Math4202 Topology II (Lecture 7)

Algebraic Topology

Classify 2-dimensional topological manifolds (connected) up to homeomorphism/homotopy equivalence.

Use fundamental groups.

We want to show that:

The fundamental group is invariant under the equivalence relation.
develop some methods to compute the groups.
2-dimensional topological spaces with the same fundamental group are equivalent (homeomorphism).

Homotopy of paths

Definition of path

If f and f' are two continuous maps from X to Y, where X and Y are topological spaces. Then we say that f is homotopic to f' if there exists a continuous map F:X\times [0,1]\to Y such that F(x,0)=f(x) and F(x,1)=f'(x) for all x\in X.

The map F is called a homotopy between f and f'.

We use f\simeq f' to mean that f is homotopic to f'.

Definition of homotopic equivalence map

Let f:X\to Y and g:Y\to X be two continuous maps. If f\circ g:Y\to Y and g\circ f:X\to X are homotopic to the identity maps \operatorname{id}_Y and \operatorname{id}_X, then f and g are homotopic equivalence maps. And the two spaces X and Y are homotopy equivalent.

Note

This condition is weaker than homeomorphism. (In homeomorphism, let g=f^{-1}, we require g\circ f=\operatorname{id}_X and f\circ g=\operatorname{id}_Y.)

Example of homotopy equivalence maps

Let X=\{a\} and Y=[0,1] with standard topology.

Consider f:X\to Y by f(a)=0 and g:Y\to X by g(y)=a, where y\in [0,1].

g\circ f=\operatorname{id}_X and f\circ g=[0,1]\mapsto 0.

g\circ f\simeq \operatorname{id}_X

and f\circ g\simeq \operatorname{id}_Y.

Consider F:X\times [0,1]\to Y by F(a,0)=0 and F(a,t)=(1-t)y. F is continuous and homotopy between f\circ g and \operatorname{id}_Y.

This gives example of homotopy but not homeomorphism.

Definition of null homology

If f:X\to Y is homotopy to a constant map. f is called null homotopy.

Definition of path homotopy

Let f,f':I\to X be a continuous maps from an interval I=[0,1] to a topological space X.

Two pathes f and f' are path homotopic if

there exists a continuous map F:I\times [0,1]\to X such that F(i,0)=f(i) and F(i,1)=f'(i) for all i\in I.
F(s,0)=f(0) and F(s,1)=f(1), \forall s\in I.

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