Complex log and Winding Number in Pi_1(S^1)

[Bacle]

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. f_t(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.

 

[Hurkyl]

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.?

I'm surprised that your book said only that and nothing else.

One way to patch things up is to note that w(f) is by noting:

• That formula is well-defined for paths in simply-connected regions
• Such paths generate the fundamental groupoid
• Writing a path as a sum of generators in two different ways yields the same value for the winding number

Another way is that the derivative of log(z) is well-defined on the entire circle, and you can define w(f) via integration.


Another way is to lift paths on the circle to paths on the Riemann surface for log(z).