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I am not a topologist, so please excuse me if the question is trivial.

Suppose I am given three nice, path-connected spaces. Then I can think of two ways to wedge these spaces: join all three at a common basepoint, or have one space in the "middle" with two basepoints (so I guess there are four ways in total). Do these constructions result in the same space up to homotopy? My guess would be to take a path between the two basepoints in the "middle" space and contract it to a point, but I understand that it is sometimes a subtle matter to tell if this sort of thing is a homotopy equivalence.

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This is already an issue for two spaces, right? It's not clear that the wedge doesn't depend on the choice of basepoint in each space. –  Qiaochu Yuan Jul 30 '12 at 20:06

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Generally these things aren't homotopy-equivalent.

As Qiaochu mentions, when you wedge together only two spaces, the homotopy-type of the wedge depends on your choice of basepoint (even if they're path connected).

What you need is that your input spaces are path connected and moreover, given your two choices of base-point you want a homotopy-equivalence $(X,x_0) \simeq (X,x_1)$.

I don't know if these kinds of spaces have a name or not, it seems like homogeneous would be fairly appropriate.

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If $X$ is also locally contractible, then $(X, x_0) \simeq (X, x_1)$. But this is probably too strong a condition for the homotopy equivalence. –  Shaun Ault Jul 30 '12 at 21:38
    
Great. I was thinking about homogeneous spaces anyway... –  Justin Campbell Jul 31 '12 at 0:58

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