I know there is a relation in homotopy theory which is $\pi(G\times H) = \pi(G)\times\pi(H)$. However, does this relation still hold for $\pi_0$ which may not be a group? Moreover, is there such a relation for the semi-direct product, e.g., $\pi(G\rtimes H) = \pi(G) \rtimes \pi(H)$?
2026-05-06 04:35:12.1778042112
Homotopy and Semidirect Product
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Yes, it holds $\pi_0(X\times Y)\cong \pi_0(X)\times \pi_0(Y).$
Concerning your second question:
The homotopy groups $\pi_i(X)$ are defined for topological spaces (with base points) $X$, so they do not depend on the group structure, if $X$ is a topological group. Since semidirect products of topological groups have the direct product as underlying space, it holds $$\pi_i(G\rtimes H)=\pi_i(G\times H)\cong \pi_i(G)\times \pi_i(H).$$