Behavior of Homotopy Groups with respect to Products

This blog post is on the behavior of homotopy groups with respect to products. Proposition 4.2 of Hatcher:

For a product \prod_\alpha X_\alpha of an arbitrary collection of path-connected spaces X_\alpha there are isomorphisms \pi_n(\prod_\alpha X_\alpha)\cong\prod_\alpha \pi_n(X_\alpha) for all n.

The proof given in Hatcher is a short one: A map f:Y\to \prod_\alpha X_\alpha is the same thing as a collection of maps f_\alpha: Y\to X_\alpha. Taking Y to be S^n and S^n\times I gives the result.

A possible alternative proof is to first prove that \pi_n(X_1\times X_2)\cong\pi_n(X_1)\times\pi_n(X_2), which is the result for a product of two spaces. The general result then follows by induction.

We construct a map \psi:\pi_n(X_1\times X_2)\to\pi_n(X_1)\times\pi_n(X_2), \psi([f])=([f_1],[f_2]).

Notation: f:S^n\to X_1\times X_2, f_1=p_1\circ f:S^n\to X_1, f_2=p_2\circ f:S^n\to X_2 where p_i:X_1\times X_2\to X_i are the projection maps.

We can show that \psi ([f]+[g])=\psi([f])+\psi([g]), thus \psi is a homomorphism.

We can also show that \psi is bijective by constructing an explicit inverse, namely \phi:\pi_n(X_1)\times\pi_n(X_2)\to\pi_n(X_1\times X_2), \phi([g_1],[g_2])=[g] where g:S^n\to X_1\times X_2, g(x)=(g_1(x),g_2(x)).

Thus \psi is an isomorphism.


About mathtuition88
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