Understanding the Degree of a Continuous Map g:Circle --> Circle

[jimisrv]

Hi,

I am having some problems understanding the degree of a continuous map g:circle --> circle

I have found a definition in Munkres (pg 367) that I can't really understand (I'm an engineering student with little algebraic topology) and one in Lawson (pg 181), Topology:A Geometric approach, that makes some sense.

Lawson defines it:

For g as above, consider p : I --> circle and a lift h:I --> R
such that gp=ph

Then define the degree of g as

deg g =h(1)-h(2)

Intuitively I understand this to be the integer number of times the image of the circle wraps around the circle under g? If so, what about the continuous function g(x)=exp(i*pi*x/2) where x is in[0,2pi). The image would only wrap around half the circle, which would be homotopic to the constant map, a map of degree 0...but according to the above definition this would have degree 1/2?

Thanks for any help!

Regards,
Mike

 

[adriank]

Your example doesn't work because [0, 2pi) is not a circle. If you try to make a circle out of it by identifying 0 and 2pi, then your map would be discontinuous, since g(0) = 1 and g(2pi) = -1.

(Sorry that this isn't a very formal argument, but I think it gets the point across.)