Coverings and fundamental groups: a new approach
Classical fundamental groups behave reasonably well for Poincare spaces (i.e.. semy-locally simply connected spaces). One has a construction of the universal covering for such spaces. For arbitrary spaces it is a different matter. We define monodromy groups $\pi (p,b_{0})$ for any map $...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552771046080512 |
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| author | Dydak, J. Dydak, J. |
| author_facet | Dydak, J. Dydak, J. |
| author_institution_txt_mv | [
{
"author": "J. Dydak",
"institution": "The University of Tennessee, Knoxville"
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| author_sort | Dydak, J. |
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| datestamp_date | 2018-02-10T20:56:26Z |
| description | Classical fundamental groups behave reasonably well for Poincare spaces (i.e.. semy-locally simply connected spaces). One has a construction of the universal covering for such spaces. For arbitrary spaces it is a different matter. We define monodromy groups $\pi (p,b_{0})$ for any map $p: E \rightarrow B$ with the path lifting property and any $b_{0} \in B$. $p$ is called a $\mathcal{P}$-covering, where $\mathcal{P}$ isa class of Peano spaces (i.e., connected and locally path connected spaces), if it has existence and uniqueness of lifts of maps $f: X \rightarrow B$ for any $X \in \mathcal{P}$. For any $B$ there is the maximal $\mathcal{P}$-covering $p_{\mathcal{P}}:B_{\mathcal{P}}\rightarrow B$ and its monodromy group is called the $\mathcal{P}$-fundamental group of $(B,b_{0})$. In case of $\mathcal{P}$ consisting of all disk-hedgehogs we construct a universal covering theory of all spaces in analogy to the classical covering theory of Poincare spaces. |
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Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 7–21
Jerzy Dydak
Coverings and fundamental groups: a
new approach
Classical fundamental groups behave reasonably well for Poincaré spaces
(i.e.. semy-locally simply connected spaces). One has a construction of the
universal covering for such spaces. For arbitrary spaces it is a different
matter.
We define monodromy groups π(p, b0) for any map p : E → B with the path
lifting property and any b0 ∈ B. p is called a P-covering, where P is a class
of Peano spaces (i.e., connected and locally path connected spaces), if it has
existence and uniqueness of lifts of maps f : X → B for any X ∈ P. For
any B there is the maximal P-covering pP : BP → B and its monodromy
group is called the P-fundamental group of (B, b0). In case of P consisting
of all disk-hedgehogs we construct a universal covering theory of all spaces
in analogy to the classical covering theory of Poincaré spaces.
COVERINGS AND FUNDAMENTAL GROUPS: A NEW
APPROACH
JERZY DYDAK
Abstract. Classical fundamental groups behave reasonably well for Poincaré
spaces (i.e., semi-locally simply connected spaces). One has a construction of
the universal covering for such spaces. For arbitrary spaces it is a different
matter.
We define monodromy groups π(p, b0) for any map p : E → B with the
path lifting property and any b0 ∈ B. p is called a P-covering, where P is
a class of Peano spaces (i.e., connected and locally path connected spaces), if
it has existence and uniqueness of lifts of maps f : X → B for any X ∈ P.
For any B there is the maximal P-covering pP : B̃P → B and its monodromy
group is called the P-fundamental group of (B, b0). In case of P consisting
of all disk-hedgehogs we construct a universal covering theory of all spaces in
analogy to the classical covering theory of Poincaré spaces.
Contents
1. Introduction 1
2. Coverings and deck transformations 2
2.1. The monodromy group 2
2.2. The deck transformation group 3
3. Hedgehog coverings 4
4. The whisker topology 7
5. Supremums of coverings 8
6. Hedgehog fundamental group 10
7. Comparison to the classical fundamental group 11
References 14
1. Introduction
The traditional approach of defining the fundamental group first and then con-
structing universal coverings works well only for the class of Poincaré spaces. For
general spaces there were several attempts to define generalized coverings (see [1],
[3], and [12]), yet there is no general theory so far that covers all path connected
spaces. In this paper we plan to remedy that by changing the order of things: we
define the universal covering first and its group of deck transformations is the new
fundamental group of the base space.
Date: July 16, 2011.
2000 Mathematics Subject Classification. Primary 55Q52; Secondary 55M10, 54E15.
Key words and phrases. universal covering maps, locally path-connected spaces.
1
c© Jerzy Dydak, 2013
8 Coverings and fundamental groups: a new approach
2 JERZY DYDAK
The basic idea is that a non-trivial loop ought to be detected by a covering (not
by extension over the unit disk): a loop is non-trivial if there is a covering such
that some lift of the loop is a non-loop.
So it remains to define coverings: the most natural class is the class of maps that
have unique disk lifting property. To make the theory work one needs to add the
assumption that path components of pre-images of open sets form a basis of the
total space.
2. Coverings and deck transformations
Maps are synonymous with continuous functions.
Definition 2.1. Let P be a class of spaces. A map p : E → B has P-lifting
property if for any e0 ∈ E and any map f : (X,x0) → (B, p(e0)), where X ∈ P,
there is a map g : X → E such that p ◦ g = f and g(x0) = e0.
p is a P-covering (or a P covering) if it has the P-lifting property and all lifts
are unique. That means g = h if g, h : X → E, p ◦ g = p ◦ h, and g(x0) = h(x0) for
some x0 ∈ X ∈ P.
Of special interest are arc-coverings (P consists of the unit interval I), disk-
coverings (P consists of the unit disk D2), and hedgehog-coverings (see 3.1 for the
definition of hedgehogs).
Definition 2.2. A topological space X is an lpc-space if it is locally path-
connected. X is a Peano space if it is locally path-connected and connected.
Problem 2.3. Suppose p : E → D2 is an arc-covering for some Peano space E.
Is p a homeomorphism?
The most fundamental example of a covering is that of the identity function
id : P (X) → X from the Peanification P (X) of X to X (see [3]). P (X) is
obtained from X by changing its topology to the one whose basis consist of path-
components of open sets in X. id : P (X) → X is a P-covering for the class P of
all Peano spaces.
Proposition 2.4. If p : E → B is an arc-covering and E is path-connected, then
the fibers of p are T1 spaces.
Proof. A space F is T1 if each point is closed in it. Equivalently, for any two
different points a, b ∈ F there is an open subset of F containing a but not b.
Suppose e0, e1 ∈ p−1(b0) are two different points such that every neighborhood
of e0 contains e1. Choose a path α from e0 to e1 in E. Consider the loop β
obtained from α by changing the value at 1 from e1 to e0. Notice β is continuous
(β−1(U) = α−1(U) for all open subsets U of E) and is a lift of the same path as α,
yet ending at a different point, a contradiction. �
2.1. The monodromy group. Suppose p : E → B is an arc-covering and b0 ∈ B.
Any loop α at b0 induces a function from the fiber F = p−1(b0) to itself that we
denote by x→ α · x. Namely, we lift α to α̃ starting at x and we put α · x = α̃(1).
Notice the function x→ α · x is a bijection: it inverse is x→ α−1 · x, where α−1(t)
is defined as α(1− t) (in other words, α−1 is the reverse of α). We say that α acts
on F . Notice the composition of α acting on F and β acting on F is the action of
the concatenation α ∗ β on F . The basic idea is to identify any two loops that act
on F the same way.
Jerzy Dydak 9
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 3
Definition 2.5. Suppose p : E → B is an arc-covering and b0 ∈ B. The mon-
odromy group π(p, b0) of p at b0 is the set of equivalence classes of loops in B at
b0: α ∼ β if and only for any two lifts α̃ (of α) and β̃ (of β) one has α̃(1) = β̃(1) if
α̃(0) = β̃(0). The group operation is induced by concatenation: [α] · [β] := [α ∗ β].
Remark 2.6. Notice the above equivalence of loops can be easily extended to the
concept of equivalence of paths in B starting at b0. We will use that equivalence
throughout the paper. In particular, by α · x we mean α̃(1), where α̃ is the lift of
α starting at x.
Notice [α] is the trivial element of π(p, b0) if and only if all its lifts are loops.
Notice that, if p is a disk-covering, then any null-homotopic loop of (B, b0)
represents the trivial element of π(p, b0) and there is a natural homomorphism
π1(B, b0)→ π(p, b0) that is surjective.
It is easy to show that π(p, b0) and π(p, b1) are isomorphic just as in the case of
classical fundamental groups of spaces.
2.2. The deck transformation group.
Definition 2.7. Given a map p : E → B its deck transformation group
DTG(p) is the group of homeomorphisms h : E → E such that p ◦ h = h.
Proposition 2.8. If p : E → B is an arc-covering and E is path-connected, then
the group of deck transformations DTG(p) of p acts freely on E.
Proof. Suppose g(e) = e for a deck transformation g. For any x ∈ E pick a
path α from e to x. Both α and g ◦ α are lifts of p ◦ α originating at e. Therefore
x = α(1) = (g ◦ α)(1) = g(x) and g ≡ idE . �
Definition 2.9. An arc-covering p : E → B is regular if for any loop α in B all
its lifts are either all loops or all non-loops. This is the same as saying that π(B, b0)
acts freely on the fiber F = p−1(b0).
Notice that, if B is path-connected, regularity of p depends only on loops at a
specific point. If no loop at b0 ∈ B has mixed lifts, then no loop at another point
b ∈ B has mixed lifts.
Proposition 2.10. If p : E → B is a regular arc-covering and E is path-connected,
then for any e0 ∈ E there is a natural monomorphism DTG(p) → π(p, b0), b0 =
p(e0). The monomorphism is an isomorphism if DTG(p) acts transitively on the
fibers of p.
Proof. For any h ∈ DTG(p) choose a path αh in E from e0 to h(e0). Since
p is a regular arc-covering, the equivalence class [p ◦ αh] does not depend on the
choice of αh. If g ∈ DTG(p), then αg ∗ g(αh) is a path from e0 to g(h(e0)) and
[p(αg ∗ g(αh))] = [p(αg) ∗ p(αh)], so it is indeed a homomorphism.
If DTG(p) acts transitively on the fibers of p and [α] ∈ π(p, b0), then lift α to α̃
and pick a deck transformation h such that h(α̃(0)) = h(α̃(1)). Notice h is mapped
to α. �
Problem 2.11. Characterize continuous group actions G on a Peano space E such
that the projection p : E → E/G is an arc-covering.
Problem 2.12. Characterize continuous group actions G on a Peano space E such
that the projection p : E → E/G is a disk-covering.
10 Coverings and fundamental groups: a new approach
4 JERZY DYDAK
3. Hedgehog coverings
Definition 3.1. A directed wedge (see [3]) is the wedge
(Z, z0) =
∨
s∈S
(Zs, zs) of pointed Peano spaces indexed by a directed set S and
equipped with the following topology (all wedges in this paper are considered with
that particular topology):
(1) U ⊂ Z \ {z0} is open if and only if U ∩ Zs is open for each s ∈ S,
(2) U is an open neighborhood of z0 if and only if there is t ∈ S such that
Zs ⊂ U for all s > t and U ∩ Zs is open for each s ∈ S.
A arc-hedgehog is a directed wedge (Z, z0) =
∨
s∈S
(Zs, zs) such that each (Zs, zs)
is homeomorphic to (I, 0). The standard arc-hedgehog is the arc-hedgehog over
the set of natural numbers N .
A disk-hedgehog is a directed wedge (Z, z0) =
∨
s∈S
(Zs, zs) such that each Zs is
homeomorphic to the 2-disk D2.
A typical construction of an arc-hedgehog and its map to a space X is the
following:
Proposition 3.2. Let x0 ∈ X. Suppose {αV : IV = [0, 1]→ X}V ∈S is a family of
paths in X indexed by a basis S of open neighborhoods V of x0 in X. If αV (I) ⊂ V
and αV (0) = x0 for all V ∈ S and S is ordered by inclusion (U ≤ V means V ⊂ U),
then the natural function α =
∨
V ∈S
αV :
∨
V ∈S
(IV , 0)→ X is continuous.
Proof. α−1(U) is certainly open if x0 /∈ U . If x0 ∈ U , then IV ⊂ α−1(U) for all
V ⊂ U , so α is indeed continuous. �
Corollary 3.3. Suppose f : Y → X is a function from an lpc-space Y . f is con-
tinuous if f ◦ g is continuous for every map g : Z → Y from an arc-hedgehog Z to
Y .
Proof. Assume U is open in X and x0 = f(y0) ∈ U . Suppose for each path-
connected neighborhood V of y0 in Y there is a path αV : (I, 0)→ (V, y0) such that
αV (1) /∈ f−1(U). Notice the wedge α =
∨
V ∈S
αV is a map from an arc-hedgehog
to Y by 3.2 (here S is the family of all path-connected neighborhoods of y0 in Y ).
Hence h = f ◦g is continuous and there is V ∈ S so that IV ⊂ h−1(U). That means
f(αV (I)) ⊂ U , a contradiction. �
Remark 3.4. If X is first countable (it has a countable basis at each point) in 3.2
or Y is first countable in 3.3, then it is sufficient to use the standard arc-hedgehog
only.
Theorem 3.5. If p : E → B is an arc-covering, then the following conditions are
equivalent:
a. p is an arc-hedgehog covering,
b. given an open subset U of E containing e0, there is a neighborhood V of b0
in B such that the path component of p−1(V ) containing e0 is a subset of
U .
Proof. a) =⇒ b). Suppose, for every neighborhood V of b0 in B, there is a
path αV in p−1(V ) joining e0 with a point in E \ U . The function α =
∨
V ∈S
αV :
Jerzy Dydak 11
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 5
H =
∨
V ∈S
IV → E is continuous as p ◦α is continuous and α is the only possible lift
of p ◦ α at e0. However, the point-inverse of U under α contains e0 but none of IV
is contained in it, a contradiction.
b) =⇒ a). Suppose α =
∨
s∈S
αs :
∨
s∈S
Is → B is a map of an arc-hedgehog
with the base-point mapped to b0 = p(e0). The only possible lift β of α must be
obtained by lifting each αs separately. The only issue is the continuity of β at the
base-point. Given a neighborhood U of e0 in E, pick a neighborhood V of b0 in B
with the property that the path component P of p−1(V ) containing e0 is a subset
of U . Pick an open subset W of the base-point of H satisfying W ⊂ α−1(V ) so
that W is path-connected. Notice β(W ) ⊂ P ⊂ U , which means β is continuous at
the base-point of H. �
Corollary 3.6. If B is first countable and p : E → B is an arc-covering with E
being a Peano space, then p is an arc-hedghehog covering.
Proof. Suppose b0 ∈ B and {Un} is a decreasing basis of neighborhoods of b0 in
B. Given e ∈ F = p−1(b0) and a neighborhood V of e in E, assume that for every
n ≥ 1 there is a path αn in p−1(Un) joining e to a point en ∈ E \ V . Consider the
infinite concatenation p(α1) ∗ p(α−11 ) ∗ p(α2) ∗ p(α−12 ) ∗ . . . which we assume ends
at b0. The lift γ of β starting at e cannot be a loop as γ−1(V ) does not contain
any en. So it ends at a different point of F . Pick a neighborhood W of γ(1) not
containing e (see 2.4). γ−1(W ) is a neighborhood of 1 in [0, 1]. Therefore infinitely
many paths αn lie in W , a contradiction. �
Corollary 3.7. If p : E → B is an arc-hedgehog covering and E is a Peano space,
then the fibers of p are regular (T3-spaces) 0-dimensional spaces.
Proof. By 2.4, fibers of p are T1-spaces, so, given x /∈ A in a fiber F (and A
being closed in F ), there is an open neighborhood V of p(x) = p(y) such that the
path component W of p−1(V ) containing x does not intersect A. The restriction
W ∩ F of W to F is an open-closed subset of F containing x and missing A. �
Corollary 3.8. Arc-hedgehog coverings p : E → B are open if both E and B are
locally path-connected.
Proof. Suppose U is open in E and e0 ∈ U . Put b0 = p(e0) and F = p−1(b0). By
3.5 there is a path-connected neighborhood V of b0 such that the path-component
of e0 in p−1(V ) is a subset of U . Therefore V ⊂ p(U) (connect e0 with a path to
any point in V and then lift the path - it must be contained in U). �
Here is an important supplement to 2.10:
Theorem 3.9. Suppose p : E → B is an arc-hedgehog covering. If E is a Peano
space, then p is regular if and only if the deck transformation group DTG(p) acts
transitively on the fibers of p.
Proof. If DTG(p) acts transitively on the fibers of p, then for any two lifts α
and β of the same loop in B there is a deck transformation h such that h ◦ α = β.
Hence they are either both loops or both non-loops.
Suppose p is regular and e1, e2 ∈ E with p(e1) = e2. Given x ∈ E choose a path
αx in E from e1 to x and let βx be the path from e2 to h(x) with the property
p ◦ αx = p ◦ βx. Notice h(x) does not depend on the choice of αx as p is regular.
12 Coverings and fundamental groups: a new approach
6 JERZY DYDAK
The reason h is continuous is that h ◦ f is continuous for any map f from an
arc-hedgehog to E. Since analogous construction creates the inverse of h, it is a
homeomorphism. �
Proposition 3.10. Suppose p : E → B is an arc-hedgehog covering of Peano
spaces. If B is metrizable, then E is metrizable.
Proof. Denote r-balls in B centered at b by B(b, r). Define d(x, y) as the
infimum of r > 0 such that there is a path α from x to y in E with p(α([0, 1])) ⊂
B(p(x), r) ∩ B(p(y), r). Clearly, d is symmetric. Also, d(x, y) = 0 implies x = y.
Indeed, p(x) = p(y) and x 6= y would imply existence of a neighborhood U of p(x)
in B such that no path in U can be lifted to a path from x to y (see 3.5).
The proof of the Triangle Inequality is left to the reader.
Given x ∈ U , U open in E, find an r > 0 such that the path component of
p−1(B(p(x), r)) containing x is contained in U (see 3.5). Therefore the r-ball of
metric d centered in x is contained in U .
Consider the r-ball Bd(x, r) in d centered at x ∈ E. Look at the path-component
U of p−1(B(p(x), r/2)) containing x. It must be contained in Bd(x, r) which com-
pletes the proof. �
Proposition 3.11. If p : E → B an arc-hedgehog covering, E is Peano, and B
has a countable basis at b0, then F = p−1(b0) is a Baire space.
Proof. Let {Un} be a basis of open sets at b0 that forms a decreasing sequence.
We plan to show that, given a decreasing sequence {Vn} of path-components Vn
of p−1(Un), the intersection F ∩
∞⋂
n=1
Vn is not empty. By induction, pick points
en ∈ Vn and paths αn in Vn joining en with en+1. The infinite concatenation
p(α1) ∗ p(α2) ∗ . . . (its end-point is declared to be b0) is a path α in U1. Lift α
starting at e1 and notice the end-point of the lift belongs to F ∩
∞⋂
n=1
Vn. �
Remark 3.12. Combining the proofs of 3.10 and 3.11 one can show E is completely
metrizable if B is completely metrizable and both E and B are Peano spaces.
Definition 3.13. Suppose p : E → B is an arc-hedgehog covering of Peano spaces.
p is trivial at b0 if there is a connected neighborhood U of b0 in B such that p
maps each component of p−1(U) homeomorphically onto U .
Theorem 3.14. Suppose p : E → B is a regular arc-hedgehog covering of Peano
spaces. p is trivial at b0 if and only if the fiber F = p−1(b0) contains an isolated
point.
Proof. One direction is obvious, so assume F has an isolated point e ∈ F .
Choose a connected neighborhood U of b0 in B such that the path component V
of p−1(U) containing e does not intersect F \ {e}) (see 3.5). Notice p maps V
homeomorphically onto U . Indeed, p(V ) = U (lift a path from b0 to any x ∈ U
starting from e to arrive at y ∈ V such that p(y) = x) and p|V has to be injective:
if p(y) = p(z) = b for two different points y, z ∈ V , then there is a path β in V
from y to z such that p ◦ β is a loop and picking a path γ from e to y in V results
in a loop p(γ) ∗ p(β) ∗ p(γ−1) in U that has a lift in V starting at e and ending at
a different point contrary to V ∩ F = {e}.
Jerzy Dydak 13
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 7
Consider another component W of p−1(U). Using 3.9 one can see there is a deck
transformation h such that h(V ) = W . Therefore p|W : W → U is a homeomor-
phism as well. �
Proposition 3.15. If p : E → B is an arc-hedgehog covering, then p is a disk-
hedgehog covering if and only if it is a disk covering.
Proof. It only suffices to consider the case p is a disk-covering (the other
implication is obvious). Given a map f : H → B from a disk-hedgehog to B and
given e ∈ E in the fiber of p over the base-point there is only one candidate for
the lift of f . That candidate must be continuous as otherwise we would generate a
map from an arc-hedgehog to B that has no lift at e. �
4. The whisker topology
In this section we are generalizing the whisker topology that was introduced in
[3] in a special case.
Definition 4.1. Let B be a space and b0 ∈ B. Suppose ∼ is an equivalence
relation on the set of loops in B at b0 which induces a group structure on the set
of equivalence classes via [α] · [β] := [α ∗ β] with the constant loop at b0 being the
neutral element and [α]−1 = [α−1] for all loops α, β at b0.
The above can be summarized as follows:
1. α ∼ β and γ ∼ ω implies α ∗ γ ∼ β ∗ ω for all loops α, β, γ, ω at b0,
2. α∗α−1 ∼ const and α ∼ α∗const for all loops α, where α−1 is the reversed
path of α.
The above equivalence relation can be extended to an equivalence relation on the
set of all paths in B originating at b0: α ∼ β means α(1) = β(1) and α∗β−1 ∼ const.
By the whisker topology on the space P (B, b0,∼) of equivalence classes [α] we
mean the topology with the basis N([α], U), U an open set in B containing α(1),
consisting of all [β] such that β ∼ α ∗ γ for some path γ in U
Theorem 4.2. a. P (B, b0,∼) is a Peano space and the end-point projection
p : P (B, b0,∼)→ B has arc-lifting property.
b. p is an arc-hedgehog covering if and only if it is an arc-covering.
c. p is a disk-hedgehog covering if and only if it is an arc-covering and α ∼
const for every loop α at b0 that is null-homotopic.
Proof. a. Notice λ ∈ N([α], U)∩N([β], V ) implies N([λ], U ∩V ) ⊂ N([α], U)∩
N([β], V ), so it is indeed a topology.
Given α at any point of B let αt be the path equal to α on the interval [0, t]
and then being a constant path. If γ is a path in U originating at α(1), then
each [α ∗ γt] ∈ N([α], U) and t → [α ∗ γt] is continuous (indeed, the inverse of
N([α ∗ γt], V ) contains the interval around t that is mapped under γ to V ). That
means P (B, b0,∼) is a Peano space. At the same time it implies p has arc-lifting
property.
b. Suppose p is an arc-covering, U is open in B, and α is a path in B starting at
b0 and ending at a point in U . It suffices to show N([α], U) is the path component of
p−1(U) containing [α]. Suppose γ̃ is a path in p−1(U) starting at [α]. Put γ = p◦ γ̃
and notice t → [α ∗ γt] is another lift of γ. Thus γ̃(t) = [α ∗ γt] for all t proving
that γ̃ is a path in N([α], U). In view of 3.5, p is an arc-hedgehog covering.
14 Coverings and fundamental groups: a new approach
8 JERZY DYDAK
c. Assume p is an arc-covering and α ∼ const for every loop α at b0 that is
null-homotopic. In view of b) and 3.15 it suffices to show p is a disk-covering.
Suppose f : D2 → B and α is a path in B from b0 to f(d) for some d ∈ D2.
Given x ∈ D2 let βx be a path in D2 from d to x. Define g(x) ∈ P (B, b0,∼) as
g(x) = [α ∗ (f ◦ βx)] and notice g(x) does not depend on the choice of βx. Given
a map u : H → D2 from an arc-hedgehog, g ◦ u is the lift of f ◦ u, hence it is
continuous. Therefore g is continuous. �
Here is an inner description of arc-hedghehog coverings:
Theorem 4.3. Suppose E is a Peano space. If p : E → B is an arc-hedgehog
covering and b0 ∈ B, then p is equivalent to the end-point projection P (B, b0,∼)→
B, where P (B, b0,∼) is equipped with the whisker topology.
Proof. Pick e0 ∈ E with p(e0) = b0 and declare two paths α and β in B
originating at b0 equivalent if α · b0 = β · b0.
Given a point x ∈ E choose a path αx in E from e0 to x and define h : E →
P (B, e0) by h(x) = [p ◦ αx].
Since h−1(N([αx], U)) is the path-component of p−1(U) containing x, it is open
in E and h is continuous.
If U is an open neighborhood of x in E, choose an open neighborhood V of
p(x) with the property that the path component W of x in p−1(V ) is contained in
U . Notice N([αx], V ) ⊂ h(W ) ⊂ h(U), so h is open. Since h is bijective, it is a
homeomorphism. �
5. Supremums of coverings
Two coverings p1 : E1 → B and p2 : E2 → B are said to be equivalent if there
is a homeomorphism h : E1 → E2 satisfying p2 ◦ h = p1. It turns out there is a
set of coverings over B such that any disk-hedgehog covering over B is equivalent
to one from that set. In that sense we may talk about the set of all disk-hedge
coverings over B.
In this section we define a partial order on the set of all disk-hedgehog coverings
over a fixed path-connected space B and we show this set has a maximum. That
maximum plays the role of the universal covering space.
Definition 5.1. Suppose E1, E2 are Peano spaces and p1 : E1 → B, p2 : E2 → B
are disk-hedgehog coverings. We define the inequality (p1, e1) ≥ (p2, e2) of pointed
coverings as follows: p1(e1) = p2(e2) and there is a continuous function f : E1 → E2
satisfying p2 ◦ f = p1 and f(e1) = e2.
We define the inequality of unpointed coverings p1 ≥ p2 as follows: for every
points e1 ∈ E1 and e2 ∈ E2 such that p1(e1) = p2(e2) we have (p1, e1) ≥ (p2, e2).
Lemma 5.2. If (p1, e1) ≥ (p2, e2) and (p2, e2) ≥ (p1, e1), then there is a homeo-
morphism h : E2 → E1 such that h(e2) = e1 and p1 ◦ h = p2.
Proof. Choose continuous functions f : E1 → E2 and g : E2 → E1 such that
p2 ◦ f = p1, p1 ◦ g = p2 and f(e1) = e2, g(e2) = e1. As p1 ◦ (g ◦ f) = p1 and
(g ◦ f)(e1) = e1, we get g ◦ f = idE1
. Similarly, f ◦ g = idE2
. �
Lemma 5.3. If p1 is a regular disk-hedgehog covering and (p1, e1) ≥ (p2, e2), then
p1 ≥ p2.
Jerzy Dydak 15
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 9
Proof. Choose a continuous function f : E1 → E2 such that p2 ◦f = p1. Notice
f is surjective. Given x1 ∈ E1 and x2 ∈ E2 satisfying p1(x1) = p2(x2) choose a
deck transformation h : E1 → E1 so that h(x1) ∈ f−1(x2) (see 3.9). Put g = f ◦ h
and notice p2 ◦ g = p1, g(x1) = x2. �
Corollary 5.4. p ≥ p if and only if p is regular.
Proof. In view of 5.3 it suffices to show p is regular if p ≥ p. That follows from
3.9 as any f : E → E satisfying p ◦ f = p must be a homeomorphism. �
Definition 5.5. Suppose {ps : Es → B}s∈S is a family of disk-hegehog coverings
of Peano spaces over a path-connected B and es ∈ Es so that ps(es) = b0 for all
s ∈ S. (p, e) is the supremum of {(ps, es)}s∈S if (p, e) ≥ (ps, es) for all s ∈ S and
(p, e) is the smallest pointed covering with that property.
Definition 5.6. Suppose {ps : Es → B}s∈S is a family of disk-hegehog coverings
of Peano spaces over a path-connected B and es ∈ Es so that ps(es) = b0 for all
s ∈ S.
The Peano fibered product of {(ps, es)}s∈S is the pair (p, e), where p : E → B,
e = {es}s∈S , and E is the Peanification of the path-component of e in the subset of∏
s∈S
Es consisting of points {xs}s∈S such that ps(xs) = pt(xt) for all s, t ∈ S. The
projection p is defined by p({xs}s∈S) = pt(xt) for any t ∈ S.
Proposition 5.7. Peano fibered product of a family of pointed disk-hedgehog cov-
erings is the supremum of that family.
Proof. If q : E′ → B and (q, e′) ≥ (ps, es) for all s ∈ S, then there are maps
gs : E′ → Es so that q = ps ◦ gs and gs(e
′) = es for each s ∈ S. The collection
{gs}s∈S induces a map g : E′ → E satisfying g(e′) = e and p ◦ g = q. Thus
(q, e′) ≥ (p, e).
Suppose b0 = p({es}s∈S), {es}s∈S ∈ E, and f : (H, 0) → (B, b0) is a map from
a disk-hedgehog. Create lifts fs : (H, 0) → (Es, es) of f with respect to ps. That
defines a map f : H → E by f(x) = {fs(x)}s∈S that is a lift of f with respect to p.
That proves existence of lifts - a proof of uniqueness is obvious. �
Proposition 5.8. If p : E → B is a disk-hedgehog covering and e0 ∈ E, then the
Peano fibered product of all p : (E, e)→ (B, p(e0)), e ranging over all points in the
fiber F of p containing e0, is regular.
Proof. Suppose α is a loop in B at b0 = p(e0) such that for some {xe}e∈F in
the fiber of q, α · {xe}e∈F = {xe}e∈F . That means α · xe = xe for all e ∈ F .
Since both {xe}e∈F and {e}e∈F can be joined by a path in the Peano fibered
product, there is a loop β at b0 in B such that β ·{e}e∈F = {xe}e∈F . Thus β ·e = xe
and (α ∗ β) · e = β · e for all e ∈ F . Plugging in β−1 · e ∈ F for e in the equation
(α ∗ β) · e = β · e gives α · e = e for all e ∈ F . That implies α · {ye}e∈F = {ye}e∈F
for all {ye}e∈F in the fiber of q, i.e. q is regular. �
Notice the Peano fibered product of all z → zn is the covering t→ exp(2πti) of
reals over the unit circle.
Corollary 5.9. Every path-connected space B has a maximal disk-hedgehog cover-
ing among those with total space being Peano. It is a regular covering.
Proof. Pick b0 and consider the space of paths P (B, b0) in B starting at b0.
For every disk-hedgehog covering p : E → B, E is an image of a function from
16 Coverings and fundamental groups: a new approach
10 JERZY DYDAK
P (B, b0) obtained by lifting paths (the lifts start at e0 ∈ p−1(b0). That means
there is a set {ps : Es → B}s∈S of disk-hedgehog coverings with the property that
for any disk-hedgehog covering p : E → B there is s ∈ S and a homeomorphism
h : E → Es such that p = ps ◦ h. We only consider disk-hedgehog with Peano total
space. Take the Peano fibered product of {ps : Es → B}s∈S . It must be a regular
disk-hedge covering but it is easier to use 5.8 and produce the maximal covering
that is regular. �
6. Hedgehog fundamental group
Definition 6.1. Given a path-connected space B and b0 ∈ B define the hedgehog
fundamental group π(B, b0) of (B, b0) as the monodromy group π(p, b0), where
p : E → B is the maximal disk-hedgehog covering over B.
Proposition 6.2. Any map f : B1 → B2 of path-connected spaces induces a natural
homomorphism from π(B, b1) to π(B2, f(b1)).
Proof. Let f(b1) = b2. Consider the maximum disk-hedgehog covering p2 :
E2 → B2 and pick e2 ∈ p−12 (b2). Take the path-component of (b1, e2) in {(x, y) ∈
B1 × E2|f(x) = p2(y)}, Peanify it to get E and let q : E → B1 be the projection
onto the first coordinate. Notice q is a disk-hedgehog covering. Let p : E1 → E be
the maximum disk-hedgehog covering over E. Notice p1 = q ◦ p is the maximum
disk-hedgehog covering over E1. If a loop α in B1 at b1 has all lifts to E1 that
are loops, then all lifts of α to E must be loops. Given a lift β in E2 of f ◦ α,
the map t → (α(t), β(t)) is a lift of α in E. As it is a loop, β must be a loop as
well. Consequently, if two loops in B1 at b1 are similar, so are their images in B2
which is sufficient to conclude there is a natural homomorphism from π(B, b1) to
π(B2, f(b1)). �
Proposition 6.3. If p : E → B is a regular disk-hedgehog covering and p(e0) = b0,
then one has a natural exact sequence
1→ π(E, e0)→ π(B, b0)→ π(p, b0)→ 1
Proof. Choose a maximal disk-hedgehog covering p1 : E1 → E over E, where
E1 is a Peano space. Notice p ◦ p1 is a maximal disk-hedgehog covering over B.
The kernel of π(B, b0)→ π(p, b0) consists exactly of loops whose all lifts to E are
loops. In particular, the kernel is contained in the image of π(E, e0) → π(B, b0).
Obviously, the image of π(E, e0)→ π(B, b0) is contained in that kernel.
Any loop in E at e0 that becomes trivial in π(B, b0) must have all lifts in E1 as
loops. That means π(E, e0)→ π(B, b0) is a monomorphism. �
Theorem 6.4. Suppose p : E → B is a disk-hedgehog covering of path connected
spaces. Suppose f : X → B is a map from a Peano space, x0 ∈ X and e0 ∈ E with
f(x0) = b0 = p(e0). f has a lift g : (X,x0) → (E, e0) if and only if the image of
π(X,x0)→ π(B, b0) is contained in the image of π(E, e0)→ π(B, b0).
Proof. Only one implication is of interest, so assume the image of π(X,x0) →
π(B, b0) is contained in the image of π(E, e0)→ π(B, b0).
Given a point x ∈ X pick a path αx in X from x0 to x and define g(x) as αx · e0.
g(x) does not depend on the choice of αx: choosing a different path βx leads to a
loop γ in E at e0 such that [βx∗α−1x ] = [p◦γ] in π(B, b0). Therefore βx ∼ (p◦γ)∗αx
and βx · e0 = ((p ◦ γ) ∗ αx) · e0 = αx · e0 = g(x).
Jerzy Dydak 17
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 11
Given any map q : H → X from a disk-hedgehog H to X, the composition
g ◦ q : H → E is the only possible lift of f ◦ q, hence it is continuous. By 3.3, g is
continuous. �
Corollary 6.5. Suppose p : E → B is a disk-hedgehog covering with E being
Peano and e0 ∈ E. π(E, e0) = 0 if and only if p : E → B is the maximal disk
hedgehog-covering over B.
Proof. If p is maximal, then E does not admit any non-trivial disk-hedgehog
covering and π(E, e0) = 0. If π(E, e0) = 0, then given any other disk-hedgehog
covering q : E1 → B there is a lift g : E → E1 of q proving p is maximal. �
7. Comparison to the classical fundamental group
As the natural homomorphism π1(B, b0) → π(B, b0) is an epimorphism, there
are two natural questions:
Problem 7.1. Characterize the kernel of π1(B, b0)→ π(B, b0) for path-connected
spaces B.
Problem 7.2. Characterize path-connected spaces B such that π1(B, b0)→ π(B, b0)
is an isomorphism.
Since the identity map P (B)→ B from the Peanification of B to B induces iso-
morphisms of both the classical fundamental group and the hedgehog fundamental
group, we will consider both Problems 7.1 and 7.2 for Peano B spaces only. In
particular, we differ with [10] in that regard.
Recall B is shape injective if the natural homomorphism π1(B, b0)→ π̌1(B, b0)
from the classical fundamental group to the Čech fundamental group is a monomor-
phism. Papers [11], [7, Corollary 1.2 and Final Remark], [6], and [9] contain results
that various classes of spaces are shape injective. We will generalize the concept of
shape injectivity as follows:
Definition 7.3. B is residually Poincaré if for every loop α in B that is not
null-homotopic there is a map f : B → P such that P is a Poincaré space and f ◦α
is not null-homotopic.
Proposition 7.4. If B is residually Poincaré, then π1(B, b0) → π(B, b0) is an
isomorphism.
Proof. Clearly, it is so if B is a Poincaré space as it has the classical universal
cover that is simply connected. Given a non-trivial element [α] ∈ π1(B, b0) choose
f(B, b0)→ (P, p0) such that f ◦α is not null-homotopic. If α represents the neutral
element of π(B, b0), then [f ◦ α] is neutral in π(P, p0) = π1(P, p0), a contradiction.
�
Theorem 7.5. Suppose U is an open cover of a paracompact space B consisting of
path-connected sets. If, for each x ∈ B, the inclusion st(x,U)→ B of the star of U
at x induces the trivial homomorphism of π(st(x,U), x) → π(B, x), then π(B, b0)
is isomorphic to the fundamental group of the nerve of U for all b0 ∈ B.
Proof. Pick V0 ∈ U containing b0. For each V ∈ U pick bV ∈ V (bV = b0 if
V = V0).
18 Coverings and fundamental groups: a new approach
12 JERZY DYDAK
Define a map α from the 1-skeleton of the nerve N(U) to B as follows: each
vertex V of the nerve is mapped to bV and each edge VW is mapped to a path
αVW in V ∪W joining bV and bW .
Given an edge-path in the nerve from V0 to V followed by a loop around a triangle
that belongs to the nerve, then followed back by the path-edge results in a loop
that is mapped to the star st(bV ,U) of bV in U , hence α induces a homomorphism
j from π1(N(U), V0) to π(B, b0).
Given a loop λ in B at b0, we can represent it as the concatenation of paths γi,
0 ≤ i ≤ n, such that the carrier of γi is contained in V (i) ∈ U , and V (0) = V0 =
V (n). Pick a path ωi in V (i) joining γi(1) and bV (i). Notice each γi is equivalent to
ωi−1 ∗ αV (i−1)V (i) ∗ ω−1i , so replacing it by that path results in a loop in the image
of j that is equivalent to λ. That proves j is an epimorphism.
To show it is a monomorphism, assume there is an edge-loop in the nerve that
is mapped to a loop in B being trivial in π(B, b0). Choose a partition of unity
φ : B → N(U) sending b0 to V0 ∈ U . The composition of j : π(N(U)) → π(B, b0)
and the homomorphism induced by φ is the identity.
Indeed, for each V ∈ U choose a path βV in N(U) from φ(bV ) to V that lies
in the open star st(V ) of V in N(U). Notice, if V ∩ W 6= ∅, then βV ∗ VW ∗
β−1W ∗ (φ(αVW ))−1 lies in the union st(V ) ∪ st(W ) of open stars in N(U). As their
intersection is contractible, the union is simply connected and the composition of
j : π(N(U))→ π(B, b0) and the homomorphism induced by φ is the identity. �
Corollary 7.6. If B is a paracompact Peano space and π(B, b) is discrete for all
b ∈ B, then for every sufficiently small open cover U of B, π(B, b) is isomorphic
to the fundamental group of the nerve of U for all b ∈ B.
Proof. By 3.14 every point b ∈ B has a path-connected neighborhood Ub such
that the maximal disk-hedgehog covering p : E → B has a section over Ub. That
implies π(Ub, b) → π(B, b) is trivial. Choose a star-refinement V of {Ub}b∈B and
apply 7.5 to any refinement U of V. �
Let us show that the analog of the famous result of Shelah [19] (see also [18])
stating that the fundamental group of a Peano continuum is finitely generated if it
is countable not only holds for the hedgehog fundamental groups but it also has a
much simpler proof.
Corollary 7.7. Suppose B is a Peano continuum. If π(B, b0) is countable for some
b0 ∈ B, then it is finitely presented.
Proof. Consider the maximal disk-hedgehog covering p : E → B. 3.11 says its
fibers are Baire spaces. Since they are countable, they must be discrete. Apply 7.6.
�
Let’s turn to Problem 7.1. First, let us show that every small loop belongs to
the kernel of π1(B, b0)→ π(B, b0). It shows that the hedgehog fundamental group
eliminates some of the pathologies of the classical fundamental group.
Recall (see [21]) that a loop α at b0 in B is called small if it can be homotoped
relative to b0 into any neighborhood U of b0 in B.
Proposition 7.8. Suppose B is path-connected. If p : E → B is a disk-hedgehog
covering, then [α] is the neutral element of π(p, b0) for every small loop α at b0.
Jerzy Dydak 19
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 13
Proof. We may assume E is Peano by switching to its Peanification. Suppose
α is a small loop at b0 in B so that [α] is not the neutral element of π(p, b0). There
is a lift α̃ of α with e0 = α̃(0) 6= e1 = α̃(1).
Choose a path-connected neighborhood U of b0 in B such that the path com-
ponent V of e0 in p−1(U) is different from path-component W of e1 in p−1(U).
Suppose there is a loop β in U homotopic to α rel.b0 in B. Its lift β̃ would join e0
and e1, a contradiction. �
Let’s consider a more general question than 7.1: Characterize kernels of π1(B, b0)→
π(p, b0), where p : E → B is a disk hedgehog covering over a Peano space B.
As in [20, p.81], given an open cover U of X, π(U , x0) is the subgroup of π1(X,x0)
generated by elements of the form [α ∗ γ ∗ α−1], where γ is a loop in some U ∈ U
and α is a path from x0 to γ(0).
Lemma 7.9. Suppose P (B, b0,∼) has a whisker topology such that α ∼ const
implies [α] ∈ π(U , b0) for some open cover U of B. If β(t), t ∈ [0, 1], are paths in
P (B, b0,∼) forming a lift of a path γ starting at [α], then β(t)∗γ−1t ∗α−1 ∈ π(U , b0)
for all t ∈ [0, 1].
Proof. Let S = {t ∈ [0, 1]|β(t) ∗ γ−1t ∗ α−1 ∈ π(U , b0)}. Clearly, 0 ∈ S. It
suffices to show that for any t ∈ S, t < 1, there is s > t such that [t, s] ⊂ S and
that S contains its supremum. Given t ∈ S, t < 1, pick V ∈ U containing γ(t) and
choose a closed interval W = [t, u] in [0, 1], u > t, such that β(s) ∈ N(β(s), V ) for
s ∈ W and γ(s) ∈ V for s ∈ W . Therefore, given s ∈ W , there is a path ω in V
satisfying β(s) ∼ β(t) ∗ ω. Notice ω joins γ(t) and γ(s).
The loop λ = ω∗(γ|[t, s])−1 lies in V and β(s)∗γ−1s ∗α−1 ∼ β(t)∗ω∗γ−1s ∗α−1 ∼
β(t)∗λ∗(γ|[t, s])∗γ−1s ∗α−1 ∼ β(t)∗λ∗γ−1t ∗α−1 ∼ (β(t)∗γ−1t ∗α−1)∗α∗γt∗λ∗γ−1t ∗α−1
and the last loop belongs to π(U , b0).
The same argument proves that the supremum of S belongs to S (we only used
that s and t are sufficiently close). �
Proposition 7.10. Let B be a Peano space. If p : E → B is a covering projection,
then the kernel of π1(B, b0) → π(p, b0) contains π(U , b0), where U consists of all
open subsets U of B that are evenly covered.
Given an open cover U of B, the set of covering projections q : E → B for which
each U ∈ U is evenly covered has a maximum p and the kernel of π1(B, b0) →
π(p, b0) is exactly π(U , b0).
Proof. Obviously, elements of the form [α ∗ γ ∗ α−1], where γ is a loop in some
U ∈ U and α is a path from b0 to γ(0) have a lift to E that is a loop, so they are
trivial in π(p, b0).
Consider the end-point projection p : P (B, b0,∼) → B (α ∼ β if and only if
[α ∗ β−1] ∈ π(U , b0)). It is a classical covering with each member of U being evenly
covered (see [3] or use 7.9 to deduce it has unique path-lifting property and then
construct sections over members of U). Notice the kernel of π1(B, b0) → π(p, b0)
is exactly π(U , b0). Indeed, if a loop γ in B lifts to a loop in P (B, b0,∼), then 7.9
says the loop must belong to π(U , b0).
Given any classical covering projection q : E → B with each member of U being
evenly covered one can construct f : P (B, b0,∼)→ E such that q ◦ f = p by lifting
paths. That proves maximality of p. �
Definition 7.11. The intersection of all π(U , b0), U ranging over all open covers
of B, is called the Spanier group of (B, b0) (see [10]).
20 Coverings and fundamental groups: a new approach
14 JERZY DYDAK
By a medium loop we mean a loop α at b0 that is not small and its homotopy
class [α] belongs to the Spanier group. By a big loop we mean a loop α at b0 that
is neither medium nor small.
Proposition 7.12. Let B be a Peano space. If p is the supremum of all classical
coverings over B, then the kernel of π1(B, b0) → π(p, b0) is exactly the Spanier
group.
Proof. Consider the end-point projection p : P (B, b0,∼) → B (α ∼ β if and
only if [α ∗ β−1] ∈ π(U , b0) for all open covers U of B). Use 7.9 to deduce it has
unique path-lifting property and then use 4.2 to show it is a disk-hedgehog covering.
Notice the kernel of π1(B, b0) → π(p, b0) is exactly the Spanier group. Indeed, if
a loop γ in B lifts to a loop in P (B, b0,∼), then 7.9 says the loop must belong to
π(U , b0) for all open covers U of B.
Given any classical covering projection q : E → B with each member of U being
evenly covered one can construct f : P (B, b0,∼)→ E such that q ◦ f = p by lifting
paths.
Suppose q : E → B, E Peano, is a disk-hedgehog covering with q(e) = b0 and
maps fU : E → P (B, b0,∼U ) such that fU ◦ pU = q for each open cover U of B.
Here α ∼U β if and only if [α ∗ β−1] ∈ π(U , b0) and pU is the end-point projection.
Given x ∈ E and two path αx, βx from e to x, the loop αx ∗ β−1x must belong
to the Spanier group as it can be factored through all P (B, b0,∼U ), therefore the
function f(x) = [γαx] (γ a fixed loop at b0 in B) is well-defined and is continuous
as p is an arc-hedgehog covering. As p ◦ f = q, q ≥ p. That proves maximality of
p. �
Corollary 7.13. The the kernel of π1(B, b0) → π(B, b0) contains all small loops
and is contained in the union of small loops and medium loops.
Let us show how direct wedge can be used to construct interesting spaces.
First of all, one can change the topology of the standard arc-hedgehog
∨
n∈N
(In, 0n)
by requiring open neighborhoods of the base-point to contain all but finitely many
0n’s (instead of all but finitely many In’s) and get a connected space that is not
locally connected (a modified topologist’s sine curve).
Second, one can change the topology of the standard disk-hedgehog
∨
n∈N
(D2
n, 0n)
by requiring open neighborhoods of the base-point to contain all but finitely many
∂D2
n’s (instead of all but finitely many D2
n’s) and get a space with properties similar
to Harmonic Archipelago [2]: every loop is small.
It is easy to construct examples of medium loops by connecting two Harmonic
Archipelagos by an arc. However, there is a more interesting example of Fischer-
Zastrow [12] that can be used for that purpose. What is not clear is if that example
does not become trivial once we kill all small loops.
Problem 7.14. Construct a medium loop in a Peano space that does not belong to
the normalizer of all small loops.
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Jerzy Dydak 21
COVERINGS AND FUNDAMENTAL GROUPS: A NEW APPROACH 15
[3] N.Brodskiy, J.Dydak, B.Labuz, A.Mitra, Covering maps for locally path-connected spaces,
http://front.math.ucdavis.edu/0801.4967
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University of Tennessee, Knoxville, TN 37996
E-mail address: dydak@math.utk.edu
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| id | oai:trim.imath.kiev.ua:article-270 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:04:43Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
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| spelling | oai:trim.imath.kiev.ua:article-2702018-02-10T20:56:26Z Coverings and fundamental groups: a new approach Накриття та фундаментальні групи: новий підхід Dydak, J. Dydak, J. Classical fundamental groups behave reasonably well for Poincare spaces (i.e.. semy-locally simply connected spaces). One has a construction of the universal covering for such spaces. For arbitrary spaces it is a different matter.&nbsp;We define monodromy groups $\pi (p,b_{0})$ for any map $p: E \rightarrow B$ with the path lifting property and any $b_{0} \in B$. $p$ is called a $\mathcal{P}$-covering, where $\mathcal{P}$ isa class of Peano spaces (i.e., connected and locally path connected spaces), if&nbsp;it has existence and uniqueness of lifts of maps $f: X \rightarrow B$ for any $X \in \mathcal{P}$.&nbsp;For any $B$ there is the maximal $\mathcal{P}$-covering $p_{\mathcal{P}}:B_{\mathcal{P}}\rightarrow B$ and its monodromy group is called the $\mathcal{P}$-fundamental group of $(B,b_{0})$. In case of $\mathcal{P}$ consisting of all disk-hedgehogs we construct a universal covering theory of all spaces in analogy to the classical covering theory of Poincare spaces. Класичні фундаментальні групи достатньо добре себе поводять для просторів Пуанкаре (напівлокальних однозв'язних просторів). Можна побудувати універсальне накриття для таких просторів. Інша справа для довільних просторів. Ми визначаємо монодромні групи $ \pi (p, b_ {0}) $ для будь-якого відображення $ p: E \rightarrow B $ з властивістю підняття шляху та будь-якого $ b_ (0) \in B $. $ p $ називається $ \mathcal {P} $- накриттям, де $ \mathcal (P) $&nbsp;клас просторів Пеано (тобто зв'язних і локально лінійно звя'зних просторів), якщо в ньому існує єдине відображення підніття $ f: X \rightarrow B $ для будь-якого $ X \in \mathcal (P) $.Для будь-якого $ B $ існує максимальне $ \mathcal {P} $, що покриває $p_{\mathcal{P}}:B_{\mathcal{P}}\rightarrow B$ і його група монодромій називається $ \mathcal {P} $ - фундаментальною групою $ (B, b_ (0)) $. У випадку $ \mathcal (P) $, що складається з усіх диск-їжаків, ми побудуємо універсальну теорію накриття всіх просторів за аналогією з класичною теорією накриття просторів Пуанкаре. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/270 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 7-21 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 7-21 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 7-21 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/270/277 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Dydak, J. Dydak, J. Coverings and fundamental groups: a new approach |
| title | Coverings and fundamental groups: a new approach |
| title_alt | Накриття та фундаментальні групи: новий підхід |
| title_full | Coverings and fundamental groups: a new approach |
| title_fullStr | Coverings and fundamental groups: a new approach |
| title_full_unstemmed | Coverings and fundamental groups: a new approach |
| title_short | Coverings and fundamental groups: a new approach |
| title_sort | coverings and fundamental groups: a new approach |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/270 |
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