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...

Using universal constructions of topological groups, one can endow the fundamental group of a space with a topology and obtain a topological group. Additionally, the fundamental groupoid of a space becomes enriched over Top when the homsets are endowed with similar topologies. This paper is devoted to a generalization of classical covering theory in the context of these constructions.

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.

The quasitopological fundamental group π qtop 1 (X, x 0 ) is the fundamental group endowed with the natural quotient topology inherited from the space of based loops and is typically non-discrete when X does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group π qtop 1 (X, x 0 ) and properties of the underlying space X such as 'π 1 -shape injectivity' and 'homotopically path-Hausdorff. ' A space X is π 1 -shape injective if the fundamental group canonically embeds in the first shape group so that the elements of π 1 (X, x 0 ) can be represented as sequences in an inverse limit. We show a locally path connected metric space X is π 1 -shape injective if and only if π qtop 1 (X, x 0 ) is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that X is not π 1 -shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space X is homotopically path-Hausdorff if and only if π qtop 1 (X, x 0 ) satisfies the T 1 separation axiom. qtop 1 (X, x 0 ). Examples [6] illustrate that π qtop 1 (X, x 0 ) need not be a topological group even if X is a compact metric space [20][21]. 1 arXiv:1304.6453v3 [math.AT] 26 Jun 2013 π qtop 13. Corollary 33: If X is both locally path connected and the inverse limit of nested retracts of polyhedra, then π qtop 1 (X, x 0 ) is T 1 k s + 3 π qtop 1 (X, x 0 ) is T 2 k s + 3 X is π 1 -shape injective. Theorem 34: If πqtop 1 (X, x 0 ) is T 3 , then π qtop 1 (X, x 0 ) is T 4 .

Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any pathconnected metric space X, and a subgroup H ≤ π 1 (X, x 0 ), we characterize the unique path lifting property relative to H in terms of a new closure operator on the π 1 -subgroup lattice that is induced by maps from a fixed "test" domain into X. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.