Sel. math., New ser. 10 (2004) 359 – 375 1022–1824/04/030359-17 DOI 10.1007/s00029-004-0379-1 c  Birkh¨auser Verlag, Basel, 2004 Selecta Mathematica, New Series On sequences of maps into S 1 with equibounded W 1/2 energies Mariano Giaquinta, Giuseppe Modica and Jiˇ r´ ı Souˇ cek Abstract. In the last years there has been some interest in studying mappings in the fractional Sobolev space W 1/2 (Ω,S 1 ), see e.g., [4] [3] [15] [12] and the paper [5]. Motivated by these papers, we characterize here in the framework of Cartesian currents, see [9], the class of weak limits of sequences of smooth mappings with values into S 1 with equibounded W 1/2 energies. Mathematics Subject Classification (2000). 49Q20. Key words. Sobolev spaces, trace spaces, Cartesian currents, lifting of maps. 1. Mappings in W 1/2 (Ω) We collect here a few facts about functions in the fractional Sobolev space W 1/2 which are relevant for our discussion. We recall, see e.g., [1] [13], that W 1/2 (Ω) is the Hilbert space of real valued functions that have finite W 1/2 -seminorm |u| 2 1/2 :=  Ω  Ω |u(x) − u(y)| 2 |x − y| n+1 dy dx (1) endowed with the norm ||u|| 2 1/2 := ||u|| 2 L 2 (Ω) + |u| 2 1/2 . Lipschitz functions are dense in W 1/2 (Ω) if Ω is smooth, and there is a continuous extension operator T : W 1/2 (Ω) → W 1/2 (R n ). Thus, without loss of generality we can assume Ω to be, for instance, a cube of R n . W 1/2 (R n ) can be characterized by Fourier transform: u belongs to W 1/2 (R n ) if and only if  R n (1 + |k|) |  u(k)| 2 dk < ∞. W 1/2 -functions can be also characterized in terms of 1-dimensional restrictions, see [17]. Denote by Π ⊥ j the coordinate plane x j = 0 in R n , by x ′ the points in Π ⊥ j , and by Ω j,x ′ the 1-dimensional open set intersection of Ω with the line Π j,x ′ through