Hi, everyone: I am confused about the def. of winding number of a loop on S^1 ( on elements of Pi_1(S^1,1) ). The winding number is defined by w(f):= [log(f)1)-log(f)0)]/i2Pi One of the problems is that log cannot be defined continuously on S^1, since complex logs can only be define in simply-connected regions that do not wind around the origin --and S^1 fails both these conditions. How does this definition then make sense.? 2) Also, in my old class notes, we use log to define a homotopy between any two functions f,g with the same winding number: f_t(x)=exp[tlog(f)x+ (1-t)g(x)) In which we seem to assume log is continuous. Just curious: what would happen if g,f had different winding number.?. I understand each homotopy class [n] in Pi1(S1) consists of all paths that loop around n times. ft(x) above is clearly a homotopy between g(x) and f(x) I am hoping to understand better why the classes [n] and [m] are only homotopic for m=n. I am aware that we construct the isomorphism by first using the fact that the cardinality of Pi1(S1) is the same as that of the fiber of any point under the standard cover by the reals, and that we show that the map to the deck transformation group is an isomorphism, but understanding the log issue will help me get more insights. Thanks in Advance.