§ Spaces that have same homotopy groups but not the same homotopy type
- Two spaces have the same homotopy type iff there are functions and such that and are homotopic to the identity.
- Now consider two spaces: (1) the point, (2) the topologists's sine curve with two ends attached (the warsaw circle).
- See that the second space can have no non-trivial fundamental group, as it's impossible to loop around the sine curve.
- So the warsaw circle has all trivial , just like the point.
- See that the map must send every point in the warsaw circle to the point .
- See that the map backward can send somewhere, so we are picking a point on .
- The composite smooshes all of to a single point. For this to be homotopic to the identity is to say that the space is contractible.