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In the wikipedia article about free products, it says that "the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the free product of the fundamental groups of the spaces."

That is generally not true, such as in the example of two Hawaiian earrings glued at their "special" point (the point where all circles intersect), since the wedge point has no neighborhood that is contractible (the space is not semi locally simply connected).

Do you think a change should be done to make the wikipedia page more exact? Maybe add "as long as the wedge point has a contractible neighborhood in the wedge space"?

Whyka
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  • When I read "free product Wikipedia article", I reacted thinking, "What, Wikipedia turns into a commercial site ?". I am reassured now. – Jean Marie Oct 07 '17 at 12:16
  • What is the fundamental group of the wedge sum of two Hawaiian earrings joined at their "special" point? – Tyrone Oct 07 '17 at 12:23
  • Related : https://math.stackexchange.com/questions/22733/the-fundamental-group-of-a-pair-of-hawaiian-earrings?rq=1 – Arnaud D. Oct 07 '17 at 12:44

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I believe the full statement should require the basepoints of each space in the wedge to be non-degenerate. See theorem 10.7 in Lee's "Introduction to Topological Manifolds" and the surrounding discussion, for example.

Tyrone
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  • Thanks. Why is demanding the wedge point to have a contractible neighborhood in the wedge space not enough? – Whyka Oct 07 '17 at 15:12
  • I believe that it must be pointed contractible. That is the basepoint must be a strong neighborhood deformation retract, or non-degenerate. I'm afraid I'm not going to chase through the technical details of the theorem, but obviously we want to thicken up each summand so that it is open in the wedge product (by wedging it with an appropriate contractible neighborhood of the basepoint of the other space) and we want all resulting homotopy equivalences coherent and compatible with the fundamental group functor (which is a functor on the pointed topological category). – Tyrone Oct 07 '17 at 22:13