Fréchet-Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan Sω in a topological group G admitting a functorial embedding [0, 1] ⊂ G. The latter means that each autohomeomorphism of [0, 1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing Sω. Introduction D. Shakhmatov noticed in [16] that the classical Lefschetz-Nöbeling-Pontryagin Theorem on embeddings of n-dimensional compacta into R has no categorical counterpart: one cannot embed every finite-dimensional compact space X into a finite-dimensional topological group FX so that each continuous map f : X → Y extends to a continuous group homomorphism Ff : FX → FY . The proof of this fact exploited Kulesza’s example of a pathological 1dimensional (non-metrizable) compact space that cannot be embedded into a finite-dimensional topological group [11]. However, it was discovered in [4] that the problem lies already at the level of the unit interval [0, 1]: it admits no functorial embedding into any metrizable or finitedimensional group. So, each topological group containing a functorially embedded interval is non-metrizable and thus has uncountable character. In light of this let us remark that the Markov free topological group FMI over the interval I = [0, 1] has character χ(FMI) = d (see [14], [15]), where d is the well-known uncountable small cardinal equal to the cofinality of the poset (ω,≤). This cardinal is equal to the cardinality of continuum c under Martin’s Axiom, but can also be strictly smaller than c in some models of ZFC (see [6], [18]). In this paper we show that the inequality χ(FI) ≥ d holds for many other free topological group constructions. First we give precise definitions. Let T be a subcategory of the category T op of topological spaces and their continuous maps. By a functor of a free topological group on T we understand a pair (F, i) consisting of The authors were supported by the Slovenian Research Agency grants P1-0292-0101-04 and BI-UA/04-06007.