§ Spaces that have same homotopy groups but not the same homotopy type

• Two spaces have the same homotopy type iff there are functions $f: X \to Y$ and $g: Y \to X$such that $f \circ g$ and $g \circ f$ 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 $\pi_j$, just like the point.
• See that the map $W \to \{ \star \}$ must send every point in the warsaw circle to the point $\star$.
• See that the map backward can send $\star$ somewhere, so we are picking a point on $W$.
• The composite smooshes all of $W$ to a single point. For this to be homotopic to the identity is to say that the space is contractible.