For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simply-connected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.
Deep Dive into Discreteness and Homogeneity of the Topological Fundamental Group.
For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simply-connected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.
The concept of a natural topology for the fundamental group appears to have originated with Hurewicz [8] in 1935. It received further attention by Dugundji [2] in 1950 and by Biss [1], Fabel [3,4,5,6], and others more recently. The purpose of this note is to prove the following folklore theorem.
Theorem 1. Let X be a locally path connected topological space. The topological fundamental group π top 1 (X) is discrete if and only if X is semilocally simplyconnected.
Theorem 5.1 of [1] is Theorem 1 without the hypothesis of local path connectedness. However a counterexample of Fabel [4] shows that this stronger result is false. Fabel [4] also proves a weaker version of Theorem 1 assuming that X is locally path connected and a metric space. In this note we remove the metric hypothesis.
Our proof proceeds from first topological principles making no use of rigid covering fibrations [1] nor even of classical covering spaces. We make no use of the functoriality of the topological fundamental group, a property which was also a main result in [1,Cor. 3.4] but in fact is unproven [6, pp. 188-189]. Beware that the misstep in the proof of [1,Prop. 3.1], namely the assumption that the product of quotient maps is a quotient map, is repeated in [7,Thm. 2.1].
In general the homeomorphism type of the topological fundamental group depends on a choice of basepoint. We say that π top 1 (X) is discrete without reference to basepoint provided π top 1 (X, x) is discrete for each x ∈ X. If x and y are connected by a path in X, then π top 1 (X, x) and π top 1 (X, y) are homeomorphic. This fact was proved in [1,Prop. 3.2] and a detailed proof is in Section 4 below for completeness. Theorem 1 now immediately implies the following.
Corollary. Let X be a path connected and locally path connected topological space. The topological fundamental group π top 1 (X, x) is discrete for some x ∈ X if and only if X is semilocally simply-connected. As mentioned above it is open whether π top 1 is a functor from the category of pointed topological spaces to the category of topological groups. The unsettled question is whether multiplication
is continuous. By Theorem 1, if X is locally path connected and semilocally simplyconnected, then π top 1 (X, x), and hence π top 1 (X, x) × π top 1 (X, x), is discrete and so µ is trivially continuous. Continuity of µ in general remains an interesting question.
Lemma 4 below shows that if (X, x) is an arbitrary pointed topological space, then left and right multiplication by any fixed element in π top 1 (X, x) are continuous self maps of π top 1 (X, x). Therefore π top 1 (X, x) acts on itself by left and right translation as a group of self homeomorphisms. Clearly these actions are both transitive. Thus we obtain the following result.
Theorem 2. If (X, x) is a pointed topological space, then π top 1 (X, x) is a homogeneous space.
This note is organized as follows. Section 2 contains definitions and conventions, Section 3 proves two lemmas and Theorem 1, Section 4 addresses change of basepoint, and Section 5 shows left and right translation are homeomorphisms.
By convention, neighborhoods are open. Unless stated otherwise, homomorphisms are inclusion induced.
Let X be a topological space and x ∈ X. A neighborhood U of x is relatively inessential (in X) provided π 1 (U, x) → π 1 (X, x) is trivial. X is semilocally simplyconnected at x provided there exists a relatively inessential neighborhood U of x. X is semilocally simply-connected provided it is so at each
The fundamental group is a functor from the category of pointed topological spaces to the category of groups. Consequently if A and B are any subsets of
is trivial as well. This observation justifies the convention that neighborhoods are open.
If X is locally path connected and semilocally simply-connected, then each x ∈ X has a path connected relatively inessential neighborhood U . Such a U is necessarily a strongly relatively inessential neighborhood of x as the reader may verify (see for instance [9, Ex. 5 p. 330]).
Let (X, x) be a pointed topological space and let I = [0, 1] ⊂ R. The space
is surjective so π 1 (X, x) inherits the quotient topology and one writes π top 1 (X, x) for the resulting topological fundamental group. Let e x ∈ C x (X) denote the constant map. If f ∈ C x (X), then f -1 denotes the path defined by f -1 (t) = f (1 -t).
We prove two lemmas and then Theorem 1.
Proof. The quotient map q is continuous and
where each
subbasic open set for the compact-open topology on C x (X). We will show that
Clearly U is open in X and, by (1), x ∈ U . Finally, let f : (I, ∂I) → (U, x). For each 1 ≤ n ≤ N we have
Thus f ∈ [e x ] by (1) and so [f ] = [e x ] is trivial in π 1 (X, x).
Lemma 2. Let (X, x) be a pointed topological space and let f ∈ C x (X). If X is locally path connected and semilocally simply-connected, then
For each t ∈ I let U t be a path connected relatively inessential neighborhood of g(t) in X. The sets g -1 (U
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