Talk:Fundamental group

torus
Could someone add the surface of a torus in the example section, in depth? Cesiumfrog (talk) 11:47, 6 October 2010 (UTC)

not so obvious
"if f and g : X → Y are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f* = g*. As a consequence (???), two homotopy equivalent path-connected spaces have isomorphic fundamental groups:"

I think we need an homotopy $(X, x_0) \times I \to (Y, y_0)$ (which is very different from an homotopy $X \times I \to Y$) in order to imply "as a consequence.."--193.205.210.32 (talk) 17:07, 6 May 2012 (UTC)

Ok! I found the answer in spanier (algebraic topology), although I think that all books about AT work as well.. But I confirm the title of this section after all! --193.205.210.32 (talk) 09:15, 8 May 2012 (UTC)

preservation of coproducts
The article claims that the fundamental group functor preserves all coproducts. This is, in general, false: a counterexample is given by the Hawaiian earring. Michael Lee Baker (talk) 05:36, 3 April 2014 (UTC)
 * Ouch, from the Hawaiian earring article:
 * G contains additional elements which arise from loops whose image is not contained in finitely many of the Hawaiian earring's circles; in fact, some of them are surjective. For example, the path that on the interval [2&minus;n, 2&minus;(n&minus;1)] circumnavigates the nth circle.
 * I guess for the coproduct to result you need base points to have contractible neighborhoods or something like that? This error goes way back to 2006. --Ørjan (talk) 09:18, 3 April 2014 (UTC)
 * Oh well, I've changed it to say locally contractible, I'm pretty sure that's enough. --Ørjan (talk) 09:23, 6 April 2014 (UTC)