This paper constructs a space where the canonical homomorphism from the fundamental group to the cech homotopy group is not injective, despite the space having many properties that typically relate these groups.
Contribution
It provides a counterexample demonstrating the failure of the first cech homotopy group to detect certain fundamental group elements, challenging assumptions about their relationship.
Findings
01
The space onstructed has a non-injective canonical homomorphism.
02
emonstrates the failure of cech homotopy group to register all fundamental group elements.
03
The space exhibits properties like being homotopically Hausdorff and having a simply connected generalized covering space.
Abstract
We construct a space P for which the canonical homomorphism π1(P,p)→πˇ1(P,p) from the fundamental group to the first \v{C}ech homotopy group is not injective, although it has all of the following properties: (1) P∖{p} is a 2-manifold with connected non-compact boundary; (2) P is connected and locally path connected; (3) P is strongly homotopically Hausdorff; (4) P is homotopically path Hausdorff; (5) P is 1-UV0; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π1(P,p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) π1(P,p) is locally free.
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Full text
On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
Jeremy Brazas
Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, USA
We construct a space P for which the canonical homomorphism π1(P,p)→πˇ1(P,p) from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) P∖{p} is a 2-manifold with connected non-compact boundary; (2) P is connected and locally path connected; (3) P is strongly homotopically Hausdorff; (4) P is homotopically path Hausdorff; (5) P is 1-UV0; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π1(P,p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) π1(P,p) is locally free.
Key words and phrases:
First Čech homotopy group; first shape group; strongly homotopically Hausdorff; homotopically path Hausdorff; 1-UV0; generalized covering projection; discrete monodromy property; inverse limit of free monoids
2010 Mathematics Subject Classification:
55Q07, 55Q05, 57M12, 57M05
1. Introduction
The geometric significance of the elements of the fundamental group π1(X,x) of a connected and locally path-connected metric space X is most prominently on display in the context of a simply connected covering space (if it exists) where these elements comprise the group of deck transformations of the covering projection. In this situation, the canonical homomorphism π1(X,x)→πˇ1(X,x) to the first Čech homotopy group (also called the first shape group [26]) is an isomorphism [22].
In fact, as long as all fundamental group elements are accounted for, that is, if π1(X,x)↪πˇ1(X,x) is injective, the standard covering construction yields a generalized covering projection p:X→X with connected, locally path-connected and simply connected X [22]. It is characterized by the usual unique lifting property and we have π1(X,x)≅Aut(X→pX). Examples of spaces for which π1(X,x)↪πˇ1(X,x) is injective include all one-dimensional separable metric spaces [11, 17], all planar spaces [21], the Pontryagin sphere, the Pontryagin surface Π2, and similar inverse limits of higher-dimensional manifolds [19].
Several weaker properties that quantify the geometric relevance of the elements of π1(X,x) can be found in the literature. The strongly homotopically Hausdorff property, for example, stipulates that for every essential loop in X there should be a limit to how small it can be made at a particular point by a free homotopy [9]. The homotopically path Hausdorff property, on the other hand, calls for π1(X,x) to be T1 in the quotient topology induced by the compact-open topology on the loop space Ω(X,x) [2]. Both of these properties are implied by π1(X,x)↪πˇ1(X,x) being injective [20].
Then there are properties that guarantee, in and of themselves, the existence of a simply connected generalized covering space. These include
the homotopically path Hausdorff property above and the 1-UV0 property, which requires small null-homotopic loops to contract via small homotopies [4, 20].
Even if a generalized covering projection with simply connected domain exists, it might not be a fibration and the monodromies between fibers might not be continuous [18]. (Such is the case for the Hawaiian Earring.) However, for all one-dimensional spaces and for all planar spaces, these monodromies have discrete graphs; a fact implicitly used in the work of Eda [14, 15] and Conner-Kent [8]. If any two spaces with this property (cf. Definition 9.2) are homotopy equivalent, then their respective wild sets (points at which they are not semilocally simply connected) are homeomorphic (Theorem 9.15).
As far as the algebraic structure of fundamental groups of low-dimensional spaces is concerned, we recall that the fundamental group of an arbitrary planar Peano continuum (not necessarily homotopy equivalent to a one-dimensional space) is isomorphic to a subgroup of the fundamental group of some one-dimensional planar Peano continuum [6].
In turn,
fundamental groups of one-dimensional path-connected separable metric spaces are locally free [11] and, in the compact case, have structures similar to that of the Hawaiian Earring H, where the injective function π1(H)↪πˇ1(H) factors through the limit M of an inverse system M1⟵R1M2⟵R2M3⟵R3⋯ of free monoids Mn on {ℓ1±1,ℓ2±1,⋯,ℓn±1} with Rn−1 deleting the letters ℓn and ℓn−1 from every word [12, 24].
This raises the following
Question:Is there a space X for which π1(X)→πˇ1(X) is not injective, but with all of the other properties discussed above: low-dimensional, strongly homotopically Hausdorff, homotopically path Hausdorff, 1-UV0, admits a generalized universal covering p:X→X whose monodromies between fibers have discrete graphs, admits a natural injective function π1(X)→M, and has locally free fundamental group?
We present a relatively simple and prototypical construction of a two-dimensional space that yields a positive answer to this question.
Specifically, in Section 2, we define the space P mentioned in the abstract, by attaching countably many “pairs of pants” to the Hawaiian Earring H, and identify its fundamental group as a direct limit of groups each isomorphic to π1(H) with injective bonding homomorphisms. In particular, π1(P) is locally free (Proposition 2.6).
We show that π1(P)→πˇ1(P) is not injective (Theorem 3.6), but that
P is both strongly homotopically Hausdorff (Theorem 4.5) and homotopically path Hausdorff (Theorem 6.4); as far as the authors know, it is the first such example (Remarks 4.2 and 6.3). The proof of the latter property hinges on the fact that π1(P) naturally injects into the inverse limit of monoids associated with the Hawaiian Earring(Theorem 5.3), despite π1(P) not being isomorphic to a subgroup of an inverse limit of free groups (Remark 5.4). Moreover, we show that P is 1-UV0 (Theorem 7.3) and that π1(P) is locally free (Proposition 2.6).
After a brief review of generalized covering space theory (Section 8) we show that the monodromies for the simply connected generalized covering space of P have discrete graphs (Theorem 9.14). We also discuss some general aspects of this property, such as its relationship to the homotopically Hausdorff property relative to a subgroup of the fundamental group (Proposition 9.11) and its impact on the stability of wild subsets under homotopy equivalence (Theorem 9.15).
2. The Hawaiian pants P
Let Cn⊆R2 be the circle of radius n1 centered at (n1,0) and let H=⋃i=1∞Ci⊆R2 be the usual Hawaiian Earring with basepoint b0=(0,0). Define ℓn:[0,1]→Cn by ℓn(t)=(n1(1−cos2πt),n1sin2πt)).
For n∈N, let Dn,1 and Dn,2 be two disjoint disks in the interior Dn∘ of a disk Dn⊆R2 and consider the “pair of pants” Pn=Dn∖(Dn,1∘∪Dn,2∘). Let αn,βn,γn:[0,1]→Pn be parametrizations of the boundaries ∂Dn, ∂Dn,1, and ∂Dn,2, respectively, with clockwise orientation.
Let P be the space obtained from H by attaching all Pn via maps fn:∂Pn→H such that fn∘αn=ℓn, fn∘βn=ℓ2n and fn∘γn=ℓ2n+1. That is, we put P=H∪f(∐n∈NPn) where f:∐n∈N∂Pn→H and f∣∂Pn=fn. We refer to P as the “Hawaiian Pants”. (See Figure 2.)
We observe that exactly one pair of pants is attached to C1, namely P1 via f1∣∂D1, and that for each n⩾2, exactly two pairs of pants are attached to Cn, namely Pn via fn∣∂Dn, and either Pn/2 via fn/2∣∂Dn/2,1 (if n is even) or P(n−1)/2 via f(n−1)/2∣∂D(n−1)/2,2 (if n is odd). In particular:
Proposition 2.1**.**
(a)
P∖{b0}* is a 2-manifold with boundary C1∖{b0}.*
(b)
P* is connected and locally path connected.*
We have fn−1(b0)={an,bn,cn} with an∈∂Dn, bn∈∂Dn,1 and cn∈∂Dn,2.
Choose arcs An,1,An,2⊆Pn such that An,1∩∂Pn=∂An,1={an,bn}, An,2∩∂Pn=∂An,2={an,cn}, and An,1∩An,2={an}, configured as in Figure 2. Let Bn,j be the image of An,j in P when attaching Pn. Then Bn,j is homeomorphic to a circle. We regard Pn∘=Dn∘∖(Dn,1∪Dn,2) as a subspace of P.
Let Hn=⋃i=n∞Ci and let Pn be the space obtained from Hn by attaching Pn,Pn+1,Pn+2,…. Define Pn+=⋃i=1n−1(Bi,1∪Bi,2)∪Pn for n⩾2 and put P1+=P1=P. Then P=P1+⊇P2+⊇P3+⊇⋯.
Since each Pn deformation retracts onto ∂Dn,1∪An,1∪An,2∪∂Dn,2, there are deformation retractions ϕn:Pn+×[0,1]→Pn+ such that
(i)
for all p∈Pn+, we have ϕn(p,0)=p;
(ii)
for all p∈Pn+, we have ϕn(p,1)∈Pn+1+;
(iii)
for all p∈Pn+1+ and all t∈[0,1], we have ϕn(p,t)=p;
(iv)
for all p∈Pn+∖Pn+1+ and all t∈[0,1], we have that ϕn(p,t) lies in the image of Pn in the quotient Pn+.
Put Hn,k=⋃i=nkCi (where Hn,k=∅ for k<n) and Hn,k+=⋃i=1n−1(Bi,1∪Bi,2)∪Hn,k. Likewise, put Hn+=⋃i=1n−1(Bi,1∪Bi,2)∪Hn. Note that Hn,n−1+=⋃i=1n−1(Bi,1∪Bi,2) and H1+=H.
Defining dn(p)=ϕn(p,1) and letting rn,k:Hn+→Hn,k+ denote the canonical retractions with rn,k(⋃i=k+1∞Ci)={b0}, we have the following commutative diagrams:
[TABLE]
We may assume that there are parametrizations ρn,j:([0,1],{0,1})→(Bn,j,b0) such that
[TABLE]
Here, “⋅” denotes the usual concatenation of paths and ρn,j− denotes
the reverse path of ρn,j, given by ρn,j−(t)=ρn,j(1−t). Taking path homotopy classes, and noting that [ℓn]=[dn∘ℓn]∈π1(P,b0), we have
[TABLE]
Lemma 2.2**.**
For every k⩾2n+1, dn#:π1(Hn,k+,b0)→π1(Hn+1,k+,b0) is injective.
Proof.
For m∈{n,n+1}, the group π1(Hm,k+,b0) is free on the set
[TABLE]
So, the claim follows from Equation (2.1) and the fact that dn∘ρi,j=ρi,j for 1⩽i⩽n−1, 1⩽j⩽2 and dn∘ℓi=ℓi for n+1⩽i⩽k.
∎
Notation**.**
We will denote functions σ:L→N to inverse limits
[TABLE]
as sequences σ=(σn)n⩾1 with σn=μn∘σ:L→Nn, where μn:N→Nn are the projections.
Lemma 2.3**.**
For every n, dn#:π1(Hn+,b0)→π1(Hn+1+,b0) is injective.
Proof.
Let 1=[α]∈π1(Hn+,b0). Since
[TABLE]
we have that
[TABLE]
is injective [5].
Therefore, there is a k⩾2n+1 such that (rn,k)#([α])=1.
The claim now follows from Lemma 2.2 and applying π1 to the diagram above.
∎
Lemma 2.4**.**
π1(P,b0)* is isomorphic to the direct limit*
[TABLE]
with canonical homomorphisms ιn#:π1(Hn+,b0)→π1(P,b0) induced by inclusion ιn:Hn+↪P.
Proof.
For n∈N and [α]∈π1(Hn+,b0), we have [dn∘α]=[α]∈π1(P,b0). Hence ιn+1#∘dn#=ιn#. In order to verify the universal property, let hn:π1(Hn+,b0)→G be a homomorphisms with hn+1∘dn#=hn. Let [α]∈π1(P,b0). Since α([0,1]) is compact, there is an n∈N such that α([0,1])∩Pi∘=∅ for all i⩾n. Put β=dn−1∘dn−2∘⋯∘d1∘α. Then β:([0,1],{0,1})→(Hn+,b0) and ιn#([β])=[α]∈π1(P,b0). Moreover, hn+1([dn∘β])=hn([β]). Put h([α])=hn([β]). Once we show that h:π1(P,b0)→G is well-defined, it is clear that h is a homomorphism with h∘ιn#=hn and that h is the unique homomorphism with this property. To this end, suppose that [α]=[α]∈π1(P,b0).
Let H:[0,1]×[0,1]→P be a homotopy with H(t,0)=α(t), H(t,1)=α(t), and H(0,t)=H(1,t)=b0 for all t∈[0,1]. Since H([0,1]×[0,1]) is compact, for n∈N sufficiently large, we have dn−1∘⋯∘d2∘d1∘H([0,1]×[0,1])⊆Hn+ so that [dn−1∘⋯∘d2∘d1∘α]=[dn−1∘⋯∘d2∘d1∘α]∈π1(Hn+,b0).
∎
Lemma 2.5**.**
For every n, ιn#:π1(Hn+,b0)→π1(P,b0) is injective.
Recall that a group G is called locally free if every finitely generated subgroup of G is free.
Proposition 2.6**.**
π1(P,b0)* is locally free.*
Proof.
Since π1(Hn+,b0) is isomorphic to a subgroup of an inverse limit of free groups of finite rank, it is locally free [11, Theorem 1].
Then, by Lemma 2.4, π1(P,b0) is a direct limit of locally free groups, and thus locally free [7, Lemma 24].
∎
Remark 2.7** (Metric Hawaiian Pants P∗⊆R3).**
While P is not metrizable (it is not first countable at b0), it is naturally homotopy equivalent to the metrizable space formed by attaching the pants Pn to the union ⋃i=1∞Zi⊆R3 of the cylinders Zn=Cn×[1−n,n−1] via identifying ∂Dn with Cn×{1−n}, ∂Dn,1 with C2n×{2n−1}, and ∂Dn,2 with C2n+1×{2n}.
If we change the embedding H⊆R2, this procedure yields a subspace of R3 that is homotopy equivalent to P: For each n∈N, choose a triangle Cn′⊆R2 of diameter less than 1/n such that for all i=j, Ci′∩Cj′={b0} and the bounded components of R2∖Ci′ and R2∖Cj′ are disjoint. Then ⋃i=1∞Ci′⊆R2 is homeomorphic to H. The adjunction space resulting from attaching the pants Pn to the union ⋃i=1∞Zi′⊆R3 of the corresponding cylinders Zn′ can now readily be implemented in R3 by forming the union of ⋃i=1∞Zi′ with appropriate sets Pn′⊆R3 that are homeomorphic to Pn.
To obtain a subspace P∗⊆R3 which is semilocally simply connected at all but one point and homotopy equivalent to P, slightly deform the cylinders Zn′ so that their only common point of contact is the origin.
Then there is a bijective homotopy equivalence h:P→P∗ such that h(Cn)=Cn′×{0} for all n∈N with homotopy inverse g:P∗→P collapsing the cylinders such that g∣Pn′:Pn′→Pn is a homeomorphism, g(Zn′)=Cn, g∣Cn′×{0}=(h∣Cn)−1, and g(h(Bn,j))=Bn,j for all n∈N and j∈{1,2}.
The resulting deformation retraction of the cylinders allows the proofs in this paper for those properties of P that are not homotopy invariant (such as the 1-UV0 property and the discrete monodromy property) to go through for P∗ with only minor changes. However, working with P is conceptually simpler.
3. Non-injectivity into the first Čech homotopy group
Let X be a path-connected topological space and x0∈X. For a point x∈X, let Tx denote the set of all open neighborhoods of x in X. For U∈Tx and a path α in X from x0 to x, consider the subgroup
[TABLE]
Let π(x,U) denote the normal closure of π(α,U) in π1(X,x0), i.e., the subgroup generated by all π(β,U), with β a path from x0 to x:
[TABLE]
Remark 3.1**.**
Suppose U∈Tx is path connected and let [γ]∈π1(X,x0). Then [γ]∈π(x,U) if and only if there is a map g:D∖(D1∘∪D2∘∪⋯∪Dj∘)→X from a “disk with holes” to X with g∣∂D=γ and g(∂Di)⊆U for all i∈{1,2,…,j}. (Here, D1,D2,…,Dj are pairwise disjoint disks in the interior D∘ of the disk D.)
Let Cov(X) denote the set of all open covers of X. For U∈Cov(X), let π(U,x0) denote the subgroup of π1(X,x0) generated by all π(x,U) with U∈U and x∈U.
of the fundamental group π1(X,x0) is called the Spanier group of X.
Remark 3.3**.**
If X is locally path connected, then for a given H⩽π1(X,x0), there is a (classical) covering projection p:(X,x)→(X,x0) with p#π1(X,x)=H if and only if there is a U∈Cov(X) such that π(U,x0)⩽H [29].
Remark 3.4**.**
The Spanier group πs(X,x0) is contained in the kernel of the natural homomorphism ΨX:π1(X,x0)→πˇ1(X,x0) to the first Čech homotopy group [22]. If X is locally path connected and metrizable, then πs(X,x0) equals this kernel [3].
Let ι=ι1:H↪P denote inclusion.
Lemma 3.5**.**
ι#π1(H,b0)⩽πs(P,b0).
Proof.
Let [α]∈π1(H,b0) and U∈Cov(P). Choose U∈U with b0∈U and fix n∈N with ι(Hn)⊆U. Express [α]=[γ1][δ1][γ2][δ2]⋯[γm][δm] with loops γj in C1∪C2∪⋯∪Cn−1 and loops δj in Hn. Then ι#([δj])∈π(b0,U) for every j. Also, for every j, we have ι#([γj])∈⟨[ℓ1],[ℓ2],…,[ℓn−1]⟩⩽π1(P,b0), so that repeated application of Equation (2.2) yields ι#([γj])∈π(b0,U). Hence, ι#([α])∈πs(P,b0).
∎
Theorem 3.6**.**
ΨP:π1(P,b0)→πˇ1(P,b0)* is not injective.*
Proof.
Combining Lemma 2.5 (with n=1), Lemma 3.5, and Remark 3.4, we obtain 1=ι#π1(H,b0)⩽πs(P,b0)⩽kerΨP.
∎
Recall that if the fundamental group of a Peano continuum does not (canonically) inject into the first Čech homotopy group, then it is not residually n-slender [16]. However, since P is not a Peano continuum, we verify this separately:
A group G is called noncommutatively slender (n-slender for short) if for every homomorphism h:π1(H,b0)→G, there is
a k∈N such that h([α])=1 for all loops α:([0,1],{0,1})→(Hk,b0).
Recall that a group G is called residually n-slender (respectively residually free) if for every 1=g∈G there is an n-slender (respectively free) group S and a homomorphism h:G→S, such that h(g)=1.
Proposition 3.8**.**
π1(P,b0)* is not residually n-slender.*
Proof.
Consider 1=[ℓ1]∈π1(P,b0). Let h:π1(P,b0)→S be a homomorphism to an n-slender group S. It suffices to show that h([ℓ1])=1. If we precompose h with the homomorphism ι#:π1(H,b0)→π1(P,b0), induced by inclusion ι:H↪P, and note that S is n-slender, we see that h([ℓk])=h∘ι#([ℓk])=1 for all but finitely many k. However, by Equation (2.2), we have [ℓn]=[ρn,1][ℓ2n][ρn,1]−1[ρn,2][ℓ2n+1][ρn,2]−1 in π1(P,b0) for all n. Hence, h([ℓn])=1 for all n.
∎
Remark 3.9**.**
Every free group is n-slender [13]. So, π1(P,b0) is not residually free.
A path-connected space X is called strongly homotopically Hausdorff atx∈X if for every essential loop γ in X, there is an open neighborhood U of x in X such that γ cannot be freely homotoped into U, that is, if
[TABLE]
(If U is path connected, we may replace “α(1)∈U” by “α(1)=x”.)
The space X is called strongly homotopically Hausdorff if it is strongly homotopically Hausdorff at every point x∈X.
Remark 4.2**.**
If the natural homomorphism π1(X,x)↪πˇ1(X,x) is injective, then X is strongly homotopically Hausdorff [20]. However, the converse does not hold in general [20, Example Z′].
Remark 4.3** (The Hawaiian Mapping Torus).**
Let f:H→H be the map given by f∘ℓn=ℓn+1, n∈N. The Hawaiian Mapping Torus is the space Mf=H×[0,1]/∼, where (x,0)∼(f(x),1) for all x∈H. Identifying H with the image of H×{0} in Mf, the inclusion i:H↪Mf induces an injection i#:π1(H,b0)→π1(Mf,b0). Consider the loop ρ:([0,1],{0,1})→(Mf,b0) where ρ(s) is the image of (b0,s) in Mf and put t=[ρ]∈π1(Mf,b0). From two applications of van Kampen’s Theorem, we get that π1(Mf,b0) is isomorphic to the quotient of π1(H,b0)∗⟨t⟩ by the relations g=tf#(g)t−1, g∈π1(H,b0) (see [28]). Iterating these relations, we see that each [γ]∈π1(H,b0) factors in π1(Mf,b0), for every n∈N, as a conjugate [ρ]n[fn∘γ][ρ]−n where the diameter of the loop fn∘γ shrinks to [math].
While Mf is a Peano continuum that embeds into R3 and has many of the same properties as P, it is not strongly homotopically Hausdorff, since ℓ1 is freely homotopic to ℓn for all n∈N. Our detailed treatment of P is motivated by the fact that π1(P,b0) exhibits a somewhat more intricate algebraic phenomenon: in order to write an element g∈ι#π1(H,b0)⩽π1(P,b0) as a product of conjugates of homotopy classes of arbitrarily small loops (as in the proof of Lemma 3.5), it takes an exponentially growing number of distinct conjugating elements, namely products of [ρi,j].
A path α:[a,b]→X is called reduced if for every a⩽s<t⩽b with α(s)=α(t), the loop α∣[s,t] is not null-homotopic in X. For a one-dimensional metric space X, every path α:[a,b]→X is homotopic (relative to endpoints) within α([a,b]) to either a constant path or a reduced path, which is unique up to reparametrization [14]. A path α:[a,b]→X is called cyclically reduced if α⋅α is reduced.
Let λ:([0,1],0)→(X,x0) be a reduced path in a one-dimensional metric space X, δ:[0,1]→X be a reduced loop based at λ(1), and γ be a reduced representative of [λ⋅δ⋅λ−]. Then there exist s,t∈[0,1] such that
λ∣[0,t]∘ϕ=γ∣[0,s], for some increasing homeomorphism ϕ:[0,s]→[0,t],
and λ([t,1])⊆δ([0,1]).
Theorem 4.5**.**
P* is strongly homotopically Hausdorff.*
Proof.
For n∈N, we define Un=H∩{(x,y)∈R2∣x<n(n+1)2n+1}. Since diam(Cn+1)=n+12<n(n+1)2n+1<n2=diam(Cn), we see that the sequence U1⊇U2⊇U3⊇⋯ forms a neighborhood basis for H at b0. For every pair n,k∈N, fk−1(Un) has three components Lk,n0, Lk,n1, and Lk,n2, each of which is an open arc or a circle (there are four cases based on the position of n relative to k, 2k, and 2k+1), such that ak∈Lk,n0⊆∂Dk, bk∈Lk,n1⊆∂Dk,1 and ck∈Lk,n2⊆∂Dk,2. Since, for a given k, Lk,n+1i⊆Lk,ni, we may choose three pairwise disjoint open neigborhoods Nk,n0, Nk,n1, Nk,n2 of Lk,n0, Lk,n1, Lk,n2 in Pk, respectively, such that Nk,n+1i⊆Nk,ni, Nk,ni∩∂Pk=Lk,ni, and Nk,ni deformation retracts onto Lk,ni. Define Vk,n=Nk,n0∪Nk,n1∪Nk,n2.
(See Figure 3.) Put Vn=⋃k∈Nfk(Vk,n). Then Vn is an open neighborhood of b0 in P, Vn+1⊆Vn, and Vn deformation retracts onto Un. (Note that V1⊇V2⊇V3⊇⋯ does not form a neighborhood basis for P at b0.)
Now suppose, to the contrary, that P is not strongly homotopically Hausdorff. Then P is not strongly homotopically Hausdorff at b0, since P∖{b0} is locally contractible. Hence, there is a 1=[γ]∈π1(P,b0) such that for every n∈N, there is a path λn:[0,1]→P from λn(0)=b0 to λn(1)∈Vn with [γ]∈π(λn,Vn).
Since each Vn is path connected, we may assume that λn(1)=b0. Choose loops δn in Vn with [γ]=[λn][δn][λn]−1∈π1(P,b0). Since Vn deformation retracts onto Un, we may assume that δn lies in Un. We may also assume that each δn is reduced in H.
This implies that δn lies in Hn+1, because Cm is not fully contained in Un if m⩾n.
Replacing each λn by λ1−⋅λn, we may assume that γ=δ1, which lies in U1.
There is a maximal s0∈[0,1] such that γ∣[0,s0] is a reparametrization of (γ∣[t0,1])− for some t0∈(s0,1]. Then γ(s0)=γ(t0)=b0, γ∣[s0,t0] is cyclically reduced, and [γ]=[γ∣[0,s0]][γ∣[s0,t0]][γ∣[0,s0]]−1. Replacing each λn by γ∣[0,s0]−⋅λn and replacing γ by γ∣[s0,t0], we may assume that γ is cyclically reduced.
Let m be minimal such that γ([0,1]) intersects Cm∖{b0}. Then γ fully traverses Cm, at least once,
and γ lies in Hm. Let F be a homotopy from γ to λm⋅δm⋅λm− (relative to endpoints) within P. Choose n>m such that the image of F misses all Pi∘ with i>n. Then F′=dn∘dn−1∘⋯∘d1∘F is a homotopy from γ′=dn∘dn−1∘⋯∘d1∘γ to λ⋅δ⋅λ− (relative to endpoints) within Hn+1+, where λ=dn∘dn−1∘⋯∘d1∘λm and δ=dn∘dn−1∘⋯∘d1∘δm.
On one hand, γ′ is a cyclically reduced loop in Hn+1+ which traverses Bm,1. On the other hand,
the image of δ is disjoint from Bm,1∖{b0}. (In particular, λ is not null-homotopic in Hn+1+.)
Therefore, using reduced representatives λ′ and δ′ of [λ] and [δ], respectively, we have s,t>0 in Lemma 4.4. Applying Lemma 4.4 to γ′−, as well, we see that γ′ is not cyclically reduced; a contradiction.
∎
5. An inverse limit of finitely generated free monoids
Let Wn,k+ denote the set of finite words (including the empty word) over the alphabet An,k={ρi,j±1∣1⩽i⩽n−1,1⩽j⩽2}∪{ℓi±1∣n⩽i⩽k}.
Then Wn,k+ forms a free monoid on the set An,k under concatenation.
The deletion of a subword of the form ρi,j+1ρi,j−1, ρi,j−1ρi,j+1, ℓi+1ℓi−1, or ℓi−1ℓi+1 from an element of Wn,k+ is called a cancellation. A word is called reduced if it does not allow for any cancellation. Recall that, starting with a fixed element of Wn,k+, every maximal sequence of cancellations results in the same reduced word.
Let Fn,k+⊆Wn,k+ be the subset of all reduced words. Then Fn,k+ forms a free group on the set {ρi,j∣1⩽i⩽n−1,1⩽j⩽2}∪{ℓi∣n⩽i⩽k} under concatenation, followed by maximal cancellation. Moreover, we have isomorphisms hn,k:π1(Hn,k+,b0)→Fn,k+, mapping [ρi,j]↦ρi,j and [ℓi]↦ℓi.
Let Rn,k:Wn,k+1+→Wn,k+ denote the function that deletes every occurrence of the letters ℓk+1 and ℓk+1−1 from a word. Let Sn,k:Wn,k+→Fn,k+ be the function that maximally cancels words and put Tn,k=Sn,k∘Rn,k∣Fn,k+1+:Fn,k+1+→Fn,k+.
Then Rn,k, Sn,k, and Tn,k are monoid/group homomorphisms.
Define gn,k=hn,k∘rn,k#:π1(Hn+,b0)→Fn,k+⊆Wn,k+.
Since Tn,k∘gn,k+1=gn,k:π1(Hn+,b0)→Fn,k+ and
Tn,k∘hn,k+1=hn,k∘rn,k#:π1(Hn,k+1+,b0)→Fn,k+, we have, for each n,
as in the proof of Lemma 2.3, an injective homomorphism into an inverse limit Fn+ of free groups:
[TABLE]
(Note that gn is not surjective and that Fn+ is not free.)
Remark 5.1**.**
The image of gn equals the locally eventually constant sequences, where (yn,k)k⩾n−1∈Fn+ is called locally eventually constant if for every m⩾n−1, the sequence (Rn,m∘Rn,m+1∘⋯∘Rn,k−1(yn,k))k⩾m is eventually constant [27].
For every n∈N, x∈π1(Hn+,b0), and m⩾n−1, the sequence (Rn,m∘Rn,m+1∘⋯∘Rn,k−1(gn,k(x)))k⩾m of (unreduced) words is eventually constant, so that we may define functions
ωn,m:π1(Hn+,b0)→Wn,m+ by
[TABLE]
for sufficiently large k, and obtain commutative diagrams
[TABLE]
with injective functions ωn=(ωn,k)k⩾n−1 that output stabilized word sequences. (See [12] or [24], for example.)
For k⩾2n+1, let Dn:Wn,k+→Wn+1,k+ be the monomorphism that replaces every occurrence of the letter ℓn by ρn,1ℓ2nρn,1−1ρn,2ℓ2n+1ρn,2−1 and every occurrence of the letter
ℓn−1 by ρn,2ℓ2n+1−1ρn,2−1ρn,1ℓ2n−1ρn,1−1. Then, for every k⩾2n+1, we obtain the following commutative diagram:
[TABLE]
Note: For every ω∈Fn,k+, the word Dn(ω) is reduced.
Remark 5.2**.**
In view of the above, the direct limit structure of Lemma 2.4 suggests the possibility of labelling the elements of π1(P,b0) using sequences of finite words over finite alphabets that gradually exclude all letters ℓn±1 in favor of only using the conjugating letters ρn,j±1. Since such a shift causes conjugating pairs to become adjacent in words at later levels, we use monoid structures to prevent their cancellation. This, in turn, requires us to work with Dn:Wn,k+→Wn+1,k+ in what follows, at a level just deep enough to stabilize the appropriate word sequences.
Recall that Wn+1,n+ is the set of finite words over {ρi,j±1∣1⩽i⩽n,1⩽j⩽2}. In particular, the set W1,0+ contains only one element: the empty word.
Let En−1:Wn+1,n+→Wn,n−1+ denote the epimorphism deleting every occurrence of the letters ρn,1±1 and ρn,2±1. Then, for k⩾2n+1, we obtain commutative trapezoids:
[TABLE]
Theorem 5.3**.**
There is a well-defined injective function
[TABLE]
defined as follows: for a given [α]∈π1(P,b0) and sufficiently large n⩾2, choose [β]∈π1(Hn+,b0) with ιn#([β])=[α] and put χn−1([α])=ωn,n−1([β])∈Wn,n−1+.
Proof.
First we show that χ is well-defined. By Lemmas 2.4 and 2.5 it suffices to show that for any [β]∈π1(Hn+,b0), we have En−1(ωn+1,n(dn#([β])))=ωn,n−1([β]), making the following diagram commute:
[TABLE]
(Recall that the underlying set of a direct limit of groups is the direct limit of the underlying sets.) To this end, put β′=dn∘β. Then [β′]∈π1(Hn+1+,b0).
By definition of ωn,n−1 and ωn+1,n, respectively, for sufficiently large k⩾2n+1, we have (cf. Remark 5.1)
[TABLE]
and
[TABLE]
Noting that gn+1,k([β′])=hn+1,k∘rn+1,k#([β′])=hn+1,k∘rn+1,k#∘dn#([β])=hn+1,k∘dn#∘rn,k#([β])=Dn∘hn,k∘rn,k#([β])=Dn∘gn,k([β])
and that
[TABLE]
(see Diagram (5.3)) we obtain the desired equality:
[TABLE]
Now we show that χ is injective. Suppose [α(1)]=[α(2)]∈π1(P,b0). Choose n⩾2 sufficiently large (as in the proof of Lemma 2.4) so that β(s)([0,1])⊆Hn+, where β(s)=dn−1∘dn−2∘⋯∘d1∘α(s) for s∈{1,2}.
Then ιn#([β(s)])=[α(s)] and [β(1)]=[β(2)]∈π1(Hn+,b0). Hence, there is an m⩾n such that ωn,m−1([β(1)])=ωn,m−1([β(2)]). We may assume that m is even. Put γ(s)=dm−1∘dm−2∘⋯∘dn∘β(s). Choose k⩾2(m−1)+1 sufficiently large, such that for s∈{1,2}, we have:
[TABLE]
For n⩽j⩽m−1, let Dj,m−1:Wj,m−1+→Wj+1,m−1+ be the monomorphism that replaces every occurrence of the letter ℓj (respectively ℓj−1) by ρj,1ℓ2jρj,1−1ρj,2ℓ2j+1ρj,2−1 (respectively ρj,2ℓ2j+1−1ρj,2−1ρj,1ℓ2j−1ρj,1−1) if 2j+1⩽m−1, but instead replaces it by ρj,1ρj,1−1ρj,2ρj,2−1 (respectively ρj,2ρj,2−1ρj,1ρj,1−1) if 2j⩾m.
Since each Dj,m−1 with n⩽j⩽m−1 is injective, so is their composition D=Dm−1,m−1∘Dm−2,m−1∘⋯∘Dn,m−1. Moreover, the following diagram commutes:
[TABLE]
Hence, for s∈{1,2}, we have
[TABLE]
Since ωn,m−1([β(1)])=ωn,m−1([β(2)]) and D is injective, we have χm−1([α(1)])=χm−1([α(2)]).
∎
Remark 5.4**.**
Although ⋃i=1∞(Bi,1∪Bi,2)⊆P is a bouquet of circles that is not homeomorphic to H, one can algebraically set up a commutative diagram as follows:
[TABLE]
However, there does not exist an injective homomorphism
[TABLE]
because π1(P,b0) is not residually free (Remark 3.9).
A path-connected space X is called homotopically path Hausdorff if for every two paths α,β:[0,1]→X with α(0)=β(0) and α(1)=β(1) such that α⋅β− is not null-homotopic,
there is a partition 0=t0<t1<⋯<tn=1 of [0,1] and open subsets U1,U2,…,Un of X with α([ti−1,ti])⊆Ui for all 1⩽i⩽n and with the property that if γ:[0,1]→X is any path with γ([ti−1,ti])⊆Ui for all 1⩽i⩽n and with γ(ti)=α(ti) for all 0⩽i⩽n, then γ⋅β− is not null-homotopic.
Remark 6.2**.**
We recall from [2] that a connected and locally path-connected space X is homotopically path Hausdorff if and only if π1(X,x) is T1 in the quotient topology induced by the compact-open topology on the loop space Ω(X,x).
Remark 6.3**.**
If the natural homomorphism π1(X,x)↪πˇ1(X,x) is injective, then X is homotopically path Hausdorff [20]. However, the converse does not hold in general [20, Example Y′].
Theorem 6.4**.**
P* is homotopically path Hausdorff.*
Proof.
Let 1=[α]∈π1(P,b0). We wish to find a partition 0=t0<t1<⋯<ts=1 and open subsets U1,U2,…,Us⊆P with α([ti−1,ti])⊆Ui for all 1⩽i⩽s such that the following property holds: if γ:[0,1]→P is any loop with γ(ti)=α(ti) for all 0⩽i⩽s and γ([ti−1,ti])⊆Ui for all 1⩽i⩽s, then [γ]=1∈π1(P,b0). (This property is preserved if we add a subdivision point t′∈(ti−1,ti) and choose U′=Ui. Therefore, checking this statement for all essential loops α based at b0, validates Definition 6.1 for all essential loops α⋅β−.)
By Theorem 5.3, there is an n∈N such that for β=dn−1∘dn−2∘⋯∘d1∘α, we have β([0,1])⊆Hn+ and χn−1([α])=ωn,n−1([β])∈Wn,n−1+ is not the empty word. Choose k∈N sufficiently large, so that
[TABLE]
Consider β:[0,1]→Hn+⊆Pn+ and rn,k:Hn+→Hn,k+.
Since Hn,k+ is a finite bouquet of circles, there is an open cover B of Hn,k+
with the following property: if η,τ:([0,1],{0,1})→(Hn,k+,b0) are two loops that are B-close, i.e., if for every t∈[0,1], there is a B∈B with {η(t),τ(t)}⊆B, then [η]=[τ]∈π1(Hn,k+,b0). Choose an open cover {Wj∣j∈J} of Hn+ and a cover {Vj∣j∈J} of Hn+ by open subsets of Pn+ such that {Wj∣j∈J} refines {(rn,k)−1(B)∣B∈B} and such that each Vj deformation retracts onto Wj. (See proof of Theorem 4.5.) Choose a partition 0=t0<t1<⋯<ts=1 and indices j1,j2,…,js∈J such that β([ti−1,ti])⊆Vji for 1⩽i⩽s. Put Ui=(dn−1∘dn−2∘⋯∘d1)−1(Vji) for 1⩽i⩽s.
Now, let γ:[0,1]→P be a loop with γ(ti)=α(ti) for all 0⩽i⩽s and γ([ti−1,ti])⊆Ui for all 1⩽i⩽s. Then γ:[0,1]→P is homotopic (relative to its
endpoints) to a loop γ′:[0,1]→Hn+⊆P such that rn,k∘γ′:[0,1]→Hn,k+ and rn,k∘β:[0,1]→Hn,k+ are B-close. Hence, [rn,k∘γ′]=[rn,k∘β]∈π1(Hn,k+,b0), so that gn,k([γ′])=hn,k([rn,k∘γ′])=hn,k([rn,k∘β])=gn,k([β]).
Choose m⩾k sufficiently large, so that
[TABLE]
which, as a word in Wn,n−1+, has at least as many letters as
[TABLE]
Hence, χn−1([γ′])=ωn,n−1([γ′]) is not the empty word. We conclude that χ([γ])=χ([γ′]) is not trivial so that [γ]=1∈π1(P,b0).
∎
We say that X is 1-UV0 atx∈X if for every neighborhood U of x in X, there is an open subset V in X with x∈V⊆U such that for every map f:D2→X from the unit disk with f(S1)⊆V there is a map g:D2→U with f∣S1=g∣S1.
We say that X is 1-UV0 if X is 1-UV0 at every point x∈X.
Proposition 7.2**.**
All one-dimensional spaces and all planar spaces are 1-UV0.
Proof.
In a one-dimensional space, every null-homotopic loop contracts within its own image [10, Lemma 2.2]. In a planar space, every null-homotopic loop has a contraction whose diameter equals that of the image of the loop [21, Lemma 13].
∎
Theorem 7.3**.**
P* is 1-UV0.*
Proof.
It suffices to show that P is 1-UV0 at b0. Let U be an open neighborhood of b0 in P.
Recall Un and fk−1(Un)=Lk,n0∪Lk,n1∪Lk,n2⊆∂Pk from the proof of Theorem 4.5. Fix m∈N such that b0∈Um⊆U∩H. For each k∈N, choose three pairwise disjoint open neighborhoods Mk0,Mk1,Mk2 of Lk,m0,Lk,m1,Lk,m2 in Pk, respectively, such that, for i∈{0,1,2}, Mki∩∂Pk=Lk,mi, Mki∩Pk∘⊆U∩Pk∘, and Mki deformation retracts onto Lk,mi.
Define Vk,m′=Mk0∪Mk1∪Mk2 and V=⋃k∈Nfk(Vk,m′). Then V is an open subset of P with b0∈V⊆U.
Let f:D2→P be a map with f(S1)⊆V.
We will show that α=f∣S1 contracts within V. Since V is path connected, we may assume that α is a loop based at b0 and show that [α]=1∈π1(V,b0). Since V deformation retracts onto Um⊆H, we may assume that α lies in H. Since [α]=1∈π1(P,b0), we have [α]=1∈π1(H,b0) by Lemma 2.5. As H is one-dimensional, this implies that α contracts within its own image.
∎
8. Generalized covering projections and the homotopically Hausdorff property
In this section, we briefly review generalized covering projections. Let X be a path-connected topological space and H⩽π1(X,x0). Even if there is no classical covering projection p:(X,x)→(X,x0) with p#π1(X,x)=H (see Remark 3.3), there might be a generalized one:
Let X be a path-connected topological space. We call a map q:X→X a generalized covering projection if X is nonempty, connected and locally path connected and if for every x∈X, for every connected and locally path-connected space Y, and for every map f:(Y,y)→(X,q(x)) with f#π1(Y,y)⩽q#π1(X,x), there is a unique map g:(Y,y)→(X,x) such that q∘g=f.
Remark 8.2**.**
Suppose q:(X,x)→(X,x0) is a generalized covering projection. Then q#:π1(X,x)→π1(X,x0) is injective. If we put K=q#π1(X,x), then q:X→X is characterized as usual, up to equivalence, by the conjugacy class of K in G=π1(X,x0). Moreover, Aut(X→qX)≅NG(K)/K, where NG(K) denotes the normalizer of K in G. (The standard arguments apply [29, 2.3.5 & 2.6.2].)
If it exists, a generalized covering projection can be obtained in the standard way:
On the set of all paths α:([0,1],0)→(X,x0) consider the equivalence relation α∼β if and only if α(1)=β(1) and [α⋅β−]∈H. Denote the equivalence class of α by ⟨α⟩ and denote the set of all equivalence classes by XH. Let x0 denote the class containing the constant path at x0. We give XH the topology generated by basis elements of the form ⟨α,U⟩={⟨α⋅γ⟩∣γ:([0,1],0)→(U,α(1))}, where U is an open subset of X and ⟨α⟩∈XH with α(1)∈U. Then XH is connected and locally path connected and the endpoint projection pH:XH→X, defined by pH(⟨α⟩)=α(1), is a continuous surjection. Moreover, the map pH:XH→X is open if and only if X is locally path connected.
If pH:XH→X has unique path lifting, then it is a generalized covering projection and, for every ⟨α⟩∈XH,
(pH)#:π1(XH,⟨α⟩)→π1(X,α(1)) is a monomorphism onto [α−]H[α]; in particular (pH)#π1(XH,x0)=H [22].
If X admits a generalized covering projection q:(X,x)→(X,x0) such that q#π1(X,x)=H, then there is a homeomorphism h:(X,x)→(XH,x0) with pH∘h=q [1].
Remark 8.3**.**
For pH:XH→X to have unique path lifting, every fiber pH−1(x) with x∈X must be T1, but not necessarily discrete [22]. Moreover, T1 fibers are not sufficient [4, 30]. Note that these fibers are T1 if and only if for every x∈X,
[TABLE]
Definition 8.4** (Homotopically Hausdorff rel. H [22]).**
We call Xhomotopically Hausdorff relative toH⩽π1(X,x0), if every fiber of pH:XH→X is T1. We call Xhomotopically Hausdorff if it is homotopically Hausdorff relative to H={1}.
Remark 8.5**.**
If X is strongly homotopically Hausdorff, then X is homotopically Hausdorff. (Compare the formula of Remark 8.3 with that in Definition 4.1.)
Remark 8.6**.**
There are normal subgroups H\trianglelefteqslantπ1(H,b0), such that H is not homotopically Hausdorff relative to H. For example, H is not homotopically Hausdorff relative to the commutator subgroup of π1(H,b0) [4, Example 3.10].
We abbreviate X{1} by X and p{1}:X{1}→X by p:X→X.
Moreover, note that if H={1}, then ⟨α⟩=[α].
Remark 8.7**.**
If X is homotopically path Hausdorff, then p:X→X has unique path lifting [20].
Remark 8.8**.**
If X is path connected, 1-UV0 and metrizable, then p:X→X has unique path lifting [4].
Theorem 8.9**.**
There exists a generalized covering projection p:P→P with π1(P,b0)={1}.
Proof.
By Theorem 6.4 and Remark 8.7, p:P→P has unique path lifting. Hence, p:P→P is a generalized covering projection, p#π1(P,b0)={1}, and p#:π1(P,b0)→π1(P,b0) is injective.
∎
9. The discrete monodromy property
Let X be a path-connected topological space and H⩽π1(X,x0).
Even if the map pH:XH→X does not have unique path lifting, for every path β:[0,1]→X and every ⟨α⟩∈pH−1(β(0)) there is a continuous standard path liftβ:([0,1],0)→(XH,⟨α⟩) with pH∘β=β, defined by β(t)=⟨α⋅βt⟩, where βt(s)=β(ts).
Based on the standard path lift, we may define the standard monodromy for pH:XH→X, as follows. For a path β:[0,1]→X from β(0)=x to β(1)=y, we define Φβ:pH−1(x)→pH−1(y) by Φβ(⟨α⟩)=⟨α⋅β⟩.
Clearly, Φβ:pH−1(x)→pH−1(y) is a bijective function with inverse Φβ−1=Φβ−. However, Φβ need not be continuous. (Such is the case for (X,x0)=(H,b0) and H={1}, although p:X→X has unique path lifting. See [18] for a discussion.)
Remark 9.1**.**
Note that Φβ depends only on the homotopy class [β]. Moreover, ⟨α⟩∗⟨β⟩:=Φβ(⟨α⟩) is a well-defined group operation on pH−1(x0) if and only if H is a normal subgroup of π1(X,x0).
Definition 9.2** (Discrete monodromy).**
We say that X has the discrete monodromy property relative toH⩽π1(X,x0) if for every x,y∈X and for every pathβ:[0,1]→X from β(0)=x to β(1)=y, the monodromy Φβ:pH−1(x)→pH−1(y) is either the identity function or its graph is a discrete subset of pH−1(x)×pH−1(y)⊆XH×XH. We say that X has the discrete monodromy property if it has the discrete monodromy property relative to H={1}.
Remark 9.3**.**
Clearly, if every fiber of pH:XH→X is discrete, then X has the discrete monodromy property relative to H. However, the converse does not hold in general: p:H→H has the discrete monodromy property (see Proposition 9.13), but p−1(b0) is not discrete.
Remark 9.4**.**
(a)
Φβ:pH−1(x)→pH−1(y) is the identity function if and only if x=y and [β]∈[α−]H[α] for all paths α in X from x0 to x.
(b)
The graph of the identity function idpH−1(x):pH−1(x)→pH−1(x) is discrete if and only if pH−1(x) is discrete.
(c)
pH−1(x) is discrete if and only if for every path α in X from x0 to x, there is a U∈Tx such that π(α,U)⊆H, i.e., Hπ(α,U)=H.
Lemma 9.5**.**
Let H⩽π1(X,x0).
The graph of Φβ:pH−1(x)→pH−1(y) is discrete if and only if for every path α in X from x0 to x, there are U∈Tx and V∈Ty such that Hπ(α,U)∩Hπ(α⋅β,V)=H.
Proof.
First, observe that if f:A→B is an injective function between topological spaces, then its graph Γ={(a,b)∈A×B∣f(a)=b} is a discrete subset of A×B if and only if for every a∈A there are U∈Ta and V∈Tf(a) such that f(U)∩V={f(a)}.
Now, suppose that the graph of Φβ:pH−1(x)→pH−1(y) is discrete and let ⟨α⟩∈pH−1(x). Choose U∈Tx and V∈Ty with
[TABLE]
Let g∈Hπ(α,U)∩Hπ(α⋅β,V). Then g=h1[α⋅γ⋅α−] for some h1∈H and some loop γ in U, and g=h2[α⋅β⋅δ⋅β−⋅α−] for some h2∈H and some loop δ in V. Hence, Φβ(⟨α⋅γ⟩)=⟨α⋅γ⋅β⟩=⟨α⋅β⋅δ⟩∈⟨α⋅β,V⟩∩pH−1(y). Also, Φβ(⟨α⋅γ⟩)∈Φβ(⟨α,U⟩∩pH−1(x)). Therefore, ⟨α⋅γ⋅β⟩=Φβ(⟨α⋅γ⟩)=⟨α⋅β⟩, so that g=h1[α⋅γ⋅α−]∈H.
Conversely, let ⟨α⟩∈pH−1(x) and suppose U∈Tx and V∈Ty are such that Hπ(α,U)∩Hπ(α⋅β,V)=H. Let
[TABLE]
Then x=⟨α⋅γ⋅β⟩ for some loop γ in U, and x=⟨α⋅β⋅δ⟩ for some loop δ in V. Then [α⋅γ⋅α−][α⋅β⋅δ−⋅β−⋅α−]∈H, so that [α⋅γ⋅α−]∈Hπ(α,U)∩Hπ(α⋅β,V)=H. Hence, x=⟨α⋅γ⋅β⟩=⟨α⋅β⟩.
∎
Remark 9.6**.**
In order to apply Lemma 9.5, there is no need to check every path α: if α and α′ are two paths from x0 to x such that [α′⋅α−]H=H[α′⋅α−], then we have Hπ(α,U)∩Hπ(α⋅β,V)=H if and only if Hπ(α′,U)∩Hπ(α′⋅β,V)=H.
Remark 9.7**.**
The space (X,x0)=(H×[0,1],(b0,0)) is the prototypical example of a space that does not have the discrete
monodromy property, although p:X→X has unique path lifting. Observe that for the path β(t)=(b0,t) from x=(b0,0) to y=(b0,1), the graph of Φβ:p−1(x)→p−1(y) is not discrete.
A subgroup H⩽π1(X,x0) is called locally quasinormal if for every x∈X, for every path α in X from α(0)=x0 to α(1)=x, and for every U∈Tx, there is a V∈Tx such that x∈V⊆U and Hπ(α,V)=π(α,V)H.
Remark 9.9**.**
Clearly, every normal subgroup of π1(X,x0) is locally quasinormal. Combining [23, Lemma 5.2] with Remark 9.4(c), we see that if X is locally path connected, then every open subgroup of π1(X,x0) (in the topology of Remark 6.2) is locally quasinormal. For example, the nontrivial subgroup K⩽π1(H,b0) from [23] is open, while it does not contain any nontrivial normal subgroup of π1(H,b0).
The following is a straightforward variation on [18, Lemma 3.2]:
Lemma 9.10**.**
Let H⩽π1(X,x0), x∈X, α be a path in X from x0 to x, and U∈Tx.
Then the following are equivalent:
(a)
Hπ(α,U)=π(α,U)H.
(b)
For every Φβ:pH−1(x)→pH−1(x) with Φβ(⟨α⟩)∈⟨α,U⟩∩pH−1(x), we have Φβ(⟨α,U⟩∩pH−1(x))⊆⟨α,U⟩∩pH−1(x).
We include the proof for completeness.
Proof.
(i) First, assume that Hπ(α,U)=π(α,U)H and Φβ(⟨α⟩)∈⟨α,U⟩∩pH−1(x). Then ⟨α⋅β⟩=Φβ(⟨α⟩)=⟨α⋅δ⟩ for some loop δ in U.
So, [β]=[α−]h[α⋅δ] for some h∈H. Let ⟨γ⟩∈⟨α,U⟩∩pH−1(x). Then [γ]=h′[α⋅δ′] for some h′∈H and some loop δ′ in U. Since [α⋅δ′⋅α−]h∈π(α,U)H=Hπ(α,U), we have [α⋅δ′⋅α−]h=h′′[α⋅δ′′⋅α−] for some h′′∈H and some loop δ′′ in U. Therefore, we have [γ⋅β]=h′[α⋅δ′⋅α−]h[α⋅δ]=h′h′′[α⋅δ′′⋅α−][α⋅δ]=h′h′′[α⋅δ′′⋅δ]. Hence, Φβ(⟨γ⟩)=⟨γ⋅β⟩=⟨α⋅δ′′⋅δ⟩∈⟨α,U⟩∩pH−1(x).
(ii) Now, assume that Φβ(⟨α,U⟩∩pH−1(x))⊆⟨α,U⟩∩pH−1(x) whenever Φβ(⟨α⟩)∈⟨α,U⟩∩pH−1(x). It suffices to show that π(α,U)H⊆Hπ(α,U).
Let [τ]∈π(α,U)H. Then [τ]=[α⋅δ⋅α−][γ] for some loop δ in U and some [γ]∈H. Put β=α−⋅γ⋅α. Then Φβ(⟨α⟩)=⟨α⟩. Hence, ⟨α⋅δ⋅β⟩=Φβ(⟨α⋅δ⟩)=⟨α⋅δ′⟩ for some loop δ′ in U. Therefore, [τ]=[α⋅δ⋅β⋅(δ′)−⋅α−][α⋅δ′⋅α−]∈Hπ(α,U).
∎
Proposition 9.11**.**
Let H⩽π1(X,x0) be locally quasinormal.
If X has the discrete monodromy property relative to H, then every fiber of pH:XH→X is T1.
Proof.
Suppose there is an x∈X such that pH−1(x) is not T1. Then there are ⟨α⟩,⟨γ⟩∈pH−1(x) with ⟨α⟩=⟨γ⟩ such that for every W∈Tx we have ⟨γ⟩∈⟨α,W⟩∩pH−1(x). Let any U∈Tx be given. Choose V∈Tx with x∈V⊆U and Hπ(α,V)=π(α,V)H. Put V=⟨α,V⟩∩pH−1(x) and β=α−⋅γ. Then Φβ(⟨α⟩)=⟨γ⟩∈V. By Lemma 9.10, Φβ(⟨γ⟩)∈V. Hence, (⟨α⟩,Φβ(⟨α⟩))=(⟨γ⟩,Φβ(⟨γ⟩)) are both elements of V×V. We conclude that Φβ:pH−1(x)→pH−1(x) is not the identity function and that its graph is not discrete.
∎
Corollary 9.12**.**
If X has the discrete monodromy property, then X is homotopically Hausdorff.
Proof.
Since H={1}⩽π1(X,x0) is locally quasinormal, this follows from Proposition 9.11.
∎
The proof of the following proposition is modelled on [14] and [8].
Proposition 9.13**.**
All one-dimensional metric spaces and all planar spaces have the discrete monodromy property.
Proof.
Let x,y∈X and let β:[0,1]→X be a path from β(0)=x to β(1)=y.
If β is a loop, we assume that it is essential. Let α be any path from x0 to x. We wish to find open neighborhoods U and V of x and y, respectively, such that
π(α,U)∩π(α⋅β,V)={1}⩽π1(X,x0).
(a) Suppose X⊆R2. If x=y, choose ϵ>0 such that the loop β cannot be homotoped within X into Nϵ(x)={z∈R2∣∥x−z∥<ϵ}, relative to its endpoints.
(Here we use the fact that X is homotopically Hausdorff; see Remark 8.3.) If x=y, choose any ϵ with 0<ϵ<∥x−y∥/2.
Suppose that ∂Nr(x)⊆X for all 0<r<ϵ. Then Nϵ(x)⊆X.
In this case, taking U=Nϵ(x) gives π(α,U)={1}.
So, by making ϵ smaller, if necessary, we may assume that X∩∂Nϵ(x)=∂Nϵ(x).
Put U=X∩Nϵ(x) and V=X∩Nϵ(y). Suppose, to the contrary, that there are essential loops δ and τ in U and V,
respectively, with [α⋅δ⋅α−]=[α⋅β⋅τ⋅β−⋅α−], i.e., [δ]=[β⋅τ⋅β−]. Then there is a map h:A→X from an annulus A whose
boundary components J1 and J2 map to δ and τ, respectively, along with a diametrical arc a⊆A connecting J1 to J2 that maps to β.
If x=y, then h−1(X∩∂Nϵ(x)) clearly separates J1 from J2 in A.
However, this is also true if x=y; for otherwise there is an arc a′⊆A connecting J1 to J2 which h maps to a path β′ in U, so that β is homotopic within X, relative to its endpoints, to the concatenation of an initial subpath δ′ of δ, the path β′, a terminal subpath τ′ of τ, and a path of the form τ⋅τ⋯τ or τ−⋅τ−⋯τ−, all of which lie in U, violating the choice of ϵ.
Therefore, as in the proof of [8, Lemma 5.5], the loop δ contracts within X; a contradiction.
(b) Suppose X is a one-dimensional metric space. We may assume that β is a reduced non-degenerate path (possibly a loop). Choose open neighborhoods U and V of x and y in X, respectively, such that β is not contained in U∪V. Suppose, to the contrary, that there are essential loops δ and τ in U and V,
respectively, with [δ]=[β⋅τ⋅β−]. We may assume that both δ and τ are reduced. Then β([0,1])⊆δ([0,1])∪τ([0,1])⊆U∪V (see Lemma 4.4); a contradiction.
∎
Theorem 9.14**.**
P* has the discrete monodromy property.*
Proof.
Let 1=[β]∈π1(P,b0). In view of Remark 9.4(a), Lemma 9.5 and Remark 9.6, and since P locally contractible at every point other than b0, it suffices to find an open neighborhood V of b0 in P such that π(cb0,V)∩π(β,V)={1}, where cb0 denotes the constant path at b0.
Choose n∈N such that β([0,1])∩Pi∘=∅ for all i>n. Put β′=dn∘dn−1∘⋯∘d1∘β. Then β′([0,1])⊆Hn+1+ and [β]=[β′]∈π1(P,b0). So, we may assume that β is a reduced loop in Hn+1+. Increasing n if necessary, we may assume that β traverses one of the circles Bi,j with i∈{1,2,…,n} and j∈{1,2}. As in the proof of Theorem 7.3, we may construct an open neighborhood V of b0 in P that does not fully contain Bi,j, such that V deformation retracts onto Hn+1⊆H.
Suppose, to the contrary, that there are essential loops δ and τ in V such that [δ]=[β⋅τ⋅β−]∈π1(P,b0). We may assume that both δ and τ are reduced loops in Hn+1. Let F be a homotopy from δ to β⋅τ⋅β− (relative to endpoints) within P. Choose k⩾n such that the image of dk∘dk−1∘⋯∘d1∘F is contained in Hk+1+. Let β′, δ′ and τ′ be the composition of dk∘dk−1∘⋯∘d1 with β, δ and τ, respectively. Then β′, δ′ and τ′ are reduced loops in Hk+1+ such that [δ′]=[β′⋅τ′⋅β′−]∈π1(Hk+1+,b0). However, neither δ′ nor τ′ traverses Bi,j, while β′ does; a contradiction. (See Lemma 4.4.)
∎
Consider the subspace w(X) of “wild” points of X, defined by
[TABLE]
The following is the main utility for spaces satisfying the discrete monodromy property, as implicitly used in [14] and [8]. The proof is given after Corollary 9.18 below.
Theorem 9.15**.**
Suppose both X and Y have the discrete monodromy property. If f:X→Y is a homotopy equivalence with homotopy inverse g:Y→X, then f maps w(X) homeomorphically onto w(Y), with inverse g∣w(Y).
Remark 9.16**.**
In order to see the necessity of the assumptions in Theorem 9.15, consider X=H and Y=H×[0,1]. Then X has the discrete monodromy property, X and Y are homotopy equivalent, but w(X)={b0} and w(Y)={b0}×[0,1] are not homeomorphic.
For a path β:[0,1]→X, we let φβ:π1(X,β(0))→π1(X,β(1)) be the base point
changing isomorphism defined by φβ([δ])=[β−⋅δ⋅β].
Lemma 9.17**.**
Suppose Y has the discrete monodromy property.
Let f,g:X→Y be maps such that φβ∘f#=g#:π1(X,x)→π1(Y,g(x)) for some x∈X and some path β in Y from f(x) to g(x).
If f(x)=g(x), then there is a W∈Tx such that f#:π1(W,x)→π1(Y,f(x)) is trivial.
Proof.
Suppose f(x)=g(x). Let cf(x) be the constant path at f(x). By Lemma 9.5, there are U∈Tf(x) and V∈Tg(x) such that π(cf(x),U)∩π(β,V)={1}⩽π1(Y,f(x)). Choose W∈Tx with f(W)⊆U and g(W)⊆V. Let ℓ be a loop in W, based at x. Then f#([ℓ])=[f∘ℓ]=[β⋅(g∘ℓ)⋅β−]∈π(cf(x),U)∩π(β,V)={1}.
∎
Corollary 9.18**.**
Suppose X is path connected and Y has the discrete monodromy property.
If f,g:X→Y are homotopic maps and f#:π1(X,x0)→π1(Y,f(x0)) is injective, then f∣w(X)=g∣w(X).
Proof.
Let F:X×[0,1]→Y be a map with F(x,0)=f(x) and F(x,1)=g(x) for all x∈X. Fix x∈w(X) and let β:[0,1]→Y be given by β(t)=F(x,t). Then φβ∘f#=g#:π1(X,x)→π1(Y,g(x)). Since x∈w(X) and since f#:π1(X,x)→π1(Y,f(x)) is injective, its follows from Lemma 9.17 that f(x)=g(x).
∎
Let f:X→Y and g:Y→X be a pair of homotopy inverses. Then, for every x∈X, f#:π1(X,x)→π1(Y,f(x)) is an isomorphism; in particular, it is injective. Therefore, f(w(X))⊆w(Y). Similarly, g(w(Y))⊆w(X). Since g∘f is homotopic to the identity it follows from Corollary 9.18 that g(f(x))=x for all x∈w(X). Similarly, f(g(y))=y for all y∈w(Y).
∎
When working with spaces for which homomorphisms between fundamental groups are induced by continuous maps up to base point change, as is the case among all one-dimensional and planar Peano continua [8, 15, 25], the following provides additional utility:
Corollary 9.19**.**
Suppose both X and Y have the discrete monodromy property.Let ϕ:π1(X,x0)→π1(Y,y0) be an isomorphism with ϕ=φα∘f# and ϕ−1=φβ∘g# for some maps f:X→Y and g:Y→X and some paths α and β.
Then f maps w(X) homeomorphically onto w(Y), with inverse g∣w(Y).
Proof.
For every x∈X, f#:π1(X,x)→π1(Y,f(x)) is injective and for every y∈Y, g#:π1(Y,y)→π1(X,g(y)) is injective. Therefore, f(w(X))⊆w(Y) and g(w(Y))⊆w(X). Let x∈w(X). Choose a path γ in X from x0 to x. Put δ=(g∘f∘γ)−⋅(g∘α)⋅β⋅γ.
Since φ(g∘α)⋅β∘(g∘f)#=φβ∘g#∘φα∘f#=id:π1(X,x0)→π1(X,x0), we have φδ∘(g∘f)#=id:π1(X,x)→π1(X,x). By Lemma 9.17, g(f(x))=x. Similarly, f(g(y))=y for all y∈w(Y).
∎
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