MEASURABLE CARDINALS AND FUNDAMENTAL GROUPS OF COMPACT SPACES ADAM J. PRZE ´ ZDZIECKI 1 Institute of Mathematics, Warsaw University, Warsaw, Poland Email address: adamp@mimuw.edu.pl June 22, 2006 Abstract. We prove that that all groups can be realized as fun- damental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group G is nonmeasurable then the compact space K such that G = π 1 K may be chosen so that it is path connected. MSC: 03E55 (55Qxx) We construct a group whose cardinality equals the least measurable cardinal and which cannot be realized as the fundamental group of a compact Hausdorff space (Theorem 4.2). Since Keesling and Rudyak proved [6] that every group of smaller cardinality is the fundamental group of some compact space, we see that the large cardinal axiom about existence of measurable cardinals is equivalent to the statement that all groups can be obtained as fundamental groups of compact spaces. The last section gives an affirmative answer to the question, asked in [6], whether each group of nonmeasurable cardinality is the fundamental group of a path connected compact space. All spaces considered below are completely regular, I is the closed interval [0, 1] and S 1 is a circle. If X is a space then we denote its Stone- ˇ Cech compactification by βX and its Hewitt realcompactification by υX . If f : X → Y is a map then f denotes the induced map between the compactifications: βX → βY or υX → υY . A cardinal κ is measurable if it admits a countably complete ultra- filter which is not fixed [4]. The least measurable cardinal is denoted by m and the same symbol is used to denote the least ordinal and the discrete space of cardinality m. Note that m is also the least measur- able cardinal in the more restrictive sense [5, after 2.7] of admitting an m-complete ultrafilter. Remark 0.1. The set υm \ m is nonempty. 1 The author was partially supported by grant 1 P03A 005 26. 1