A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

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A simply connected universal ﬁbration with

unique path lifting over a Peano continuum with

non-simply connected universal covering space

JEREMY BRAZAS AND HANSPETER FISCHER

Abstract. We present a 2-dimensional Peano continuum T⊆R3

with the following properties: (1) There is a universal covering

projection q:T→Twith uncountable fundamental group π1(T);

(2) For every 1 ̸= [α]∈π1(T,∗), there is a covering projection

r: (E, e)→(T,∗) such that [α]̸∈ r#π1(E, e); (3) There is no

universal covering projection r:E→T; (4) The universal object

p:e

T→Tin the category of ﬁbrations with unique path lifting

(and path-connected total space) over Thas trivial fundamental

group π1(e

T) = 1; (5) p:e

T→Tis not a path component of an

inverse limit of covering projections over T.

1. introduction

Many foundational results of covering space theory can be developed

in the more general context of (Hurewicz) ﬁbrations with unique path

lifting. This is done, for example, in Spanier’s classical textbook [9],

where in §2.5 one also ﬁnds a sketch of the proof of the existence

of a universal object p:e

X→Xin the category of ﬁbrations with

unique path lifting (and path-connected total space) over a ﬁxed path-

connected base space Xand a proof of the following inclusion, assuming

Xis also locally path connected:

(S)p#π1(e

X, ex)⩽\

U∈Cov(X)

π(U, x)

Here, Cov(X) denotes the collection of all open covers of Xand π(U, x)

denotes the subgroup of the fundamental group π1(X, x) generated by

all elements of the form [γ·δ·γ−] for some path γ: ([0,1],0) →(X, x),

its reverse γ−(t) = γ(1−t), and some loop δ: ([0,1],{0,1})→(U, γ(1))

Date: May 10, 2024.

2020 Mathematics Subject Classiﬁcation. Primary 55R05; Secondary 57M10.

arXiv:2210.12567v2 [math.AT] 17 Aug 2024

2 JEREMY BRAZAS AND HANSPETER FISCHER

in some U∈ U. We will denote this intersection by

πs(X, x) := \

U∈Cov(X)

π(U, x)

and refer to it as the Spanier group of Xat x. Since p#is injective, e

is simply connected if πs(X, x) is trivial.

Given a connected and locally path-connected space X, we recall

from [9, 2.5.11 & 2.5.13] that for a loop α: ([0,1],{0,1})→(X, x),

we have that [α]∈πs(X, x) if and only if for every covering projection

q: (E, e)→(X, x), the unique lift α′: ([0,1],0) →(E, e) of αwith

q◦α′=αis a loop, i.e. α′(1) = e.

This raises the following natural question.

Question 1.1. Does equality hold in (S)?

Since ﬁbrations have continuous lifting of paths, it seems reasonable

to speculate that the answer to Question 1.1 is “yes”, as was recently

claimed in [4]. However, we will show that this is not the case, in

general, by presenting a Peano continuum Tfor which πs(T,∗) is un-

countable, although the universal object p:e

T→Tin the category

of ﬁbrations with unique path lifting (and path-connected total space)

over Tsatisﬁes p#π1(e

T,∗) = 1; in fact, Tadmits a (categorical) univer-

sal covering projection q: (T,∗)→(T,∗) with q#π1(T,∗) = πs(T,∗)

and πs(T,∗) = 1, while no universal covering projection over Texists.

To put this result into perspective, we mention that the Spanier

group πs(X, x) is always contained in the kernel of the canonical ho-

momorphism π1(X, x)→ˇπ1(X, x) to the ﬁrst ˇ

Cech homotopy group [8],

which is injective for spaces that are one-dimensional [6] or planar [7],

and it equals this kernel when Xis locally path connected and metriz-

able [1]. Therefore, πs(X, x) = 1 if Xis either one-dimensional or pla-

nar. Our Peano continuum is a 2-dimensional subset of R3. Also, if Xis

locally path connected, semilocally simply connected, and metrizable,

then π1(X, x)→ˇπ1(X, x) is an isomorphism [8]. Our Peano continuum

contains a π1-injectively embedded shrinking wedge of countably many

circles.

In contrast, the universal ﬁbration with unique path lifting p:e

X→

Xover the Griﬃths space Xsatisﬁes p#π1(e

X, ex) = πs(X, x) = π1(X, x)

with e

X=Xand p=idX[9, 2.5.18], and H1(X) = ZN/LNZ[5].

For a comparison with other universal simply connected lifting ob-

jects, see Remark 5.2.

A SIMPLY CONNECTED UNIVERSAL FIBRATION 3

2. The space Tand its universal covering q:T→T

For each m∈N, let Cm⊆R2be the circle of radius 1

mcentered at

m,0) and consider the “inﬁnite earring” E=Sm∈NCm⊆R2with

basepoint ∗= (0,0) ∈E. Let ℓm: [0,1] →Cmbe the loop at ∗deﬁned

ℓm(t)=(1

m(1 −cos 2πt),1

msin 2πt))

which traverses Cmonce clockwise. Let f:E→Ebe the unique “circle

shifting” map with

f◦ℓm=ℓm+1

for all m∈N. Let Tbe the mapping torus of f:E→E. That is, let T

be the quotient space E×[0,1]/∼with (x, 1) ∼(f(x),0) for all x∈E.

Let h:E×[0,1] →Tdenote the quotient map. We will identify E

with h(E× {0})⊆Tand let ι: (E,∗)→(T,∗) denote inclusion. Let

η: [0,1] →E×[0,1] be given by η(t) = (∗, t) and put λ=h◦η. Then

[λ]∈π1(T,∗).

Lemma 2.1. The space Tis a Peano continuum.

Proof. Note that Tcan be realized in R3as the quotient of the Peano

continuum E×[0,1]. □

For each i∈Z, let Xibe a copy of Eand let fi=f:Xi→Xi+1.

Let Tbe the mapping telescope of the bi-inﬁnite sequence

· · · f−2

−→ X−1

f−1

−→ X0

−→ X1

−→ X2

−→ · · · .

That is, let T=Pi∈ZXi×[0,1]/∼be the quotient of the disjoint

union of the spaces Xi×[0,1] with (x, 1) ∼(fi(x),0) for all i∈Zand

x∈Xi. Let h:Pi∈ZXi×[0,1] →Tdenote the quotient map. We

will identify each Xiwith h(Xi× {0})⊆T. As the basepoint for Twe

take ∗ ∈ X0.

Let q: (T,∗)→(T,∗) be the unique map making the following

diagram commute, where Pi∈Zid|Xi×[0,1] =idE×[0,1] for all i∈Z:

Pi∈ZXi×[0,1] Pi∈Zid //



E×[0,1]



Tq//T

Observe that q:T→Tis a covering projection. (For every t∈(0,1),

the set U=h(E×([0,1] \ {t})) is open in T, its preimage q−1(U) equals

the union of the disjoint open subsets Vi=h((Xi−1×(t, 1])∪(Xi×[0, t)))

of Twith i∈Z, and q|Vi:Vi→Uis a homeomorphism for all i∈Z.)

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摘要：

AsimplyconnecteduniversalfibrationwithuniquepathliftingoveraPeanocontinuumwithnon-simplyconnecteduniversalcoveringspaceJEREMYBRAZASANDHANSPETERFISCHERAbstract.Wepresenta2-dimensionalPeanocontinuumT⊆R3withthefollowingproperties:(1)Thereisauniversalcoveringprojectionq:T→Twithuncountablefundamentalgrou...

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A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space

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