N × E) ≤ dimB(N × E) ≤
≤ dimB(N) + dimB(E) < 2 + n− 3 = n− 1.
By Step 2, there is a path σ : [0, 1] → Rn, such that σ(0) = o and for each t ∈ (0, 1],
σ(t) ∈ (Rn −K). Let θ : I × I → [0, 1] be a continuous function such that
θ(s, t) = 0 if and only if (s, t) belongs to the boundary of I × I.
Since σ is continuous, there is a δ > 0 such that
|σ(δθ(s, t))| < ε, (s, t) ∈ I × I.
Now, put
G : I × I → Rn, G(s, t) = F (s, t) + σ(δθ(s, t)).
We have
G(s, 0) = F (s, 0) = α(s), G(s, 1) = F (s, 1) = β(s), s ∈ I,
in similar way
G(0, t) = G(1, t) = x0, t ∈ I.
Thus, G is a homotopy equivalence between α and β. Also we obtain
G(s, t) /∈ E, (s, t) ∈ I × I.
Because, if G(s, t) ∈ E then
(F (s, t), F (s, t) + σ(δθ(s, t)) ∈ N × E ⇒ (F (s, t) + σ(δ(θ(s, t)))− F (s, t)) ∈ K.
Therefore, σ(δθ(s, t)) ∈ K, which is contradiction. This means that G : I × I → (Rn−
− E) is a homotopy equivalence between α and β in (Rn − E). Also we have
|G(s, t)− F (s, t)| = |σ(δθ(s, t))| < ε.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON FUNDAMENTAL GROUP OF RIEMANNIAN MANIFOLDS WITH OMITTED . . . 857
Step 4. Let U be an open subset of Rn, n > 3, E ⊂ U and dimB(E) < n − 3.
Then π1(U) is isomorphic to π1(U − E).
Proof. Let x0 ∈ (U −E) and for each loop α : I → (U −E) at x0, denote by [α]1
and [α]2 the elements of π1(U − E, x0) and π1(U, x0) generated by α. Put
φ : π1(U − E)→ π1(U), φ([α]1) = [α]2.
We show that φ is one to one and onto. Let [α]1, [β]1 ∈ π1(U −E). If [α]2 = [β]2 then
there is a differentiable homotopy equivalence F : I × I → U between α and β in U.
By Step 3, for each ε > 0, there is a homotopy equivalence G : I × I → (Rn − E)
between α and β such that
|G(s, t)− F (s, t)| < ε, (s, t) ∈ I × I.
Since for each (s, t), F (s, t) ∈ U, we can choose ε sufficiently small, such that G(s, t) ∈
∈ U (i.e., G(s, t) ∈ U − E). Thus G will be a homotopy equivalence between α and β
in U − E. Then [α]1 = [β]1 and consequently φ is one to one.
Now, we show that φ is onto. let [γ] ∈ π1(U, x0) and suppose that γ is a differen-
tiable representative of [γ] and let L = {γ(t) : t ∈ [0, 1]}. Consider the following metric
on L×Rn:
d
(
(x1, y1), (x2, y2)
)
= |x1 − x2|+ |y1 − y2|.
Put φ : L×Rn → Rn, φ(x, y) = y − x and let K = φ(L× E). φ is Lipschitz, so
dimBK ≤ dimB(L× E) ≤ dimBL+ dimBE < 1 + n− 3 = n− 2.
Thus, as like as the proof of Step 2, we can find a path σ : [0, 1] → Rn such that
σ(0) = o and
σ(t) /∈ K, t ∈ (0, 1].
Let θ : [0, 1]→ [0, 1] be a continuous function such that
θ(s) = 0 if and only if s ∈ {0, 1}.
For each ε > 0, there is a δ > 0 such that
|σ(δθ(s))| < ε, s ∈ [0, 1].
Put
α : [0, 1]→ Rn, α(s) = γ(s) + σ(δθ(s))
and let
H(s, t) = γ(s) + σ(δtθ(s)).
Sine for each s ∈ [0, 1], γ(s) ∈ U, we can choose the number ε, so small that
α(s) ∈ U, H(s, t) ∈ U.
Also we have α(s) /∈ E (because, if α(s) ∈ E then (γ(s), α(s)) ∈ L × E, so α(s) −
− γ(s) ∈ K, then σ(δθ(s)) ∈ K, which is contradiction). Since H : I × I → U, is a
homotopy equivalence between γ and α in U, we get that
φ[α]1 = [α]2 = [γ].
Thus φ is onto.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
858 R. MIRZAIE
Step 5. By Nash’s embedding theorem, Mm can be embedded in Rn for sufficiently
large n. Consider the normal vector bundle M → TM⊥ : p → (TpM)⊥ over the
submanifold M of Rn (i.e., TM⊥ = {(p, v) : p ∈ M, v ∈ TpM
⊥}). There exists a
neighborhood U0 of the null section OM in (TM)⊥ such that the map exp (see [3] for
definition of exp) is a diffeomorphism of U0 on to an open subset U ⊂ Rn (U is called
a tubular neighborhood of M in Rn)
exp: U0 → U, exp(p, v) = expp(v).
The following map Ψ is a deformation retract of U0 on to OM :
Ψ: U0 × I → U0,
Ψ((p, v), t) = (p, (1− t)v).
Thus, the following map is a deformation retract of U on to M (i.e., π1(M) is isomor-
phic to π1(U)).
Φ: U × I → U, Φ(x, t) = exp(Ψ(exp−1(x), t)).
Consider the map ς : U →M defined by ς(x) = Φ(x, 1) and put K̂ = ς−1(K) . It easy
to show that
dimB(K̂) ≤ dimB(K) + (n−m) < (m− 3) + (n−m) < n− 3.
Now, we can use Step 4, to get that π1(U) is isomorphic to π1(U − K̂). Since M is
a deformation retract of U, it is easy to show that M − K is a deformation retract of
U − K̂. Thus π1(U − K̂) is isomorphic to π1(M − K). Therefore, π1(M − K) is
isomorphic to π1(M).
1. Matheus C., Olivera K. K. Geometrical versus topological properties of manifols // J. Inst. Math. Jusseiu.
– 2005. – 4, № 4. – P. 639 – 651.
2. Falconer K. Fractal geometry: mathematical foundations. – New York: Jon Wiley and Sons, 1990.
3. Do Carmo M. P. Riemannian geometry. – Boston; Berlin, 1992.
Received 13.11.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
|
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| spelling | umjimathkievua-article-27692020-03-18T19:35:28Z On fundamental group of Riemannian manifolds with ommited fractal subsets Про фундаментальну групу рiманових многовидiв з пропущеними фрактальними пiдмножинами Mirzaie, R. Мірзай, Р. We show that if $K$ is a closed and bounded subset of a Riemannian manifold $M$ of dimension $m > 3$, and the fractal dimension of $K$ is less than $m - 3$, then the fundamental groups of $M$ and $M - K$ are isomorphic. Показано, що якщо $K$ — замкнена й обмежена пiдмножина рiманового многовиду M розмiрностi $m > 3$, а фрактальна розмiрнiсть K менша за $m - 3$, то фундаментальнi групи $M$ i $M - K$ є iзоморфними. Institute of Mathematics, NAS of Ukraine 2011-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2769 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 6 (2011); 854-858 Український математичний журнал; Том 63 № 6 (2011); 854-858 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2769/2294 https://umj.imath.kiev.ua/index.php/umj/article/view/2769/2295 Copyright (c) 2011 Mirzaie R. |
| spellingShingle | Mirzaie, R. Мірзай, Р. On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title | On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title_alt | Про фундаментальну групу рiманових многовидiв з пропущеними фрактальними пiдмножинами |
| title_full | On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title_fullStr | On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title_full_unstemmed | On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title_short | On fundamental group of Riemannian manifolds with ommited fractal subsets |
| title_sort | on fundamental group of riemannian manifolds with ommited fractal subsets |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2769 |
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