\magnification=\magstep1 \input amstex \documentstyle{amsppt} \NoBlackBoxes %\NoPageNumbers \loadbold \pageheight{8.5truein} \pagewidth{6.5truein} \def\Mint{\diagup\hskip-.50truecm\int} \def\mint{\diagup\hskip-.38truecm\int} \def\var{\varepsilon} \def\om{\Omega} \def\br{\Bbb R} \def\bc{\Bbb C} \def\bz{\Bbb Z} \def\ga{\Gamma} \def\la{\Lambda} \def\le{\leqslant} \def\ge{\geqslant} \def\W{W^{s,p}} \def\fc{\tfrac} \def\twoint{\int\!\int} \def\fourint{\int\!\int\!\int\!\int} \def\os{\,(\Omega ; S^1)\,} \topmatter \title{On some questions of topology for $S^1$-valued fractional Sobolev spaces}\endtitle \author{HAIM BREZIS$^{(1),(2)}$ AND PETRU MIRONESCU$^{(3)}$}\endauthor \medskip \address (1) \,\, ANALYSE NUM\'ERIQUE\endgraf \qquad UNIVERSIT\'E P. ET M. CURIE, B.C. 187\endgraf \qquad 4 PL. JUSSIEU\endgraf \qquad 75252 PARIS CEDEX 05\endgraf \endaddress \address (2) \,\, RUTGERS UNIVERSITY\endgraf \qquad DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS\endgraf \qquad 110 FRELINGHUYSEN RD, PISCATAWAY, NJ 08854\endgraf \endaddress \email brezis\@ccr.jussieu.fr; brezis\@math.rutgers.edu\endemail \null \address (3) \,\, D\'EPARTEMENT DE MATH\'EMATIQUES\endgraf \qquad UNIVERSIT\'E PARIS-SUD\endgraf \qquad 91405 ORSAY\endgraf \endaddress \email : Petru.Mironescu\@math.u-psud.fr\endemail \medskip \medskip \endtopmatter \document \centerline{} \bigskip \noindent {\bf I. Introduction} \medskip The purpose of this paper is to describe the homotopy classes (i.e., path-connected components) of the space $W^{s,p}\,(\om ; S^1)$. Here, $0 < s < \infty,\, 1 < p < \infty,\, \om$ is a smooth, bounded, connected open set in $\br^N$ and $$ W^{s,p}\,(\om ; S^1) = \{\,u \in W^{s,p}\,(\om ; S^1) ; \,|u| = 1\, a.e.\}. $$ Our main results are \medskip \proclaim{Theorem 1} If $sp <2$, then $W^{s,p}\,(\om ; S^1)$ is path-connected. \endproclaim \medskip \proclaim{Theorem 2} If $sp \ge 2$, then $W^{s,p}\,(\om ; S^1)$ and $C^0\,(\bar \om ; S^1)$ have the same homotopy classes in the sense of [7]. More precisely: \medskip \noindent a) each $u \in W^{s,p}\,(\om; S^1)$ is $W^{s,p}$-homotopic to some $v \in C^\infty\,(\bar {\om}; S^1)$; \medskip \noindent b) two maps $u,v \in C^\infty\,(\bar \om ; S^1)$ are $C^0$-homotopic if and only if they are $W^{s,p}$-homotopic. \endproclaim \medskip Here a simple consequence of the above results \medskip \proclaim{Corollary 1} If $0 < s < \infty, 1 < p < \infty$ and $\om$ is simply connected, then $\W\,(\om ; S^1)$ is path-connected. \endproclaim \medskip Indeed, when $sp < 2$ this is the content of Theorem 1. When $sp \ge 2$, we use a) of Theorem 2 to connect $u_1,\, u_2 \in \W\,(\om ; S^1)$ to $v_1,\, v_2 \in C^\infty\,(\bar \om ; S^1)$; since $\om$ is simply connected, we may write $v_j = e^{i\varphi_j}$ for $\varphi_j \in C^\infty\,(\bar \om ; \br)$ and then we connect $v_1$ to $v_2$ via $e^{i\,[(1 - t) \varphi_1 + t \varphi_2]}$. \medskip When $M$ is a compact connected manifold, the study of the topology of $W^{1,p}\,(\om ; M)$ was initiated in Brezis - Li [7] (see also White [26] for some related questions). In particular, these authors proved Theorems 1 and 2 in the special case $s = 1$. The analysis of homotopy classes for an arbitrary manifold $M$ and $s = 1$ was subsequently tackled by Hang - Lin [15]. The passage to $\W$ introduces two additional difficulties: \medskip \noindent a) when $s$ is not an integer, the $\W$ norm is not ``local''; \medskip \noindent b) when $s \ge 2$ (or more generally $s > 1 + \fc 1 p$), gluing two maps in $\W$ does not yield a map in $\W$. \medskip In our proofs, we exploit in an essential way the fact that the target manifold is $S^1$. (The case of a general target is widely open.) In particular, we use the existence of a lifting of $\W$ unimodular maps when $s \ge 1$ and $sp \ge 2$ (see Bourgain - Brezis - Mironescu [4]). Another important tool is the following \medskip \proclaim{Composition Theorem (Brezis - Mironescu [10])} If $f \in C^\infty\,(\br ;\br)$ has bounded derivatives and $s \ge 1$, then $\varphi \longmapsto f \circ \varphi$ is continuous from $\W \cap W^{1,sp}$ into $\W$. \endproclaim \medskip \noindent {\bf Remark 1}. A very elegant and straightforward proof of this Composition Theorem has been given by V.Maz'ya and T.Shaposhnikova [18]. \medskip A related question is the description, when $sp \ge 2$, of the homotopy classes of $\W\,(\om ; S^1)$ in terms of lifting. Here is a partial result \medskip \proclaim{Theorem 3} We have \medskip \noindent a) if $s \ge 1,\,N \ge 3$, and $2 \le sp < N$, then $$ [u]_{s,p} = \{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\,\cap W^{1,sp}\,(\om ; \br)\}; $$ \noindent b) if $sp \ge N$, then $$ [u]_{s,p} = \{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}. $$ \endproclaim \medskip Theorem 3 is due to Rubinstein - Sternberg [21] in the special case where $s = 1,\, p = 2$ and $\om$ is the solid torus in $\br^3$. \medskip When $0 < s < 1,\,N \ge 3$ and $2 \le sp < N$, there is no such simple description of $[u]_{s,p}$. For instance, using the ``non-lifting'' results in Bourgain - Brezis - Mironescu [4], it is easy to see that $$ [1]_{s,p} \underset \neq \to \supset \{e^{i\varphi} ; \varphi \in \W\,(\om; \br)\}. $$ Here is an example: if $N = 3,\, \om = B_1,\, 0 < s < 1,\, 1 < p < \infty,\, 2 \le sp < 3$, then \medskip \noindent a) $u (x) = e^{1/|x|^\alpha} \in [1]_{s,p}$; \medskip \noindent b) there is no $\varphi \in \W\,(B_1; \br)$ such that $u = e^{i\varphi}$ \medskip \noindent for $\alpha$ satisfying $\tfrac{3 - sp}{p} \le \alpha < \tfrac{3 - sp}{sp}$. \medskip However, we conjecture the following result \medskip \proclaim{Conjecture 1} Assume that $0 < s < 1,\,1 < p < \infty,\,N \ge 3$ and $2 \le sp < N$. Then \endproclaim $$ [u]_{s,p} = u \overline{\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. $$ We will prove below (see Corollary 2) that ``half'' of Conjecture 1 holds, namely $$ [u]_{s,p} \supset u \overline{\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. $$ \medskip In a different but related direction, we establish some partial results concerning the density of $C^\infty\,(\bar \om ; S^1)$ into $\W\,(\om ; S^1)$. \medskip \proclaim{Theorem 4} We have, for $0 < s < \infty,\, 1 < p < \infty$: \medskip \noindent a) if $sp < 1$, then $C^\infty\,(\bar \om ; S^1)$ is dense in $\W\,(\om ; S^1)$; \medskip \noindent b) if $1 \le sp < 2, N \ge 2$, then $C^\infty\,(\bar \om ; S^1)$ is not dense in $\W\,(\om ; S^1)$; \medskip \noindent c) if $sp \ge N$, then $C^\infty\,(\bar \om ; S^1)$ is dense in $\W\,(\om; S^1)$; \medskip \noindent d) if $s \ge 1$ and $sp \ge 2$, then $C^\infty\,(\bar \om ; S^1)$ is dense in $\W\,(\om ; S^1)$. \endproclaim \medskip There is only one missing case for which we make the following \medskip \proclaim{Conjecture 2} If $0 < s < 1,\, 1 < p < \infty,\, N \ge 3,\, 2 \le sp < N$, then $C^\infty\,(\bar \om ; S^1)$ is dense in $\W\,(\om ; S^1)$.\endproclaim \medskip \noindent This problem is open even when $\om$ is a ball in $\br^3$. We will prove below the equivalence of Conjectures 1 and 2. \medskip Parts of Theorem 4 were already known. Part a) is due to Escobedo [14]; so is part b), but in this case the idea goes back to Schoen - Uhlenbeck [24] (see also Bourgain - Brezis - Mironescu [5]). For $s = 1$, part c) is due to Schoen - Uhlenbeck [24]; their argument can be adapted to the general case (see, e.g., Brezis - Nirenberg [12] or Brezis - Li [7]). The only new result is part d). The proof relies heavily on the Composition Theorem and Theorems 2 and 3. We do not know any direct proof of d). We also mention that for $s = 1$ and $\om = B_1$, Theorem 4 was established by Bethuel - Zheng [3]. For a general compact connected manifold $M$ and for $s = 1$, the question of density of $C^\infty\,(\bar \om ;M)$ into $W^{1,p}\,(\om ;M)$ was settled by Bethuel [1] and Hang - Lin [15]. \medskip \noindent {\bf Remark 2.} In Theorems 2 and 4, one may replace $\om$ by a manifold with or without boundary. The statements are unchanged. However, the argument in the proof of Theorem 1 does not quite go through to the case of a manifold without boundary. Nevertheless, we make the following \medskip \proclaim{Conjecture 3} Let $\om$ be a manifold without boundary with $\dim \om \ge 2$. Then $\W\,(\om ; M)$ is path-connected for every $0 < s < \infty, 1 < p < \infty$ with $sp < 2$, and for every compact connected manifold $M$. \endproclaim \medskip Note that the condition $\dim \om \ge 2$ is necessary, since $\W\,(S^1;S^1)$ is not path-connected when $sp \ge 1$. \medskip Finally, we investigate the local path-connectedness of $\W\,(\om ; S^1)$. Our main result is \proclaim{Theorem 5} Let $0 < s < \infty, 1 < p < \infty$. Then $\W\,(\om ; S^1)$ is locally path-connected. Consequently, the homotophy classes coincide with the connected components and they are open and closed. \endproclaim \medskip The heart of the matter in the proof is the following \medskip \noindent {\bf Claim}. Let $0 < s < \infty, 1 < p < \infty$. Then there is some $\delta > 0$ such that, if $||u - 1||_{\W} < \delta$, then $u$ may be connected to $1$ in $\W$. \medskip As a consequence of Theorem 5, we have \medskip \proclaim{Corollary 2} Let $0 < s < 1, 1 < p < \infty$. Then \endproclaim $$ [u]_{s,p}\supset \, \overline{\{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}\,\, = u \, \overline{\{ e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. $$ \medskip Equality in Corollary 2 follows from the well-known fact that $\W \cap L^\infty$ is an algebra. The inclusion is a consequence of the fact that, clearly, we have $$ [u]_{s,p} \,\,\supset \,\,\{u e^{i\varphi};\,\varphi \in \W\,(\om ; \br)\} $$ and of the closedness of the homotopy classes. \medskip Another consequence of Theorem 5 is \medskip \proclaim{Corollary 3} Conjecture 1 $\Leftrightarrow$ Conjecture 2. \endproclaim \medskip {\bf Proof.} By Corollary 2, we have $$ [u]_{s,p} \supset u \overline {\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. $$ We prove that the reverse inclusion follows from Conjecture 1. By Proposition 1 a) below, we may take $u = 1$. Let $v \in [1]_{s,p}$. By Theorem 5, there is some $\var > 0$ such that $||v-w||_{\W} < \var \,\Rightarrow \, w \in [1]_{s,p}$. Let $(w_n) \subset C^\infty\,(\bar \om ; S^1)$ be such that $w_n \to v$ in $\W$ and $||w_n - v||_{\W} < \var$. By Theorem 2 b), we obtain that $w_n$ and $1$ are homotopic in $C^0\,(\bar \om ; S^1)$. Thus $w_n = e^{i\varphi_n}$ for some {\bf globally} defined smooth $\varphi_n$. Hence $$ v \in \overline {\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}. $$ \medskip Conversely, assume that Conjecture 2 holds. Let $u \in \W\,(\om ; S^1)$. By Theorem 2 a), there is some $w \in C^\infty\,(\bar \om ; S^1)$ such that $w \in [u]_{s,p}$. By Proposition 1 b), we have $u \bar w \in [1]_{s,p}$. Thus $u \bar w \in\,\overline {\{e^{i\varphi} ; \varphi \in \W\,(\om ; \br)\}}^{\W}$, so that clearly $u \bar w \in\,\overline {\{e^{i\varphi} ; \varphi \in C^\infty(\bar \om ; \br)\}}^{\W}$. Finally, $u \in \overline {\{w e^{i\varphi} ; \varphi \in C^\infty\,(\bar \om ; \br)\}}^{\W}$, i.e. $u$ may be approximated by smooth maps. \medskip In the same vein, we raise the following \medskip \noindent {\bf Open Problem 1}. Let $\om$ be a manifold with or without boundary. Is $\W\,(\om ; M)$ locally path-connected for every $s,p$ and every compact manifold $M$? \medskip The case $s = 1$ can be settled using the methods of Hang - Lin [15]. We will return to this question in a subsequent work; see Brezis - Mironescu [11]. \medskip The reader who is looking for more open problems may also consider the following \medskip \noindent {\bf Open Problem 2}. Let $\om \subset \br^2$ be a smooth bounded domain. Assume $0 < s < \infty,$ \newline $1 < p < \infty$ and $1 \le sp < 2$ (this is the range where $C^\infty\,(\bar \om ; S^1)$ is not dense in $\W\,(\om ; S^1)$). Set $$ \Cal R_0 = \{ u \in \W\,(\om ; S^1); u \text{ is smooth except a finite number of points}\}. $$ (Here, the number and location of singular points is left free). Is $\Cal R_0$ dense in $\W\,(\om ; S^1)$? \medskip \noindent {\bf Comment.} $\Cal R_0$ is known to be dense in $\W\,(\om ; S^1)$ in many cases, e.g.: \medskip \noindent a) $s = 1$ and $1 \le p < 2$; see Bethuel-Zheng [3] \medskip \noindent b) $s = 1 - 1/p$ and $ 2 < p < 3$; see Bethuel [2] \medskip \noindent c) $s = 1/2$ and $p = 2$; see Rivi\`ere [20]. \bigskip The paper is organized as follows \bigskip \noindent I. Introduction \medskip \noindent II. Proof of Theorem 1 \medskip \noindent III. Proof of Theorems 2 and 3 \medskip \noindent IV. Proof of Theorem 4 \medskip \noindent V. Proof of Theorem 5 \bigskip \noindent Appendix A. An extension lemma \medskip \noindent Appendix B. Good restrictions \medskip \noindent Appendix C. Global lifting \medskip \noindent Appendix D. Filling a hole - the fractional case \medskip \noindent Appendix E. Slicing with norm control \bigskip \noindent {\bf II. Proof of Theorem 1} \medskip \noindent {\bf Case 1:} $sp < 1$ \medskip \noindent When $sp < 1$, we have the following more general result \proclaim{Theorem 6} If $s > 0, \,1 < p < \infty, \,sp < 1$ and $M$ is a compact manifold, then $\W\,(\om ; M)$ is path-connected. \endproclaim \medskip {\bf Proof.} Fix some $a \in M$. For $u \in \W\,(\om ; M)$, let $$ \tilde u \,=\,\cases u, &\text{ in } \om \\ a, &\text{ in } \br^N \backslash \om \endcases . $$ \noindent Since $sp < 1$, we have $\tilde u \in W^{s,p}_{loc}\,(\br^N ; M)$. Let $U\,(t,x)\,=\, \tilde u\,(x/(1 - t)) , \,0 \le t < 1, \,x \in \om$ and $U\,(1,x)\, \equiv a$. Then clearly $U \in C\,([0,1] ; \W\,(\om ; M))$ and $U$ connects $u$ to the constant $a$ (here we use only $sp < N$). \medskip \noindent {\bf Case 2:} $1 < sp < 2, N \ge 2$ \medskip In this case one could adapt the tools developed in Brezis - Li [7], but we prefer a more direct approach. \medskip Let $\var > 0$ be such that the projection onto $\partial \om$ be well-defined and smooth in the region $\{ x \in \br^N ;\text{ dist } (x, \partial \om)\, < 2 \var\}$. Let $\omega = \{ x \in \br^N \backslash \bar \om ;\text{ dist }\,(x, \partial \om) < \var\}$. We have $\partial \omega \, = \, \partial \om \cup \Lambda$, where $\la \, = \, \{ x \in \br^N \backslash \om ; \text{ dist }\,(x, \partial \om)\, = \var \}$. \medskip Since $1 < sp < 2$, we have $1/p < s < 1 + 1/p$; thus, for $u \in \W$ we have tr $u \in W^{s - 1/p,p}$. Let $u \in \W\,(\om ; S^1)$. Fix some $ a \in S^1$ and define $v \in W^{s - 1/p,p}\,(\partial \omega ; S^1)$ by $$ v\, = \, \cases \text{tr }u, \quad &\text{ on } \partial \omega\\ a, \quad &\text{ on } \Lambda \endcases . $$ \medskip We use the following extension result. (The first result of this kind is due to Hardt - Kinderlehrer - Lin [16]; it corresponds to our lemma when $\sigma = 1 - 1/p,\,p < 2$.) \medskip \proclaim{Lemma 1} Let $0 < \sigma < 1,\, 1 < p < \infty,\, \sigma p < 1$. Then any $v \in W^{\sigma,p}\,(\partial \omega ; S^1)$ has an extension $w \in W^{\sigma + 1/p,p}\,(\omega ; S^1)$. \endproclaim \medskip The proof is given in Appendix A; see Lemma A.1. It relies heavily on the lifting results in Bourgain - Brezis - Mironescu [4]. \medskip Returning to the proof of Case 2, with $w$ given by Lemma 1, set $$ \tilde u \,=\, \cases u, \,\, \quad &\text{ in } \om\\ w, \,\, \quad &\text{ in } \omega\\ a, \,\, \quad &\text{ in } \br^n \backslash\,(\om \cup \omega) \endcases . $$ \medskip Clearly, $\tilde u \in W^{s,p}_{loc}\,(\br^N ; S^1)$ and $\tilde u$ is constant outside some compact set. As in the proof of Theorem 6, we may use $\tilde u$ to connect $u$ to $a$, since once more we have $sp < N$. \medskip \noindent {\bf Case 3:} $sp = 1, \,\, N \ge 2$ \medskip The idea is the same as in the previous case; however, there is an additional difficulty, since in the limiting case $s = 1/p$ the trace theory is delicate - in particular, tr $W^{1/p,p}\, \neq\, L^p$ (unless $p = 1$). Instead of trace, we work with a notion of ``good restriction'' developed in Appendix B; when $s = 1/2, \, p = 2$, the space of functions in $H^{1/2}$ having $0$ as good restriction on the boundary coincides with the space $H^{1/2}_{00}$ of Lions - Magenes [17] (see Theorem 11.7, p. 72). \medskip Our aim is to prove that any $u \in W^{1/p,p}\,(\om ; S^1)$ can be connected to a constant $a \in S^1$. \medskip {\bf Step 1:} we connect $u \in W^{1/p,p}\,(\om ; S^1)$ to some $u_1 \in W^{1/p,p}\,(\om ; S^1)$ having a good restriction on $\partial \om$ \medskip Let $\var > 0$ be such that the projection $\Pi$ onto $\partial \om$ be well-defined and smooth in the set $\{x \in \br^N; \text{ dist } \,(x, \partial \om)\, < 2 \var)\}$. For $0 < \delta < \var$, set $\Sigma_{\delta} = \{x \in \om ; \text{ dist }\,(x, \partial \om) = \delta\}$. By Fubini, for a.e. $0 < \delta < \var$ , we have $$ u|_{\sum_{\delta}} \in W^{1/p,p}\,(\Sigma_\delta) \text{ and } \underset {\Sigma_{\delta}}\to\int\,\underset {\om}\to\int \frac{|u(x) - u(y)|^p}{|x - y|^{N + 1}}\,dy\,ds_x\, < \infty . \tag 1 $$ By Lemma B.5, this implies that $u$ has a good restriction on $\Sigma_\delta$, and that Rest $u|_{\sum_{\delta}}\, = \, u|_{\sum_{\delta}}$ a.e. on $\Sigma_{\delta}$. \medskip Let any $0 < \delta < \var$ satisfying (1). For $0 < \lambda < \delta$, let $\Psi_\lambda$ be the smooth inverse of $\Pi|_{\sum_{\lambda}} \, : \, \Sigma_{\lambda} \to \partial \om$. Let also $\om_\lambda \, = \, \{x \in \om ; \text{ dist }\,(x, \partial \om) \, > \lambda\}$. Consider a continuous family of diffeomorphisms $\Phi_t : \bar \om \to \overline{\om_{t\delta}}, 0 \le t \le 1$, such that $\Phi_0 =$ id and $\Phi_t |_{\partial \om}\, = \, \Psi_{t\delta}$. Then $t \mapsto u \circ \Phi_t$ is a homotopy in $W^{1/p,p}$. Moreover, if $u_t \, = \, u \circ \Phi_t$, then $u_0 = u$ and $u_1|_{\partial \om} = u|_{\sum_\delta} \circ \bold\Psi_\delta|_{\partial \om}$. By (1), $u_1$ has a good restriction on $\partial \om$. \medskip {\bf Step 2:} we extend $u_1$ to $\br^N$ \medskip Let $\omega = \{x \in \br^N \backslash \bar \om ; \text{ dist }\,(x ; \partial \om)\, < \var\}$. As in Case 2, we fix some $a \in S^1$ and set $$ v \, = \,\cases u_1, \quad &\text{ on } \partial \om\\ a, \quad &\text{ on } \Lambda \endcases . $$ Clearly, $v \in W^{1/p,p}\,(\partial\omega)$, so that $v \in W^{\sigma,p}\,(\partial\omega)$ for $0 < \sigma < 1/p$. We fix any $0 < \sigma < 1/p$. By Lemma 1, there is some $w \in W^{\sigma + 1/p,p}\,(\omega ; S^1)$ such that $w|_{\partial \omega} = v$. We define $$ \tilde u_1 \, = \,\cases u_1, \quad &\text{ in } \om\\ w, \quad &\text{ in } \omega\\ a, \quad &\text{ in } \br^N \backslash (\om \cup \omega) \endcases . $$ We claim that $\tilde u_1 \in W^{1/p,p}_{loc}\,(\br^N ; S^1)$. Obviously, $\tilde u \in W^{1/p,p}_{loc}\,(\br^N \backslash \om)$. It remains to check that $\tilde u_1 \in W^{1/p,p}\,(\om \cup \omega)$. This is a consequence of \medskip \proclaim{Lemma 2} Let $0 < s < 1, 1 < p < \infty, sp \ge 1$ and $\rho >s$. Let $u_1 \in \W (\om)$ and $ w \in W^{\rho,p} (\omega)$. Assume that $u_1$ has a good restriction {\rm Rest } $u_1|_{\partial \om}$ on $\partial \om$ and that $\hskip 10mm$ {\rm tr } $w|_{\partial \om} =$ {\rm Rest } $u_1|_{\partial \om}$. Then the map $$ \cases u_1,\,\, &\text{ in }\om\\ w,\,\, &\text{ in } \omega \endcases $$ belongs to $\W\,(\om \cup \omega)$. \endproclaim \medskip Clearly, in the proof of Lemma 2 it suffices to consider the case of a flat boundary. When $\om\,=\, (-1,1)^{N - 1}\, \times\,(0,1)$ and $\omega\,=\, (-1,1)^{N - 1}\, \times\,(-1,0)$, the proof of Lemma 2 is presented in Appendix B; see Lemma B.4. \medskip Returning to Case 3 and applying Lemma 2 with $s = 1/p,\, \rho = \sigma + 1/p$, we obtain that $\tilde u_1 \in W^{1/p,p}_{loc}\,(\br^N)$. As in the two previous cases, this means that $u_1$ is $W^{1/p,p}$-homotopic to a constant. \medskip \noindent {\bf Case 4:} $1 \le sp < 2,\, N=1$ \medskip \noindent In this case, $\om$ is an interval. Recall the following result proved in Bourgain - Brezis - Mironescu [4] (Theorem 1): if $\om$ is an interval and $sp \ge 1$, then for each $u \in \W (\om ; S^1)$ there is some $\varphi \in \W \,(\om ;\br)$ such that $u = e^{i\varphi}$. Recall also that, when $sp \ge N$, then $C^\infty\,(\br;\br)$ functions $f$ with bounded derivatives operate on $\W$; that is, the map $\varphi \mapsto f \circ \varphi$ is continuous from $\W$ into itself (see, e.g., Peetre [19] for $sp > N$, Runst - Sickel [23], Corollary 2 and Remark 5 in Section 5.3.7 or Brezis - Mironescu [9] when $sp = N$; this is also a consequence of the Composition Theorem). By combining these two results, we find that the homotopy $t \mapsto e^{i (1 - t) \varphi}$ connects $u = e^{i\varphi}$ to $1$. \medskip The proof of Theorem 1 is complete. \bigskip \bigskip {\bf III. Proof of Theorems 2 and 3} \medskip We start with some useful remarks. For $u \in \W\,(\om ; S^1)$, let $[u]_{s,p}$ denote its homotopy class in $\W$. \proclaim{Proposition 1} Let $0 < s < \infty,\,1 < p < \infty$. For $u, v \in \W\,(\om ; S^1)$, we have \endproclaim \medskip \noindent a) $u [v]_{s,p}\,=\, [uv]_{s,p}$; \medskip \noindent b) $[u]_{s,p}\,=\, [v]_{s,p}\,\Leftrightarrow\,[u \bar v]_{s,p}\,=\,[1]_{s,p}$; \medskip \noindent c) $[u]_{s,p}\,\,[v]_{s,p}\,=\,[uv]_{s,p}$. \medskip The proof relies on two well-known facts: $\W \cap L^\infty$ is an algebra; moreover, if $u_n \to u, \,v_n \to v$ in $\W$ and $||u_n||_{L^\infty}\,\le\,C,\,||v_n||_{L^\infty}\,\le\,C$, then $u_n\,v_n \to \,uv$ in $\W$. Here is, for example, the proof of c) (using a)). Let first $u_1 \in [u]_{s,p},\,v_1 \in [v]_{s,p}$. If $U,V$ are homotopies connecting $u_1$ to $u$ and $v_1$ to $v$, then $UV$ connects $u_1\,v_1$ to $uv$; thus $[u]_{s,p}\,[v]_{s,p} \subset [uv]_{s,p}$. Conversely, if $w \in [uv]_{s,p}$, then $w \in u [v]_{s,p}$ (by a)), so that $w \bar u \in [v]_{s,p}$. Therefore, $w = u(w \bar u) \in [u]_{s,p}\,[v]_{s,p}$. \medskip We next recall the degree theory for $\W$ maps; see Brezis - Li - Mironescu - Nirenberg [8] for the general case, White [25] when $s = 1$ or Rubinstein - Sternberg [20] for the space $H^1\,(\om ; S^1)$ and $\om$ the solid torus in $\br^3$. Let $0 < s < \infty,\, 1 < p < \infty$ be such that $sp \ge 2$. Let $ u \in \W\,(S^1 \,\times\, \la ; S^1)$, where $\la$ is some open connected set in $\br^k$. Clearly, for a.e. $\lambda \in \la, u\,(\cdot, \lambda) \in \W\,(S^1 ; S^1)$. For any such $\lambda, u\,(\cdot, \lambda)$ is continuous, so that it has a winding number (degree) deg $\bigl( {u\,(\cdot, \lambda)\bigr )}$. The main result in [8] asserts that, if $sp \ge 2$, then this degree is constant a.e. and stable under $\W$ convergence. \medskip In the particular case where $s \ge 1$, there is a formula $$ \text { deg } ({u(\cdot, \lambda))}\,=\, \frac{1}{2\pi} \underset{S^1}\to \int u\,(x,\lambda)\, \wedge\, \frac{\partial u}{\partial \tau}\, (x, \lambda)\,ds_x, $$ where $u \wedge v \,=\, u_1\,v_2 - u_2\,v_1$. It then follows that, if $s \ge 1$ and $sp \ge 2$, we have $$ \text { deg }(u|_{S^1 \times \la})\,=\, \underset\la\to \Mint \, \underset {S^1}\to \Mint \, u(x,\lambda)\, \wedge \frac{\partial u}{\partial \tau}\, (x, \lambda)\, ds_x d \lambda. $$ \medskip Clearly, the above result extends to domains which are diffeomorphic to $S^1 \times \la$. In the sequel, we are interested in the following particular case: let $\ga$ be a simple closed smooth curve in $\om$ and, for small $\var > 0$, let $\ga_\var$ be the $\var$-tubular neighborhood of $\ga$. We fix an orientation on $\ga$. \medskip Let $\Phi : S^1 \times B_\var \to \ga_\var$ be a diffeomorphism such that $\Phi|_{S^1 \times \{0\}} : S^1 \times \{0\} \to \ga$ be an orientation preserving diffeomorphism; here $B_\var$ is the ball of radius $\var$ in $\br^{N - 1}$. Then we may define deg $(u|_{\ga_\var})\,=\,$ deg $(u \circ \Phi|_{S^1 \times B_\var})$; this integer is stable under $\W$ convergence. \medskip We now prove b) of Theorem 2, which we restate as \medskip \proclaim{Proposition 2} Let $0 < s < \infty,\,1 < p < \infty,\, sp \ge 2$. Let $ u, v \in C^\infty\,(\bar \om ; S^1)$. Then $[u]_{s,p}\,=\,[v]_{s,p}$ if and only if $u$ and $v$ are $C^0$- homotopic. \endproclaim \medskip {\bf Proof.} Using Proposition 1, we may assume $v = 1$. Suppose first that $u \in C^\infty\,(\bar \om ; S^1)$ and $1$ are $C^0$-homotopic. Then $u$ and $1$ are $\W$-homotopic. Indeed, when $s = 1$, this is proved in Brezis - Li [7], Proposition A.1; however, their proof works without modification for any $s$. We sketch an alternative proof: since $u$ and $1$ are $C^0$-homotopic, there is some $\varphi \in C^\infty\,(\bar \om ; \br)$ such that $u = e^{i\varphi}$. Then $t \mapsto e^{i\,(1 - t)\,\varphi}$ connects $u $ to $1$ in $\W$. \medskip Conversely, assume that the smooth map $u$ is $\W$-homotopic to $1$. By continuity of the degree, we then have deg $(u|_{\ga_{\var}})\,=\,0$ for each $\Gamma$. Since $u$ is smooth, we obtain $$ 0 = \text{ deg }(u|_{\ga_\var})\,=\, \text { deg }(u|_\ga)\, = \frac{1}{2\pi} \underset\ga\to \int\, u \wedge \frac{\partial u}{\partial \tau} ds. $$ Thus the closed form $X = u \wedge D u$ has the property that $\underset\ga\to \int X \cdot \tau ds = 0$ for any simple closed smooth curve $\ga$. By the general form of the Poincar\'e lemma, there is some $\varphi \in C^\infty\,(\bar \om ; \br)$ such that $X = D\varphi$. One may easily check that $u = e^{i(\varphi + C)}$ for some constant $C$. Then $t \mapsto e^{i(1 - t)\,(\varphi + C)}$ connects $u$ to $1$ in $C^0\,(\bar \om ; S^1)$. \medskip We now turn to the proof of the remaining assertions in Theorems 2 and 3. \medskip \noindent {\bf Case 1:} $sp \ge N,\,N \ge 2$ \medskip {\bf Step 1:} each $u \in \W\,(\om ; S^1)$ can be connected to a smooth map $v \in C^\infty\,(\bar \om ; S^1)$ \medskip This is proved in Brezis - Li [7], Proposition A.2, for $s = 1$ and $p \ge N$; their arguments apply to any $s$ and any $p$ such that $sp \ge N$. The main idea originates in the paper Schoen - Uhlenbeck [23]; see also Brezis - Nirenberg [12], [13]. \medskip {\bf Step 2:} we have $[u]_{s,p}\,=\,\{u e^{i\varphi}; \varphi \in \W\,(\om ; \br)\}$ \medskip Let $\varphi \in \W\,(\om ; \br)$. Then $t \longmapsto u e^{i(1 - t)\varphi}$ connects $u e^{i\varphi}$ to $u$ in $\W$. (Recall that, if $f \in C^\infty\,(\br ; \br)$ has bounded derivatives and $sp \ge N$, then the map $\varphi \mapsto f \circ \varphi$ is continuous from $\W$ into itself.) This proves ``$\supset$''. To prove the reverse inclusion, by Proposition 1, it suffices to show that $[1]_{s,p}\,\subset\,\{e^{i\varphi}\,;\varphi \in \W\,(\om ; \br)\,\}$. \medskip Let $v \in [1]_{s,p}$. For each $x \in \om$, let $B_x \subset \om$ be a ball containing $x$. We recall the following lifting result from Bourgain - Brezis - Mironescu [4] (Theorem 2): if $U$ is simply connected in $\br^N$ and $sp \ge N$, then for each $w \in \W\,(U ; S^1)$ there is some $\psi \in \W\,(U ; \br)$ such that $w = e^{i\psi}$. Thus, for each $x \in \om$ there is some $\varphi_x \in \W\,(B_x ; \br )$ such that $v|_{B_x} = e^{i\varphi_x}$. Note that , in $B_x \cap B_y$, we have $\varphi_x - \varphi_y \in \W\,(B_x \cap B_y; 2 \pi \bz)$. Therefore, $\varphi_x - \varphi_y \in VMO\,(B_x \cap B_y; 2\pi \bz)$, since $sp \ge N$. It then follows that $\varphi_x - \varphi_y$ is constant a.e. on $B_x \cap B_y$; see Brezis - Nirenberg [12], Section I.5. \medskip By a standard continuation argument, we may thus define a (multi-valued) argument $\varphi$ for $v$ in the following way: fix some $x_0 \in \om$. For any $x \in \om$, let $\gamma$ be a simple smooth path from $x_0$ to $x$. Then, for $\var > 0$ sufficiently small, there is a unique function $\varphi^\gamma \in \W\,(\gamma_\var ; \br)$ such that $v|_{\gamma_\var} \,=\, e^{i\varphi^\gamma}$ and $\varphi^\gamma|_{B_\var (x_0)} \,=\, \varphi_{x_0}|_{B_\var (x_0)}$; here, $\gamma_\var$ is the $\var$-tubular neighborhood of $\gamma$. We then set $$ \varphi|_{B_{\var}(x)}\,=\,\varphi^\gamma|_{B_{\var}(x)}. $$ \medskip We actually claim that $\varphi$ is single-valued. This follows from \medskip \proclaim{Lemma 3} Assume that $0 < s < \infty,\,1 < p < \infty,\, sp \ge N,\,N \ge 2$. If $w \in \W\,(S^1 \times B_1; S^1)$ is such that {\rm deg } $\left(w|_{S^1 \times B_1}\right) = 0$, then there is some $\psi \in \W\,(S^1 \times B_1)$ such that $w = e^{i\psi}$. \endproclaim \medskip Here, $B_1$ is the unit ball in $\br^{N-1}$. The proof of Lemma 3 is presented in Appendix C; see Lemma C.1. \medskip Returning to the claim that $\varphi$ is single-valued, we have that deg $\left(v|_{\ga_{\var}}\right)\,=\,0$ for each $\ga$, since $v \in [1]_{s,p}$. By Lemma 3, a standard argument implies that $\varphi$ is single-valued. \medskip The proof of Theorems 2 and 3 when $sp \ge N$ is complete. \bigskip \noindent {\bf Case 2:} $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp < N$ \medskip {\bf Step 1:} we have $[u]_{s,p} = \{u e^{i\varphi} ; \varphi \in \W\,(\om ; \br) \cap W^{1,sp}\,(\om ; \br)\}$ \medskip For ``$\supset$'', we use the Composition Theorem mentioned in the Introduction, which implies that $t \mapsto u e^{i(1-t)\varphi}$ connects $u e^{i\varphi}$ to $u$ in $\W$. \medskip For ``$\subset$'' it suffices to prove that $[1]_{s,p}\,\subset\,\{e^{i\varphi}; \varphi \in \W\,(\om ; \br)\, \cap W^{1,sp}\,(\om ; \br)\}$. We proceed as in Case 1, Step 2. Let $v \in [1]_{s,p}$. The corresponding lifting result we use is the following (see Bourgain - Brezis - Mironescu [4], Lemma 4): if $s \ge 1,\,sp \ge 2$ and $U$ is simply connected in $\br^N$, then for each $w \in \W\,(U ; S^1)$ there is some $\psi \in \W\,(U ; \br)\, \cap W^{1,sp}\,(U ; \br)$ such that $w\,=\,e^{i\psi}$. As in Case 1, for each $x$ there is some $\varphi_x \in \W\,(B_x ; \br)\,\cap W^{1,sp}\,(B_x ; \br)$ such that $v|_{B_x}\,=\,e^{i\varphi_x}$. Since $\varphi_x\,-\,\varphi_y \in W^{1,1}\,(B_x \cap B_y; 2\pi \bz)$, we find that $\varphi_x - \varphi_y$ is constant ae. on $B_x \cap B_y$ (see [4], Theorem B.1.). These two ingredients allow the construction of a multi-valued phase $\varphi \in \W \cap W^{1,sp}$ for $v$. To prove that $\varphi$ is actually single-valued, we rely on \medskip \proclaim{Lemma 4} Assume that $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp < N$. If $w \in \W\,(S^1 \,\times\, B_1; S^1)$ is such that {\rm deg } $(w|_{S^1 \,\times\, B_1})\,=\,0$, then there is some $\psi \in \W\,(S^1 \,\times B_1; \br)\,\cap W^{1,sp}\,(S^1 \,\times \,B_1; \br)$ such that $v\,=\,e^{i\psi}$. \endproclaim \medskip The proof of Lemma 4 is given in Appendix C; see Lemma C.2. \medskip The proof of Step 1 is complete. \medskip {\bf Step 2:} assume $s \ge 1,\,1 < p < \infty,\,sp \ge 2$; then, for each $u \in \W\,(\om ; S^1)$, there is some $v \in \W\,(\om ; S^1)\,\cap C^\infty\,(\om ; S^1)$ such that $v \in [u]_{s,p}$ \medskip Consider the form $X\,=\,u \wedge Du$. Then $X \in W^{s-1,p}\,(\om)\,\cap L^{sp}\,(\om)$ (see Bourgain - Brezis - Mironescu [4], Lemmas D.1 and D.2). Let $\varphi \in \W\,(\om ; \br)\,\cap W^{1,sp}\,(\om ; \br)$ be any solution of $\Delta \varphi\,=\,$ div $X$ in $\om$. By the Composition Theorem, we then have $e^{- i\varphi} \in \W\,(\om ; S^1)$, and thus $v\,=\,u e^{- i\varphi} \in \W\,(\om ; S^1)$. We claim that $v \in C^\infty\,(\om ; S^1)$. Indeed, let $B$ be any ball in $\om$. Since $s \ge 1$ and $sp \ge 2$, there is some $\psi \in \W\,(B ; \br)\,\cap W^{1,sp}\,(B; \br)$ such that $u|_B = e^{i\psi}$. It then follows that $X|_B \,=\, D\psi$. Thus $\Delta \varphi\,=\,\Delta \psi$ in $B$, i.e., $\psi - \varphi$ is harmonic in $B$. Since in $B$ we have $v\,=\,u e^{- i\varphi}\,=\,e^{i(\psi - \varphi)}$, we obtain that $v \in C^\infty (B)$, so that the claim follows. \medskip Using Step 1 and the equality $v = u e^{-i\varphi}$, we obtain that $v \in [u]_{s,p}$. \medskip {\bf Step 3:} for each $u \in \W\,(\om ; S^1)$, there is some $w \in C^\infty\,(\bar \om; S^1)$ such that $w \in [u]_{s,p}$ \medskip In view of Step 2, it suffices to consider the case where $u \in \W \os \cap C^\infty \os$. We use the same homotopy as in Step 1, Case 3, in the proof of Theorem 1: $t \mapsto u \circ \Phi_t$, where $\Phi_t$ is a continuous family of diffeomorphisms $\Phi_t : \bar \om \to \overline{\om_{t \delta}}$ such that $\Phi_0 = id$. Clearly, $v\,=\, u \circ \Phi_1 \in C^\infty\,(\bar \om ; S^1)$. \medskip The conclusions of Theorems 2 and 3 when $s \ge 1,\,1 < p < \infty, \, N \ge 3,\, 2 \le sp < N $ follow from Proposition 2 and Steps 1 and 3. \medskip We now complete the proof of Theorem 2 with \medskip \noindent {\bf Case 3:} $0 < s < 1,\,1

0$ be small enough; for $x \in \br^N$, we set $$ \Cal C^x_N\,=\, \bigcup \{x + \var l + (0,\var)^N; l \in \bz^N \text{ and } x + \var l + (0,\var)^N \subset \Cal O\}. $$ Define also $\Cal C^x_j, j = 1, ..., N - 1$, by backward induction : $\Cal C^x_j$ is the union of faces of cubes in $\Cal C^x_{j + 1}$. \medskip By Fubini, for a.e. $x \in \br^N$, we have $u|_{\Cal C^x_j} \in \W,\,j = 1,...,\,N-1$, in the following sense: since $1/p < s < 1$, we have tr $u|_{\Cal C^x_{N - 1}} \in W^{s - 1/p,p}$ for all $x$. However, for a.e. $x$, we have the better property tr $u|_{\Cal C^x_{N-1}} = u|_{\Cal C^x_{N-1}} \in \W$. For any such $x$, we have \break\hfill\noindent tr $\left(u|_{\Cal C^x_{N-1}}\right)\big|_{\Cal C^x_{N-2}} \in W^{s - 1/p,p}$, but once more for a.e. such $x$ we have the better property tr $\left(u|_{\Cal C^x_{N-1}}\right)\big|_{\Cal C^x_{N-2}} = u|_{\Cal C^x_{N-2}} \in \W$, and so on. (See Appendix E for a detailed discussion). \medskip We fix any $x$ having the above property and we drop from now on the superscript $x$. \medskip {\bf Step 1:} we connect $u$ to some smoother map $u_1$ Let $k = [sp]$, so that $2 \le k \le N-1$. Since $u|_{\Cal C_k} \in \W$ and $sp \ge k$, there is a neighborhood $\omega$ of $\Cal C_k$ in $\Cal C_{k +1}$ and an extension $\tilde u \in W^{s + 1/p,p}\,(\omega ; S^1)$ of $u|_{\Cal C_k}$. This extension is first obtained in each cube $C \subset \Cal C_{k+1}$ starting from $u|_{\partial C}$ (see Brezis - Nirenberg [12], Appendix 3, for the existence of such an extension). We next glue together all these extensions to obtain $\tilde u;\, \tilde u$ belongs to $W^{s+1/p,p}$ since $1/p < s + 1/p < 1 + 1/p$. Moreover, the explicit construction in [12] yields some $\tilde u \in C^\infty\,(\omega \backslash \Cal C_k)$. We next extend $\tilde u$ to $\Cal C_{k+1}$ in the following way: for each $C \subset \Cal C_{k+1}$, let $\Sigma_C$ be a convex smooth hypersurface in $C \cap \omega$. Since $\Sigma_C$ is $k$-dimensional and $k \ge 2, \tilde u|_{\Sigma_C}$ may be extended smoothly in the interior of $\Sigma_C$ as an $S^1$-valued map (here, we use the fact that $\pi_k\,(S^1)\,=\,0$). Let $\tilde u_C$ be such an extension. Then the map $$ v = \cases \tilde u,\quad &\text{ outside the } \Sigma_C \text{'s}\\ \tilde u_C, \quad &\text{ inside } \Sigma_C \endcases $$ belongs to $W^{s + 1/p,p}\,(\Cal C_{k+1})$. To summarize, we have found some $v \in W^{s+1/p,p}\,(\Cal C_{k+1}; S^1)$ such that $v|_{\Cal C_k} = u|_{\Cal C_k}$. \medskip Pick any $s \, < \, s_1\, <\,$min$\, \{ s + 1/p, 1\}$ and let $p_1$ be such that $s_1 p_1 = sp + 1$ (note that $1 < p_1 <\infty$). By Gagliardo - Nirenberg (see, e.g., Runst [22], Lemma 1, p.329 or Brezis - Mironescu [10], Corollary 3), we have $W^{s + 1/p,p} \cap L^\infty \subset W^{s_1,p_1}$. Thus $v \in W^{s_1,p_1}\,(\Cal C_{k+1})$. \medskip We complete the construction of the smoother map $u_1$ in the following way: if $k = N - 1$, then $v$ is defined in $\Cal C_N$ and we set $u_1 = v$; if $k < N - 1$, we extend $v$ to $\Cal C_N$ with the help of \medskip \proclaim{Lemma 5} Let $0 < s_1 < \infty,\,1 < p_1 < \infty,\,1 0$ such that \endproclaim $$ \{ v \in \W\os ; ||v - u||_{W^{s_1,p_1}} < \delta\}\, \subset [u]_{s,p}. $$ \medskip Note that $\W \os \subset W^{s_1,p_1}\,\os$, by Gagliardo - Nirenberg and the Sobolev embeddings, so that Theorem 5 follows from Theorem 7 when $sp \ge 2$ (when $sp < 2$, there is nothing to prove, by Theorem 1). \medskip {\bf Proof of Theorem 7} \medskip {\bf Step 1:} reduction to special values of $s,s_1, p,p_1$ \medskip We claim that it suffices to prove Theorem 7 when $$ 0 < s_1 < s < 1 - (N - 1)/p,\,1 < p < \infty,\,1 < p_1 < \infty, \,s p = 2,\,s_1p_1 = 1, N \ge 2 . \tag 4 $$ Indeed, assume Theorem 7 proved for all the values of $s, s_1, p, p_1$ satisfying (4). Let $0 < s_0 < \infty,\,1 < p_0 < \infty, N \ge 2$ be such that $s_0 p_0 \ge 2$ (when $N = 1$ or $s_0 p_0 < 2$, there is nothing to prove). Let $u \in W^{s_0,p_0}$ and let $s, s_1, p, p_1$ satisfy (4) and the additional condition $s < s_0$. By Gagliardo - Nirenberg and the Sobolev embeddings, there is some $\delta_0 > 0$ such that $$ \aligned M = & \{v \in W^{s_0,p_0} \os ; ||v - u||_{W^{s_0,p_0}} < \delta_0\} \subset\\ &\{v \in \W \os; ||v - u||_{W^{s_1,p_1}} < \delta\}. \endaligned \tag 5 $$ By the special case of Theorem 7, we have $v \in M \Rightarrow v \in [u]_{s,p}$. By Corollary 5, we obtain $M \subset [u]_{s_0,p_0}$, i.e., $ [u]_{s_0,p_0}$ is open. \medskip In conclusion, it suffices to prove Theorem 7 under assumption (4). Moreover, by Proposition 1 we may assume $u = 1$. \medskip {\bf Step 2:} construction of a good covering \medskip We fix a small neighborhood $\Cal O$ of $\bar \om$. By reflections across the boundary of $\om$, we may associate to each $u \in \W \os$ an extension $\tilde u \in \W (\Cal O ; S^1)$ satisfying $$ ||\tilde u - \tilde v||_{\W (\Cal O)} \le C_1\,\, ||u - v||_{\W (\om)} \tag 6 $$ and $$ ||\tilde u - \tilde v||_{W^{s_1,p_1} (\Cal O)} \le C_1 ||u - v||_{W^{s_1,p_1} (\om)}.\tag 7 $$ \medskip \noindent In this section, $C_1,C_2,...$ denote constants independent of $u,v,...$. \medskip We fix some small $\var > 0$. By Lemma E.2 in Appendix E, for each $v \in \W \os$ there is some $x \in \br^N$ (depending possibly on $v$) such that the covering $\Cal C^x_N$ has the properties $$ v|_{\Cal C^x_j} \in \W, \,\,j = 1,..., N-1 \tag 8 $$ and $$ ||v|_{\Cal C^x_1} - 1||_{W^{s_1,p_1}\,(\Cal C^x_1)} \le C_2 ||v - 1||_{W^{s_1,p_1}\,(\Cal O)} \le C_2 C_1||v - 1||_{W^{s_1,p_1}(\om)} \tag 9 $$ (the last inequality follows from (7)). \medskip While $x$ may depend on $v$, the covering $\Cal C^x_N$ has two features independent of $v$: $$ \text{ the number of squares in } \Cal C^x_2 \text{ has a uniform upper bound } K ; \tag 10 $$ \medskip $$ \align &\text{if } C^1, C^2 \text{ are two squares in } \Cal C^x_2, \text{ there is a path of squares in } \Cal C^x_2\\ &\text{each one having an edge in common with its neighbours, connecting}\tag 11\\ & C^1 \text{ to } C^2. \endalign $$ \medskip {\bf Step 3:} choice of $\delta$ \medskip We rely on \medskip \proclaim{Lemma 7} Let $C = (0,\var)^2$ and $0 < s_1 < 1,\,1 < p_1 < \infty,\, s_1 p_1 = 1.$ Then for each $\delta_1 > 0$ there is some $\delta_2 > 0$ such that every map $v \in W^{s_1,p_1}\,(\partial C ; S^1)$ satisfying $$ ||v - 1||_{W^{s_1,p_1}(\partial C)}\, < \delta_2 \tag 12 $$ has a lifting $\varphi \in W^{s_1,p_1}\,(\partial C; \br)$ such that $$ ||\varphi||_{W^{s_1,p_1} (\partial C)}\, < \delta_1. \tag 13 $$ \endproclaim \medskip Clearly, in Lemma 7, $C$ may be replaced by the unit disc. For the unit disc, the proof of Lemma 7 is given in Appendix C; see Lemma C.3. \medskip In particular, if (12) holds, then we have $$ ||\varphi||_{L^1\,(\partial C)} < C_3 \delta_1 \tag 14 $$ for some $C_3$ independent of the $\delta^\prime$s. \medskip We now take $\delta_1$ such that $$ \delta_1 < \pi \var/C_3. \tag 15 $$ With $\delta_2$ provided by Lemma 7, we choose $$ \delta =\, \text{min}\, \{\delta_2 /C_0, \delta_2 /C_1 C_2\} . \tag 16 $$ \medskip {\bf Step 4:} construction of a global lifting for $v|_{\Cal C^x_1}$ \medskip Let $v \in \W \os$ satisfy $||v - 1||_{W^{s_1,p_1}} < \delta$. Since $\delta \le \delta_2 / C_1 C_2$, (9) implies that the conclusion of Lemma 7 holds for $v|_{\partial C}$ and every square $C$ in $\Cal C^x_2$. Thus, for every $C\in \Cal C_2^x$, $v|_{\partial C}$ has a lifting $\varphi_C$ satisfying (14) and $\varphi_C \in W^{s_1,p_1}\,(\partial C)$. \medskip We claim that $\varphi_C \in \W \,(\partial C)$. The statement being local, it suffices to prove that $\varphi_C \in \W\,(L)$, where $L$ is the union of three edges in $\partial C$. Since $L$ is Lipschitz homeomorphic with an interval, by Theorem 1 in [4] there is some $\psi \in \W\,(L)$ such that $v = e^{i\psi}$ in $L$ (here we use $0 < s < 1$ and $sp = 2 \ge 1$). In $L$, we have $\psi - \varphi_C \in (\W + W^{s_1,p_1})\,(L; 2 \pi \bz)$; thus $\psi - \varphi_C$ is constant a.e. in $L$ (see [4], Remark B.3), so that the claim follows. \medskip Since $sp > 1$ and $v|_{\Cal C^x_1} \in \W, \varphi_C \in \W$, we may redefine $v|_{\Cal C^x_1}$ and $\varphi_C$ on null sets in order to have continuous functions. We claim that the function $\varphi(y) = \varphi_C(y)$, if $y \in C$ is well-defined on $\Cal C^x_1$ (and thus continuous and $\W$). By (11), it suffices to prove that, if $C^1,C^2$ are squares in $\Cal C^x_2$ having the edge $\Cal E$ in common, then $\varphi_{C^1} = \varphi_{C^2}$ on $\Cal E$. Clearly, on $\Cal E$ we have $\varphi_{C^2} = \varphi_{C^1} + 2l \pi$ for some $l \in \bz$. Thus $$ ||\varphi_{C^1} + 2l \pi||_{L^1\,(\Cal E)} = ||\varphi_{C^2}||_{L^1\,(\Cal E)} < C_3 \delta_1, $$ by (14). It follows that $$ 2|l|\pi \varepsilon = ||2 l \pi||_{L^1\,(\Cal E)} \le ||\varphi_{C^1}||_{L^1\,(\Cal E)} + C_3 \delta_1 < 2 C_3 \delta_1, \tag 17 $$ which implies $l = 0$ by (15) and (16). \medskip In conclusion, $v|_{ \Cal C^x_1}$ has a global lifting $\varphi \in \W\,(\Cal C^x_1; \br)$. \medskip {\bf Step 5:} construction of a good extension $w$ of $v|_{\Cal C^x_1}$ \medskip Let $\varphi_2 \in W^{s + 1/p,p}\,(\Cal C^x_2; \br)$ be an extension of $\varphi$, $\varphi_3 \in W^{s + 2/p,p} \,(\Cal C^x_3; \br)$ an extension of $\varphi_2$, and so on; let $\varphi_N \in W^{s + (N-1)/p,p}\,(\Cal C^x_N; \br)$ be the final extension. Note that these extensions exist since $s < 1 + (N - 1)/p$, so that trace theory applies. We set $w = e^{i\varphi_N} \in W^{s+(N-1)/p,p}\,(\Cal C^x_N; S^1)$. Since $(s + (N-1)/p) \cdot p = N + 1 > N$, we obtain by Theorem 3 that $w \in [1]_{s + (N-1)/p,p}$. By Corollary 5, we also have $w \in [1]_{s,p}$. \medskip We complete the proof of Theorem 7 by proving \medskip {\bf Step 6:} $w \in [v]_{s,p}$ \medskip We rely on the following variant of Lemma 6 \medskip \proclaim{Lemma 8} Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\, [sp] \le j < N$. Let $v, w \in \W\,(\Cal C_N; S^1)$ be such that $v|_{\Cal C_l} \in \W, w|_{\Cal C_l} \in \W,\, l= j,..., N - 1$. Assume that $v|_{\Cal C_j}$ and $w|_{\Cal C_j}$ are $\W$-homotopic. Then $v$ and $w$ are $\W$-homotopic. \endproclaim \medskip The proof of Lemma 8 is given Appendix D; see Lemma D.5. \medskip When $N\ge 3$, we are going to apply Lemma 8 with $j=2$. In order to prove that $v|_{\Cal C_2}$ and $w|_{\Cal C_2}$ are $\W$-homotopic, it suffices to find, for each $C\in \Cal C_2$, a homotopy $U_C$ from $v|_C$ to $w|_C$ preserving the boundary condition on $\partial C$; we next glue together these homotopies (this works since $0 1 - 1/p$. \medskip \noindent {\bf Case $\sigma \le 1 - 1/p$:} since $\sigma p < 1,\,v$ may be lifted in $W^{\sigma,p}$ (see Bourgain - Brezis - Mironescu [4]), i.e. there is some $\psi \in W^{\sigma,p}\,(\partial \omega ; \br)$ such that $v = e^{i\psi}$. Let $\varphi \in W^{\sigma + 1/p,p}\,(\omega ; \br)$ be an extension of $\psi$. Then $w = e^{i\varphi} \in W^{\sigma + 1/p,p}\,(\omega ; S^1)$ (since $\sigma + 1/p \le 1$ and $x \mapsto e^{ix}$ is Lipchitz). Clearly, $w$ has all the required properties. \medskip \noindent {\bf Case $\sigma > 1 - 1/p$:} the argument is similar, but somewhat more involved. The proof in [4] actually yields a lifting which is better than $W^{\sigma,p}$; more specifically, this lifting $\psi$ belongs to $W^{t\sigma, p/t}$ for $0 < t \le 1$, see Remark 2, p.41, in the above reference. On the other hand, since $\sigma > 1 - 1/p$, we have $t = p/(\sigma p + 1) < 1$. For this choice of $t$, we obtain that $v$ has a lifting $\psi \in W^{\sigma,p} \cap W^{1 - 1/(\sigma p + 1), \sigma p + 1}$. This $\psi$ has an extension $\varphi \in W^{\sigma + 1/p,p} \cap W^{1, \sigma p + 1}$. By the Composition Theorem stated in the Introduction, the map $w = e^{i\varphi}$ belongs to $W^{\sigma + 1/p,p}\,(\omega ; S^1)$. Clearly, we have tr $w = v$. \medskip \noindent {\bf Remark A.1.} The special case $p < 2$ and $\sigma = 1 - 1/p$ was originally treated by Hardt - Kinderlehrer - Lin [16] via a totally different method. Their argument extends to the case $p < 2$ and $\sigma p < 1$, but does not seem to apply when $p \ge 2$. \bigskip {\bf Appendix B. Good restrictions} \medskip In this appendix, we describe a natural substitute for the trace theory when $s = 1/p$; it is known that the standard trace theory is not defined in this limiting case. \medskip For simplicity, we consider mainly the case of a flat boundary. However, we state Lemma B.5 (used in the proof of Theorem 1) for a general domain. We start by introducing some \medskip \noindent {\bf Notations:} let $Q =\,(0,1)^{N - 1},\, \om_+ = Q \times \,(0,1),\,\om_- = Q \times \, (-1,0),\, \om = \om_+\,\cup \om_-\,= Q \times\,(-1,1)$. If $v$ is a function defined on $Q$, we set $\tilde v\,(x^\prime,t)\,=\, v(x)$ for $(x^\prime,t) \in \om$. \medskip \proclaim{Lemma B.1} Let $0 < s < 1,\,1 < p < \infty$. Then for $u \in \W\,(\om_+)$ and for any function $v$ defined on $Q$, the following assertions are equivalent: \medskip \noindent a) $v \in \W\,(Q)$ and $$ I = \underset{\om_+}\to \int \,\,\frac{|u(x) - \tilde v(x)|^p}{x^{sp}_N}\,dx < \infty ; \tag B.1 $$ \medskip \noindent b) the map $w_1\,=\,\cases u, \quad &\text{ in } \om_+\\ \tilde v, \quad &\text{ in } \om_- \endcases$ belongs to $\W\,(\om)$; \medskip \noindent c) the map $w_2\,=\,\cases u -\tilde v, \quad &\text{ in } \om_+\\ 0, \quad &\text{ in } \om_- \endcases$ belongs to $\W\,(\om)$. \endproclaim \medskip {\bf Proof.} Recall that, if $U$ is a smooth or cube-like domain, then an equivalent (semi-) norm on $\W\,(U)$ is given by $$ f \longmapsto \left ( \sum^N_{j = 1}\,\,\int_0^\infty\,\underset{\{x \in U;\, x + t e_j \in U\}}\to \int\,\, \frac{f(x+te_j) - f(x)|^p}{t^{sp + 1}}\,dx dt \right )^{1/p} \tag B.2 $$ \noindent (see, e.g., Triebel [25]). \medskip Clearly, both b) and c) imply that $v \in \W\,(Q)$. Conversely, for $v \in \W\,(Q)$ we have to prove the equivalence of (B.1), b) and c). We consider the norm given by (B.2). Taking into account the fact that $w_1, w_2$ belong to $\W$ in $\om_+$ and $\om_-$, we see that $$ w_1 \in \W\,(\om)\,\Leftrightarrow\, J = \int_{\om_+} \int_{-1}^0 \, \frac{|u(x) - \tilde v(x)|^p}{(x_N - t)^{sp + 1}}\, dt dx < \infty \tag B.3 $$ \noindent and $$ w_2 \in \W\,(\om)\,\Leftrightarrow\, J < \infty .\tag B.4 $$ \medskip \noindent The lemma follows from the obvious inequality $$ \frac{1 - 2^{-sp}}{sp}\,I\,\le\,J\,\le \frac{1}{sp}\,I. $$ \medskip We now assume in addition that $sp \ge 1$ and derive the following \medskip \proclaim{Corollary B.1} Let $0 < s < 1,\,1 < p < \infty$ be such that $sp \ge 1$. Then, for every $u \in \W\,(\om_+)$ we have \medskip \noindent a) for each $0 \le t_0 < 1$, there is at most one function $v$ defined on $Q$ such that the maps $$ w^{t_0}_1 = \cases u, \quad &\text{ in }Q\, \times\,(t_0,1)\\ \tilde v, \quad &\text{ in }Q\,\times\,(-1,t_0) \endcases $$ and $$ w^{t_0}_2 = \cases u - \tilde v, \quad &\text{ in } Q\,\times\,(t_0,1)\\ 0, \quad &\text{ in } Q\,\times\,(-1,t_0) \endcases $$ belong to $\W\,(\om)$; \medskip \noindent b) for a.e. $0 \le t_0 < 1$, the function $v = u\,(\cdot ,t_0)$ has the property that $w^{t_0}_1, w^{t_0}_2 \in \W\,(\om)$. \endproclaim \medskip \noindent (As usual, the uniqueness of $v$ is understood a.e.) \medskip The above corollary suggests the following \medskip \noindent {\bf Definition:} let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1,\,0 \le t_0 < 1$. Let $u \in \W\,(\om_+)$ and let $v$ be a function defined on $Q$. Then $v$ is the downward good restriction of $u$ to $ \{x_N = t_0\}$ if $w^{t_0}_1, w^{t_0}_2 \in \W\,(\om)$; we then write $v = \text{ Rest } u|^-_{x_N = t_0}$. Similarly, for $0 < t_0 < 1$ we may define an upward good restriction Rest $u|^+_{x_N = t_0} = v$ as the unique function $v$ defined on $Q$ satisfying the two equivalent conditions \medskip \noindent a) $W^{t_0}_1 = \cases \tilde v, \quad &\text{ in } Q \,\times\,(t_0,1)\\ u, \quad &\text{ in } Q \,\times\,(0,t_0) \endcases \in \W\,(\om_+)$ \medskip \noindent and \medskip \noindent b) $W^{t_0}_2 = \cases 0, \quad &\text{ in } Q \,\times\,(t_0,1)\\ u - \tilde v, \quad &\text{ in } Q \,\times\,(0,t_0) \endcases \in \W\,(\om_+).$ \medskip \noindent If $v$ is both an upward and a downward good restriction, we call it a good restriction and we write $v = \text{ Rest } u|_{x_N = t_0}$. \medskip \proclaim{Corollary B.2} Let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1$. Let $u \in \W\,(\om_+)$. Then, for a.e. $0 < t_0 < 1$, we have {\rm Rest } $u|_{x_N = t_0} = u\,(\cdot,t_0)$. \endproclaim \medskip \noindent {\bf Remark B.1.} If $sp > 1$, then functions $u \in \W\,(\om_+)$ have traces for {\bf all} $0 \le t_0 \le 1$. However, these traces need not be good restrictions. Here is an example: For $N = 2$, one may prove that the map $x \mapsto (x - 1/2 e_1)/ |x - 1/2 e_1|$ belongs to $\W\,(\om)$ if $0 < s < 1,$ \newline $1 < p < \infty, sp < 2$. However, if $sp > 1$, its trace $$ \text{ tr } u|_{x_2 = 0}\,\,=\,\,\cases 1, \quad &\text{ if } x_1 > 1/2\\ -1, \quad &\text{ if } x_1 < 1/2 \endcases $$ does not belong to $\W\,(0,1)$, so that it is not a good restriction. \medskip \noindent {\bf Remark B.2.} In the limiting case $s = 1/p$, functions in $\W$ do not have traces. However, they do have good restrictions a.e. \medskip Here is yet another simple consequence of Lemma B.1 \medskip \proclaim{Corollary B.3} Let $0 < s < 1,\,1 < p < \infty,\,sp \ge 1$. Let $ u_\pm\in \W\,(\om_\pm)$ be such that {\rm Rest }$u_+|^-_{x_N = 0}\,=\,$ {\rm Rest }$u_-|^+_{x_N = 0}$. \medskip \noindent Then the map $w = \cases u_+, \quad &\text{ in } \om_+\\ u_-, \quad &\text{ in } \om_- \endcases$ belongs to $\W$. \endproclaim \medskip The following results explain the connections between good restrictions and traces. \medskip \proclaim{Lemma B.2} Let $0 < s < 1,\,1 < p < \infty,\, sp > 1$. Let $u \in \W\,(\om_+)$. Assume that there exists $v =$ {\rm Rest } $u|^-_{x_N = 0}$. Then $v = $ {\rm tr } $u|_{x_N = 0}$. \endproclaim \medskip {\bf Proof.} Let $w = \cases u - \tilde v, \quad &\text{ in } \om_+\\ 0, \quad &\text{ in } \om_- \endcases$. By Lemma B.1, we have $w \in \W\,(\om)$. By trace theory and continuity of the trace, we have $0=$ tr $w|_{x_N=0}$, so that tr $u|_{x_N=0}=v$. \medskip \proclaim{Lemma B.3} Let $0 < s < 1,\,1 < p < \infty,\, sp \ge 1$. Let $u \in W^{s+1/p,p}\,(\om_+)$. Then, considered as a $\W$ function, $u$ has a good downward restriction to $\{x_N = 0\}$ which coincides with {\rm tr }$u|_{x_N = 0}$. \endproclaim \medskip {\bf Proof.} Let $v =$ tr $u|_{x_N = 0}$. Then $v \in \W\,(Q)$, by the trace theory. By Lemma B.1, it remains to prove that $$ \int_{\om_+}\quad \frac{|u(x) - \tilde v(x)|^p}{x_N^{sp}}\, dx < \infty . \tag B.5 $$ \medskip \noindent Assume first that $ s + 1/p = 1$. Then (B.5) follows from the well-known Hardy inequality $$ \int_Q\,\int^1_0 \quad \frac{|u(x^\prime ,t) - u(x^\prime ,0)|^p}{t^p} dt dx \le C \|Du\|^p_{L^p},\,\forall u \in W^{1,p}\,(\om_+). \tag B.6 $$ \medskip \noindent Consider now the case where $s + 1/p \neq 1$. Let $\sigma = s + 1/p$. We are going to prove that $$ \int_{\om_+}\quad \frac{|u(x) - \tilde v(x)|^p}{x^{sp}_N}\, dx \le C\|u\|^p_{W^{\sigma,p}} \tag B.7 $$ for some convenient equivalent (semi-) norm on $W^{\sigma,p}$. It is useful to consider the norm $$ f \mapsto \left ( \sum^N_{j = 1} \int^\infty_0\,\underset{\{ x \in U;\,x + te_j \in U,\,x + 2te_j \in U\}}\to\int \frac{|f(x+2te_j) - 2f(x +te_j) + f(x)|^p}{t^{\sigma p+1}} dxdt \right )^{1/p} \tag B.8 $$ \noindent (see, e.g., Triebel [24]). \medskip \noindent For any $x^\prime \in Q$ such that $u_{x^\prime} = u(x^\prime,\cdot) \in W^{\sigma,p}\,(0,1)$, the map $$ f_{x^\prime} (t) = \cases u(x^\prime ,t), \quad &\text{ if } t > 0\\ v(x^\prime ), \quad &\text{ if } t<0 \endcases $$ \noindent belongs to $W^{\sigma,p}\,(-1,1)$, by standard trace theory. Moreover, for any such $x^\prime$ we have $$ \|f_{x^\prime} \|^p_{W^{\sigma,p}\,(-1,1)} \, \le \, C\|u_{x^\prime}\|^p_{W^{\sigma,p}\,(0,1)}, \tag B.9 $$ \noindent i.e. $$ \aligned &\int^\infty_0\,\,\underset{\{ h\in (-1,1);\, h+t\in (-1,1),\, h + 2t\in (-1,1)\}}\to\int\, \frac{|f_{x^\prime} (h+2t) - 2f_{x^\prime} (h+t) + f_{x^\prime} (h)|^p}{t^{\sigma p+ 1}} \,dh dt \le\\ &C \int^\infty_0 \, \underset{\{h\in (0,1);\, h + t\in (0,1),\, h + 2t\in (0,1)\}}\to\int \frac{|u_{x^\prime}(h+2t) - 2u_{x^\prime}(h+t) + u_{x^\prime}(h)|^p}{t^{\sigma p+ 1}} dh dt. \endaligned $$ \noindent In particular, $$ I = \int^{1/2}_0\,\,\int^{-t}_{-2t} \, \frac{|f_{x^\prime}(h+2t) - 2f_{x^\prime}(h+t) + f_{x^\prime} (h)|^p}{t^{\sigma p+ 1}} dh dt \le C\|u_{x^\prime}\|^p_{W\sigma,p}. \tag B.10 $$ \noindent Since $$ I \ge C \int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{\sigma p}} dt = C \int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt, \tag B.11 $$ \noindent we find that $$ \int^{1/3}_0 \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt \le C\|u_{x^\prime }\|^p_{W^{\sigma,p}}. \tag B.12 $$ \noindent On the other hand, we clearly have $$ \int^1_{1/3} \frac{|u(x^\prime ,t) - v(x^\prime )|^p}{t^{sp + 1}} dt \le C\|u_{x^\prime }\|^p_{L^p} + C|v(x^\prime )|^p. \tag B.13 $$ \noindent By combining (B.12), (B.13) and integrating with respect to $x^\prime$, we obtain (B.7). The proof of Lemma B.3 is complete. \medskip A simple consequence of Lemma B.3 is the following \proclaim{Lemma B.4} Let $0 < s < 1,\,1 < p < \infty,\, sp \ge 1$ and $\rho > s$. Let $u_1 \in \W\,(\om_+)$ and $u_2 \in W^{\rho,p}\,(\om_-)$. Assume that $u_1$ has a good downward restriction $v =$ {\rm Rest } $u_1|^-_{x_N = 0}$ and that $v =$ {\rm tr } $u_2|_{x_N = 0}$. Then the map $$ w = \cases u_1,\quad \text{ in } \om_+\\ u_2, \quad \text{ in } \om_- \endcases $$ \noindent belongs to $\W(\om)$. \endproclaim \medskip {\bf Proof.} Let $u_3 \in W^{s+ 1/p,p}\,(\om_-)$ be an extension of $v$. Then $w = w_1 + w_2$, where $$ w_1 = \cases u_1,\quad \text{ in } \om_+\\ u_3, \quad \text{ in } \om_- \endcases $$ and $$ w_2 = \cases 0, \quad &\text{ in } \om_+\\ u_2 - u_3, \quad &\text{ in } \om_- \endcases . $$ \medskip By Lemma B.3 and the assumption $v =$ Rest $u_1|^-_{x_N = 0}$, we have Rest $u_1|_{x_N = 0}^-\, =\, \text{ Rest } u_3|_{x_N = 0}^+$. By Corollary B.3, we find that $w_1 \in \W\,(\om)$. It remains to prove that $w_2 \in \W\,(\om)$. Let $\sigma =$ min $\{\rho, s + 1/p,\,1\}$. Then $w_2 \in W^{\sigma,p}\,(\om)$, by standard trace theory. Thus $w_2 \in \W\,(\om)$. \medskip We conclude this section by stating the following precised form of Corollary B.1, b) in the case of a general boundary. We use the same notations as in the proof of Theorem 1, Case 4. \medskip \proclaim{Lemma B.5} Let $u \in W^{1/p,p}\,(\om)$. Then \medskip \noindent a) for a.e. $ 0 < \delta < \var$ we have $$ u|_{\Sigma_\delta} \in W^{1/p,p}\,(\Sigma_\delta) \text{ and } \int_{\Sigma_\delta} \int _\Omega \frac{|u(x) - u(y)|^p}{|x - y|^{N + 1}} dy ds_x < \infty; \tag B.14 $$ \medskip \noindent b) for any such $\delta$, $u$ has a good restriction to $\Sigma_\delta$ which coincides (a.e. on $\Sigma_\delta$) with $u|_{\Sigma_\delta}$. \endproclaim \bigskip {\bf Appendix C. Global lifting} \medskip In this appendix, we investigate the existence of a global lifting in some domains with non-trival topology. \proclaim{Lemma C.1} Let $0 < s < \infty,\,1 < p < \infty,\,sp \ge N, N \ge 2$. Let $u \in \W\,(S^1 \times B_1; S^1)$ be such that {\rm deg} $(u|_{S^1 \times B_1}) = 0$. Then there is some $\varphi \in \W\,(S^1 \times B_1; S^1)$ such that $u = e^{i\varphi}$. \endproclaim \medskip \noindent Here, $B_1$ is the unit ball in $\br^{N-1}$. \medskip {\bf Proof.} Let $v : \br \times B_1 \to S^1,\, v(t,x) = u(e^{it},x)$. Then $v \in W^{s,p}_{loc} \, (\br \times B_1 ; S^1)$, where ``loc'' refers only to the variable $t$. By Theorem 2 in Bourgain - Brezis - Mironescu [4], there is some $\psi \in W^{s,p}_{loc}\,(\br \times B_1; \br)$ such that $v = e^{i\psi}$. We claim that $\psi$ is $2\pi$-periodic in the variable $t$. Indeed, for a.e. $x \in B_1$, we have $u \in W^{s,p}\,(S^1 \times \{x\}; S^1)$ and deg $(u|_{S^1 \times \{x\}})= 0$. In particular, for any such $x$ the map $u|_{S^1 \times \{x\}}$ has a continuous lifting $\eta_x$. On the other hand, for a.e. $x \in B_1$ we have $\psi_x = \psi(\cdot ,x) \in W^{s,p}_{loc}\,(\br \times \{x\}; \br)$. Thus, with $\lambda_x (t) = \eta_x (e^{it})$, we find that for a.e. $x \in B_1$ the function $\psi_x - \lambda_x$ is continuous and $2\pi \bz$ -valued; therefore it is a constant. Since $\lambda_x$ is $2\pi$-periodic, so is $\psi_x$ for a.e. $x \in B_1$. We obtain that $\psi$ is $2\pi$-periodic in the variable $t$. Thus the map $\varphi: S^1 \times B_1 \to \br,\, \varphi (e^{it},x) = \psi(t,x)$ is well-defined and belongs to $W^{s,p}\,(S^1 \times B_1;\br)$. Moreover, we clearly have $u = e^{i\varphi}$. \medskip In the same vein, we have \medskip \proclaim{Lemma C.2} Let $s \ge 1,\,1 < p < \infty,\,N \ge 3,\,2 \le sp < N$. Let $u \in W^{s,p}\,(S^1 \times B_1;S^1)$ be such that {\rm deg } $(u|_{S^1 \times B_1}) = 0$. Then there is some $\varphi \in W^{s,p}\,(S^1 \times B_1;\br) \cap W^{1,sp}\,(S^1 \times B_1;\br)$ such that $u = e^{i\varphi}$. \endproclaim \medskip The proof is similar to that of Lemma C.1; one has to use Lemma 4 in [4] instead of Theorem 2 in [4]. \medskip \proclaim{Lemma C.3} Let $1 < p < \infty$ and $\delta_1 > 0$. Then there is some $\delta_2 > 0$ such that every $v \in W^{1/p,p}\,(S^1;S^1)$ satisfying $\|v-1\|_{W^{1/p,p}(S^1)} < \delta_2$ has a global lifting $\varphi \in W^{1/p,p}\,(S^1;\br )$ such that $\|\varphi\|_{W^{1/p,p}(S^1)}<\delta_1$. \endproclaim \medskip {\bf Proof.} Recall that if $I$ is an interval, then every $w \in W^{1/p,p}\,(I;S^1)$ has a lifting $\psi \in W^{1/p,p}\,(I;\br)$ (see Bourgain - Brezis - Mironescu [4], Theorem 1). Moreover, this lifting may be chosen to be (locally) continuous with respect to $w$, i.e. for every $w_0 \in W^{1/p,p}(I;S^1)$ there is some $\delta_0 > 0$ such that in the set $$ \{w; \|w - w_0\|_{W^{1/p,p}(I;S^1)} < \delta_0\} $$ there is a lifting $w \mapsto \psi$ continuous for the $W^{1/p,p}$ norm. (This assertion can be established using the same argument as in Step 7 of the proof of Theorem 4 in Brezis - Nirenberg [12]; it can also be derived from the explicit construction of $\psi$ in the proof of Theorem 1 in [4]; see also Boutet de Monvel-Berthier - Georgescu - Purice [6] when $p = 2$). \medskip Let $I = [-2\pi,2\pi]$. To each $v \in W^{1/p,p}\,(S^1;S^1)$ we associate the map $w \in W^{1/p,p}\,(I;S^1)$, $w(t) = v(e^{it})$. By the above considerations, for every $\delta_3 > 0$ there is some $\delta_4 > 0$ such that, if $\|v-1\|_{W^{1/p,p}(S^1)} < \delta_4$, then $w$ has a lifting $\psi$ such that $\|\psi\|_{W^{1/p,p}(I)} < \delta_3$. We claim that $\psi$ is $2\pi$-periodic if $\delta_3$ is small enough. Indeed, the function $\xi (t)=\psi (t-2\pi )-\psi (t)$ belongs to $W^{1/p,p} ([0,2\pi ];2\pi\bz )$, so that $\xi$ is constant a.e. (see [4], Theorem B.1). Since $\|\xi\|_{L^1} \le \|\psi\|_{L^1} < C\delta_3$, we have $\xi =0$ (i.e. $\psi$ is $2\pi$-periodic) if $C\delta_3<2\pi$. \medskip Thus, for $\delta_3$ small enough, the map $\varphi (e^{it}) = \psi (t)$ is well-defined, belongs to $W^{1/p,p}$ and satisfies $\|\varphi\|_{W^{1/p,p}(S^1) }< \delta_1$ and $u = e^{i\varphi}$. \bigskip {\bf Appendix D. Filling a hole - the fractional case} \medskip We adapt to fractional Sobolev spaces the technique of Brezis - Li [7], Section 1.3. \medskip The first two results are preparations for the proofs of Lemmas 5,6 and 8 (see Lemmas D.3, D.4 and D.5 below). \medskip \proclaim{Lemma D.1} Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N$. Let $C = (-1,1)^N$ and $u \in W^{s,p}\,(\partial C)$. Then $\tilde u \in W^{s,p}\,(C)$; here, $\tilde u(x) = u(x/|x|)$ and $|\,\,\,|$ is the $L^\infty$ norm in $\br^N$. Moreover, the map $u \mapsto \tilde u$ is continuous from $W^{s,p}\,(\partial C)$ into $W^{s,p}\,(C)$. \endproclaim \medskip {\bf Proof.} Clearly, we have $\|\tilde u\|_{L^p (C)} \le C_0 \|u\|_{L^p (\partial C)}$. Thus it suffices to prove, for the Gagliardo semi-norms in $W^{s,p}$, the inequality \medskip $$ \|\tilde u\|_{W^{s,p}(C)}^p \le C_1 (\|u\|^p_{W^{s,p}(\partial C)} + \|u\|_{L^p(\partial C)}^p). \tag D.1 $$ \noindent We have $$ \underset{C}\to\int \underset{C}\to\int \frac{|\tilde u(x) - \tilde u(y)|^p}{|x-y|^{N+sp}}dxdy = \int^1_0 \int^1_0 \underset{\partial C}\to\int\,\, \underset{\partial C}\to\int \frac{|u(x) - u(y)|^p}{|\tau x - \sigma y|^{N+sp}} \tau^{N-1}\sigma^{N-1} ds_x ds_y d\tau d\sigma. \tag D.2 $$ \noindent We claim that $$ I = \int^1_0 \int^1_0 \frac{\tau^{N-1} \sigma^{N-1}}{|\tau x - \sigma y|^{N+sp}} d\tau d\sigma \le C_2/|x-y|^{N+sp}. \tag D.3 $$ \noindent Indeed, $$ \aligned I = &\int^1_0 \int^{1/\tau}_0 \,\,\frac{\tau^{N-1} (\lambda \tau)^{N-1}}{|\tau x - \lambda \tau y|^{N+sp}} d\lambda d\tau \quad = \quad \\ &\int^1_0 \int^{1/\tau}_0 \tau^{N - sp - 1} \frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau \le I_1 + I_2, \endaligned \tag D.4 $$ \noindent where $I_1 = \int^1_0 \int^2_0$ and $I_2 = \int^1_0 \int^\infty_2$. \medskip On the one hand, we have $$ \aligned I_1 &= \int^1_0 \int^2_0 \tau^{N - sp-1} \frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau\\ &\le C_3 \int^1_0 \int^2_0 \tau^{N - sp- 1} \, \frac{\lambda^{N-1}}{|x- y|^{N+sp}} d\lambda d\tau \le C_4/|x-y|^{N + sp}. \endaligned \tag D.5 $$ \noindent On the other hand, we have $$ \aligned I_2 &= \int^1_0 \int^\infty_2 \tau^{N - sp-1} \frac{\lambda^{N-1}}{|x-\lambda y|^{N+sp}} d\lambda d\tau\\ &\le C_5 \int^1_0 \int^\infty_2 \tau^{N - sp- 1} \, \frac{\lambda^{N-1}} {\lambda^{N+sp}} d\lambda d\tau = C_5 \int^1_0 \int^\infty_2 \tau^{N - sp -1} \lambda^{-sp -1}\,d\lambda d\tau \le C_6. \endaligned\tag D.6 $$ \noindent We obtain (D.3) by combining (D.4), (D.5) and (D.6). Finally, (D.1) follows from (D.2) and (D.3). \medskip The proof of Lemma D.1 is complete. \medskip \proclaim{Lemma D.2} Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N$. Let $v, w \in W^{s,p}\,(C;S^1)$ be such that $v|_{\partial C} = w|_{\partial C} \in W^{s,p}\,(\partial C)$. Then, there is a homotopy $U \in C^0 ([0,1];W^{s,p}\,(C;S^1))$ such that $U(0, \cdot) = v, \, U(1,\cdot) = w$ and $U(t, \cdot)|_{\partial C} = v|_{\partial C}, \forall t \in [0,1]$.\endproclaim \medskip {\bf Proof.} Let $u = v|_{\partial C}$. It clearly suffices to prove the lemma in the special case $w = \tilde u$. In this case, let, for $0 \le t < 1$, $$ U(t,x) = \cases v(x/(1-t)),\quad &\text{ if } |x| \le 1 - t\\ \tilde u(x),\quad &\text{ if } 1 - t < |x| \le 1 \endcases ; $$ \noindent set $U(1, \cdot) = \tilde u$. Clearly, $U \in C^0([0,1); W^{s,p}\,(C; S^1))$. It remains to prove that $U(t,\cdot) \to \tilde u$ as $t \to 1$. Let $$ f(x) = \cases v(x),\quad &\text{ if } |x| \le 1\\ \tilde u (x),\quad &\text{ if } |x| > 1 \endcases $$ \noindent and $g = f - \tilde u$. Then $f,\, \tilde u \in W^{s,p}_{loc}\,(\br^N)$, so that $g \in W^{s,p}_{loc}\,(\br^N)$. Since $g = 0$ outside $C$, we actually have $g \in W^{s,p}\,(\br^N)$. Thus $$ \aligned &\|U(t, \cdot) - \tilde u\|^p_{W^{s,p}(C)} = \|g(\cdot /(1-t))\|^p_{W^{s,p}(C)} \le\\ &\|g(\cdot/(1-t))\|^p_{W^{s,p}(\br^N)} = (1 - t)^{N - sp} \|g\|^p_{W^{s,p}(\br^N)} \to 0 \endaligned $$ \noindent as $t \to 1$. The proof of Lemma D.2 is complete. \medskip We introduce a useful notation: let $u \in W^{s_1,p_1}\,(\Cal C_k)$, where $0 < s_1 < 1,\,1 < p_1 < \infty,\,1 < s_1p_1 < N$. We extend, for each $C \in \Cal C_{k+1}, u|_{\partial C}$ to $C$ as in Lemma D.1. Let $\tilde u$ be the map obtained by gluing these extensions. We next extend $\tilde u$ to $\Cal C_{k+2}$ in the same manner, and so on, until we obtain a map defined in $\Cal C_N$; call it $H_k (u)$. \medskip \proclaim{Lemma D.3} Let $0 < s_1 < 1,\,1 < p_1 < \infty,\,1 < s_1p_1 < N,\,[s_1p_1] \le j < N$. Then every $v \in W^{s_1,p_1}\,(\Cal C_j;S^1)$ has an extension $u_1 \in W^{s_1,p_1}\,(\Cal C_N;S^1)$ such that $u_1|_{\Cal C_l} \in W^{s_1,p_1}$ for $l = j,...,N-1$. \endproclaim \medskip {\bf Proof.} We take $u_1 = H_j(v)$. We may use repeatedly Lemma D.1, since for $l = j+1,...,N$ we have $1 < s_1p_1 < l$. \medskip \proclaim{Lemma D.4} Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\, [sp] \le j < N$. If $u|_{\Cal C_l} \in W^{s,p}, u_1|_{\Cal C_l} \in W^{s,p}, l=j,..., N-1$, and $u|_{\Cal C_j} = u_1|_{\Cal C_j}$, then $u$ and $u_1$ are $W^{s,p}$-homotopic. \endproclaim \medskip {\bf Proof.} We argue by backward induction on $j$. If $j = N-1$, then for each $C \in \Cal C_N$ Lemma D.2 provides a $W^{s,p}$-homotopy of $u|_C$ and $u_1|_C$ preserving the boundary condition. By gluing together these homotopies we find that $u$ and $u_1$ are $W^{s,p}$-homotopic (here we use $1/p < s < 1$). Suppose now that the conclusion of the lemma holds for $j+1$; we prove it for $j$, assuming that $j \ge [sp]$. By assumption, $u$ and $H_{j+1}(u|_{\Cal C_{j+1}})$ are $W^{s,p}$-homotopic, and so are $u_1$ and $H_{j+1}(u_1|_{\Cal C_{j+1}})$. It suffices therefore to prove that $v = H_{j+1}(u|_{\Cal C_{j+1}})$ and $v_1 = H_{j+1}(u_1|_{\Cal C_{j+1}})$ are $W^{s,p}$-homotopic. For each $C \in \Cal C_{j+1}$, we have $v|_{\partial C} = v_1|_{\partial C} = u|_{\partial C} = u_1|_{\partial C}$. By Lemma D.2, $v|_C$ and $v_1|_C$ are connected by a homotopy preserving the trace on $\partial C$. Gluing together these homotopies, we find that $v|_{\Cal C_{j+1}}$ and $v_1|_{\Cal C_{j+1}}$ are $W^{s,p}$-homotopic. If $U$ connects $v|_{\Cal C_{j+1}}$ to $v_1|_{\Cal C_{j+1}}$, then Lemma D.1 used repeatedly implies that $t \mapsto H_{j+1} (U(t))$ connects in $W^{s,p}\,(\Cal C_N;S^1)$ the map $H_{j+1}\,(v|_{\Cal C_{j+1}})$ to $H_{j+1}\,(v_1|_{\Cal C_{j+1}})$, i.e., $v$ to $v_1$. \medskip The proof of Lemma D.4 is complete. \proclaim{Lemma D.5} Let $0 < s < 1,\,1 < p < \infty,\,1 < sp < N,\, [sp] \le j < N$. Let $v, w \in W^{s,p}\,(\Cal C_N;S^1)$ be such that $v|_{\Cal C_l} \in W^{s,p}, w|_{\Cal C_l} \in W^{s,p},l = j,...,N-1$. Assume that $v|_{\Cal C_j}$ and $w|_{\Cal C_j}$ are $W^{s,p}$-homotopic. Then $v$ and $w$ are $W^{s,p}$-homotopic. \endproclaim \medskip {\bf Proof.} By Lemma D.4, $v$ and $H_j(v|_{\Cal C_j})$ (respectively $w$ and $H^j(w|_{\Cal C_j})$) are $W^{s,p}$-homotopic. If $U$ connects $v|_{\Cal C_j}$ to $w|_{\Cal C_j}$ in $W^{s,p}$, then as in the proof of Lemma D.4, we obtain that $t \mapsto H_j(U(t))$ connects $H_j(v|_{\Cal C_j})$ to $H_j(w|_{\Cal C_j})$ in $W^{s,p}$. Thus $v$ and $w$ are $W^{s,p}$-homotopic. \bigskip \bigskip {\bf Appendix E. Slicing with norm control} \medskip In this section, we prove the existence of good coverings for $W^{s,p}$ maps. The arguments are rather standard. \medskip Without loss of generality, we may consider maps defined in $\br^N$. Throughout this section, we assume $\var = 1$, i.e. we consider a covering with cubes of size 1. We start by introducing some useful notations: for $x \in C^N = (0,1)^N$ and for $j = 1,...,N - 1$, let $$C_j = \bigcup\,\bigg\{\sum^j_{k=1}\,t_k\, e_{i_k}\, + \,\sum^{N-j}_{l=1}\, \lambda_l e_{j_l} ; \, t_k \in \br , \lambda_l \in \bz , \{e_{i_k}\} \, \cup \{e_{j_l}\} = \{e_1,...e_N\} \bigg\} $$ \noindent and $C_j (x) = x + C_j$. (With the notations introduced in Section 3, we have $C_j (x) = \Cal C_j^x$ when $\om = \br^N$). \medskip \noindent For a fixed set $\Lambda \subset \{1,..,N\}$ such that $|\Lambda| = j$, let also $$ C_j^\Lambda = \bigg\{ \sum_{i \in \Lambda} t_i e_i + \sum_{j \notin \Lambda} \lambda_j e_j ;\, t_i \in \br, \lambda_j \in \bz\bigg\}, $$ \noindent so that $$ C_j = \cup \{ C^\Lambda_j ; \Lambda \subset \{1,...,N\}, |\Lambda|=j\}, $$ \noindent and with obvious notations $$ C_j(x) = \cup \{ C^\Lambda_j(x) ; \Lambda \subset \{1,...,N\}, |\Lambda|=j\}. $$ \medskip Instead of considering a fixed (semi-) norm on $W^{s,p}, 0 < s < 1,\,1 < p < \infty$, it is convenient to consider a family of equivalent norms $$ |f|^p_j\, = \, \sum \Sb \Lambda \subset \{1,...,N\}\\ |\Lambda| = j \endSb \int_{\br^N} \, \int_{\br^j}\, \frac{|f(x + \sum_{i \in \Lambda} t_i e_i) - f(x)|^p}{|t|^{j + sp}} dt dx $$ \noindent (see, e.g., Triebel [24]). An obvious computation yields, for the usual Gagliardo \noindent (semi-) norm on $C^\Lambda_j (x)$, \medskip \proclaim{Lemma E.1} Let $0 < s < 1,\,1 < p < \infty$ and $u \in W^{s,p}$. Then $$ \sum \Sb \Lambda \subset \{1,...,N\}\\ |\Lambda| = j \endSb \,\, \int_{C^N} \, \|u\|^p_{W^{s,p}\, (C^\Lambda_j(x))} dx \le |u|_j^p $$ for some $C$ independent of $u$. \endproclaim \medskip We next define the norm $\|u\|_{W^{s,p}\,(C_j(x))}$ by the formula $$ \|u\|^p_{W^{s,p}\,(C_j(x))} \, = \, \sum_{C \in C_{j+1}(x)}\, \|u\|^p_{W^{s,p}\,(\partial C)}. $$ \medskip \proclaim{Lemma E.2} Let $0 < s < 1,\,1 < p < \infty$. Then, for $u \in W^{s,p}$, we have \medskip \noindent a) for a.e. $x \in C^N, u|_{C_j(x)} \in W^{s,p}_{loc}, j = 1,...,N - 1$; \medskip \noindent b) there is a fat set (i.e., with positive measure) $A \subset C^N$ such that $$ \|u\|^p_{W^{s,p}\,(C_j(x))} \, \, \le C \, |u|^p_j ,\quad \forall x \in A. \tag E.2 $$ \endproclaim \medskip \noindent {\bf Remark E.1.} Here, $u|_{C_j(x)}$ are restrictions, not traces. However, when $sp > 1$ we may replace restrictions by traces, by a standard argument. We obtain \medskip \proclaim{Corollary E.1} Let $0 < s < 1,\,1 < p < \infty,\, sp > 1$. Let $u \in W^{s,p}$. Then, for a.e. $x \in C^N$, {\rm tr } $u|_{C_{N-1}(x)}\in W^{s,p}$. Moreover, for a.e. $x \in C^N$, {\rm tr } $u|_{C_{N-1}(x)}$ has a trace on $C_{N-2}(x)$ which belongs to $W^{s,p}$, and so on. \endproclaim \medskip {\bf Proof of Lemma E.2.} In order to avoid long computations, we treat only the case $j=1, N=2$. The general case does not bring any additional difficulty. Let $C \in C_1(x)$; denote its lower (resp. upper, left, right) edge by $C^l$ (resp. $C^u,\,C^L,\,C^R$). By (E.1), we have $u|_{C^l } \in W^{s,p}$ for a.e. $x \in C^2$ and, for $x$ in a fat set, $\sum_{C \in C_1(x)} \|u\|^p_{W^{s,p}(C^l)} \le \text{ const. } |u|^p_1$. Similar statements hold for the other edges. \medskip It remains to control the cross - integrals in the Gagliardo norm, e.g. to prove $$ I = \int_{C^2} \sum_{C \in C_1(x)} \int_{C^l} \int_{C^L} \frac{|u (y) - u (z)|^p}{|y-z|^{2+sp}} dy dz \le \text{ const. } \|u\|^p_{W^{s,p}} \tag E.3 $$ \noindent (here, we take the usual Gagliardo norm in $W^{s,p}\,(\br^2)$). We have $$ \aligned I &= \int_{C^2} \,\, \sum_{m \in \bz^2} \int^1_0 \int^1_0 \frac{|u(x+m_1 e_1+m_2e_2+\tau e_1) - u (x + m_1e_1+m_2e_2+\sigma e_2)|^p}{|\tau e_1 - \sigma e_2|^{2+sp}} d\sigma d\tau dx\\ &= \int_{\br^2} \,\int^1_0 \int^1_0 \frac{|u(y+\tau e_1) - u(y+\sigma e_2)|^p}{|\tau e_1 - \sigma e_2|^{2+sp}} d\sigma d\tau dy\\ &= \int_{\br^2} \,\int^1_0 \int^1_0 \frac{|u(z) - u(z - \tau e_1 + \sigma e_2)|^p}{|\tau e_1 - \sigma e_2|^{2 + sp}} d\sigma d\tau dz\\ &\le \int_{\br^2} \int_{\br^2} \frac{|u(z+h) -u(z)|^p}{|h|^{2+sp}} dh dz = \|u\|^p_{W^{s,p}}. \endaligned $$ \medskip The proof of Lemma E.2 is complete. \bigskip {\bf Acknowledgements}. The first author (H.B.) warmly thanks Yanyan Li for useful discussions. He is partially supported by a European Grant ERB FMRX CT980201, and is also a member of the Institut Universitaire de France. This work was initiated when the second author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality. It was completed while both authors were visiting the Isaac Newton Institute in Cambridge, which they also wish to thank. \bigskip \centerline{\bf References} \bigskip \item{[1]} {\bf F. Bethuel}, {\it The approximation problem for Sobolev maps between two manifolds}, Acta Math., {\bf 167} (1991), 153-206. \medskip \item{[2]} {\bf F. Bethuel}, {\it Approximation in trace spaces defined between manifolds}, Nonlinear Anal. TMA, {\bf 24} (1994), 121-130. \medskip \item{[3]} {\bf F. Bethuel and X. Zheng}, {\it Density of smooth functions between two manifolds in Sobolev spaces}, J. Funct. Anal., {\bf 80} (1988), 60-75. \medskip \item{[4]} {\bf J. Bourgain, H. Brezis and P. Mironescu}, {\it Lifting in Sobolev spaces}, J. d'Analyse Math\'ematique, {\bf 80} (2000), 37-86. \medskip \item{[5]} {\bf J. Bourgain, H. Brezis and P. Mironescu}, {\it On the structure of the space $H^{1/2}$ with values into the circle}, C.R. Acad. Sci. Paris, {\bf 321} (2000), 119-124. \medskip \item{[6]} {\bf A. Boutet de Monvel-Berthier, V. Georgescu and R. Purice}, {\it A boundary value problem related to the Ginzburg-Landau model}, Comm. Math. Phys., {\bf 142} (1991), 1-23. \medskip \item{[7]} {\bf H. Brezis and Y. Li}, {\it Topology and Sobolev spaces}, J. Funct.Anal., (to appear). \medskip \item{[8]} {\bf H. Brezis, Y. Li, P. Mironescu and L. Nirenberg}, {\it Degree and Sobolev spaces}, Top. Meth. in Nonlinear Anal,. {\bf 13} (1999), 181-190. \medskip \item{[9]} {\bf H. Brezis and P. Mironescu}, {\it Composition in fractional Sobolev spaces}, Discrete and Continuous Dynamical Systems, {\bf 7} (2001), 241-246. \medskip \item{[10]} {\bf H. Brezis and P. Mironescu}, {\it Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces}, Journal of Evolution Equations, (to appear). \medskip \item{[11]} {\bf H. Brezis and P. Mironescu}, in preparation. \medskip \item{[12]} {\bf H. Brezis and L. Nirenberg}, {\it Degree Theory and BMO, Part I: Compact manifolds without boundaries}, Selecta Math., {\bf 1} (1996), 197-263. \medskip \item{[13]} {\bf H. Brezis and L. Nirenberg}, {\it Degree Theory and BMO, Part II: Compact manifolds with boundaries}, Selecta Math., {\bf 2} (1996), 309-368. \medskip \item{[14]} {\bf M. Escobedo}, {\it Some remarks on the density of regular mappings in Sobolev classes of $S^M$-valued functions}, Rev. Mat. Univ. Complut. Madrid, {\bf 1} (1988), 127-144. \medskip \item{[15]} {\bf F. B. Hang and F. H. Lin}, {\it Topology of Sobolev mappings II},( to appear). \medskip \item{[16]} {\bf R. Hardt, D. Kinderlehrer and F. H. Lin}, {\it Stable defects of minimizers of constrained variational principles}, Ann. IHP, Anal. Nonlin\'eaire, {\bf 5} (1988), 297-322. \medskip \item{[17]} {\bf J. L. Lions and E. Magenes}, Probl\`emes aux limites non homog\`enes, Dunod,1968; English translation, Springer, 1972. \medskip \item{[18]} {\bf V. Maz'ya and T. Shaposhnikova}, {\it An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces}, (to appear). \medskip \item{[19]} {\bf J.Peetre}, {\it Interpolation of Lipschitz operators and metric spaces}, Mathematica (Cluj), {\bf 12} (1970), 1-20. \medskip \item{[20]} {\bf T. Rivi\`ere}, {\it Dense subsets of $H^{1/2}(S^2;S^1)$}, Ann. of Global Anal. and Geom., (to appear). \medskip \item{[21]} {\bf J. Rubinstein and P. Sternberg}, {\it Homotopy classification of minimizers of the Ginzurg-Landau energy and the existence of permanent currents}, Comm. Math. Phys., {\bf 179} (1996), 257-263. \medskip \item{[22]} {\bf T. Runst}, {\it Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type}, Analysis Mathematica, {\bf 12} (1986), 313-346. \medskip \item{[23]} {\bf T. Runst and W. Sickel}, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, Walter de Gruyter, Berlin and New York, 1996. \medskip \item{[24]} {\bf R. Schoen and K. Uhlenbeck}, {\it Boundary regularity and the Dirichlet problem for harmonic maps}, J. Diff. Geom., {\bf 18} (1983), 253-268. \medskip \item{[25]} {\bf H. Triebel}, Theory of function spaces, Birkh\"auser, Basel and Boston, 1983. \medskip \item{[26]} {\bf B. White}, {\it Homotopy clases in Sobolev spaces and the existence of energy minimizing maps}, Acta Math., {\bf 160} (1988), 1-17. \enddocument