On nondiscreteness of a higher topological homotopy group and its cardinality

Ghane, Helen; Hamed, Z.

doi:10.36045/bbms/1235574202

Cited by 4 publications

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“…• Ω n−1 , first studied in [20,21,22], fails to take values in the category of topological abelian groups. We can then define π τ n = τ • π qtop n = π τ 1 • Ω n−1 to obtain a truly "topological" higher homotopy group.…”

Section: The Construction and Characterizations Of π τmentioning

confidence: 99%

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The fundamental group as a topological group

Brazas

2013

Topology and its Applications

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This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted π τ 1 , takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology. qtop 1: hTop * → qTopGrp from the homotopy category of based spaces to the category of quasitopological groups 1 and continuous homomorphisms [3,7], however, recent results indicate that very often π qtop 1 (X, x 0 ) fails to be a topological group. In 1 A quasitopological group is a group with topology such that inversion is continuous and multiplication is continuous in each variable. See [1] for basic theory.

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Section: The Construction and Characterizations Of π τmentioning

confidence: 99%

“…Thus the topology of π τ 1 (X) is finer than that of G. Remark 3.14. Fabel [14] has recently shown that for each n ≥ 2 the quasitopological homotopy group π [20,21,22], fails to take values in the category of topological abelian groups. We can then define…”

Section: Construction Of π τmentioning

confidence: 99%

The fundamental group as a topological group

Brazas

2013

Topology and its Applications

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The topological fundamental group and free topological groups

Brazas

2011

Topology and its Applications

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The topological fundamental group π top 1 is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space X, we compute the topological fundamental group of the suspension space Σ(X + ) and find that π top 1 (Σ(X + )) either fails to be a topological group or is the free topological group on the path component space of X. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces X for which π top 1 (Σ(X + )) is a Hausdorff topological group to some well known classification problems in topology. 1 One may similarly define the topological fundamental groupoid [20] of an unbased space, however, the same care must be taken with respect to products of quotient maps. 1 arXiv:1006.0119v3 [math.AT] 19 Jul 2010 provide preliminaries and some of the basic theory of topological fundamental groups.The counterexamples mentioned above come from a class of spaces considered in section 2. This class consists of reduced suspensions of spaces with disjoint basepoint (written Σ(X + )). The well known suspension-loop adjunction then provides an unexpected relation to the free (Markov) topological groups. In fact, π top 1 (Σ(X + )) either fails to be a topological group or is the free topological group on the path component space of X. This new connection is particularly surprising since, in general, it is difficult to describe the topological structure of both topological fundamental groups and free topological groups. We also note that much has been done to determine when free topological groups have certain quotient structures and the realization of these objects as homotopy invariants may indicate a potential application to their study. The second purpose of this paper, and our main result, is to fully describe the isomorphism class of π top 1 (Σ(X + )) in the category of quasitopological groups. Specifically, π top 1 (Σ(X + )) is the quotient (via reduction of words) of the path component space of the free topological monoid on X X −1 .Determining when a topology on a group is a group topology (is such that multiplication and inversion are continuous) is fundamental to the theory of topological groups. In section 3, we apply this theory to describe the topological properties of π top 1 (Σ(X + )) and reduce the classification of Hausdorff spaces X such that π top 1 (Σ(X + )) is a Hausdorff topological group (and necessarily a free topological group) to three separate and well known classification problems in topology. We find that π top 1 (Σ(X + )) is a Hausdorff topological group if and only if all four of the following conditions hold: top 0 (X) to be the path component of x in X. This gives an unbased and based version of the functor π top 0 , however, the presence of basepoint will be clear from context. Th...

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