이 내용은 모두 C.T.C Wall책에 있는 내용임.
1. notation >
are topological spaces.
is the unit interval on ( = [0,1] ).
is the unit circle on complex plane.
is the set of continuous map . (every subset of is open.)
is path components of .
is free abelian group of .
is set of homotopy classes of maps
2. definition > lifting maps from up to
3. theorem >
Any continuous map has a lift which is unique up to translation by integer.
4. definition > degree of map
5. a horde of theorems >
- degree 자체가 function이므로
The degree map is group isomorphism.
- The two homotopic maps have same degree.
- the following conditions are equivalent
i) is nullhomotopic
iii) has continuous lift
- If the map is not surjective, . (i.e f is nullhomotopic)
3번만 빼면 쉽다.
6. theorem > Fundamental theorem of algebra
We will suppose not, and find a contradiction.
Write the polynomial as
Without loss of generality, we may suppose
We define a map by
If we write , then provides a homotopy between the maps . Now is constant, so has zero degree. We will show that for large enough, has degree . This will give the required contradiction.
Then for ,
It follows that has a positive real part. Hence so has
So, the map has zero degree. Hence equals the degree of the map .
The theorem now follows
The Fundamental Theorem of Algebra (with the Fundamental Group) in Math ∩ Programming
 C. T. C. Wall, A geometric introduction to topology. Addison-Wesley 1972, Dover 1993.