Fundamental group

Fundamental group preserves products

Let and be pointed spaces and have the Product topology with the projections and , and let denote the projections of the product group . Then there exists a unique isomorphism such that the following diagram commutes:

https://q.uiver.app/#q=WzAsNCxbMCwyLCJcXHBpXzEoWF8xLHhfMSkiXSxbMiwyLCJcXHBpXzEoWF8yLHhfMikiXSxbMSwwLCJcXHBpXzEoWF8xLHhfMSlcXHRpbWVzXFxwaV8xKFhfMix4XzIpIl0sWzEsNCwiXFxwaV8xKFgseCkiXSxbMiwwLCJcXHZhcnBpXzEiLDJdLFsyLDEsIlxcdmFycGlfMiJdLFszLDAsIlxccGlfMShwXzEpIl0sWzMsMSwiXFxwaV8xKHBfMikiLDJdLFszLDIsIlxcUGhpIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d

which is given by

That is, the fundamental group of a Product topology is isomorphic to the direct product of fundamental groups. #m/thm/homotopy

Proof

From the universal property of the product is a unique homomorphism. Let be a a loop in with base . If then there exist homotopies and . Thus by the homotopy

and hence , hence and thus is injective. Now let be a loop in with base for . Then the following is a loop in with base

and . Hence is surjective.

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