Homotopy Group is Homeomorphism Invariant

= Theorem =

Given two manifolds $$X$$ and $$Y$$, if there is a homeomorphism $$\phi:X \to Y$$, then $$\forall n \in \mathbb{N}, \pi_n(X) = \pi_n(Y)$$, where the $$\pi_n \ $$ are the $$n^{th} \ $$ homotopy groups.

= Proof =

Let $$\phi \ $$ be any homeomorphism $$\phi:X\to Y \ $$. We must show that:


 * 1) . If $$c:[0,1]^n \to X \ $$ is a continuous mapping, then $$ \phi \circ c:[0,1]^n \to Y \ $$ is as well;
 * 2) . If $$c,d:[0,1]^n \to X \ $$ are homotopic, then $$\phi\circ c, \phi\circ d:[o,1]^n \to Y \ $$ are homotopic as well;
 * 3) . If $$c,d:[0,1]^n \to X \ $$ are not homotopic, there can be no homotopy between $$\phi\circ c \ $$ and $$\phi \circ d \ $$;
 * 4) . The image of the concatenation of two maps, $$\phi(c * d) \ $$, is the concatenation of the images, $$\phi(c)*\phi(d) \ $$.