# The Wedge Sum of path connected topological spaces

A definition of wedge sum can be found here:

http://en.wikipedia.org/wiki/Wedge_sum

My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy equivalence, independently of choice of base points x0 and y0. Base point here means the points that are identified under the equivalence relation forming the wedge product out of the disjoint union topology of X and Y.

Recall homotopy equivalence of X and Y means that there is f:X->Y and g:Y->X continuous with gf and fg homotopic to the identity.

With these definitions, please prove my professor's claim, which I have failed to do for a week. (It is left as an exercise in his lecture.)

Thanks.

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1. Mathoverflow isn't for homework. 2. This is a fun homework assignment and you should think about it more. –  Richard Kent Jul 3 '10 at 20:57
This is not homework. It is an exercise left by my professor, which was not assigned for homework. If you don't believe me, you can look at exercise 2.35 in his set of lecture notes: math.caltech.edu/~ma109a/109anotes.pdf You may notice the class is over. I want this fact proven for research I'm doing this summer. –  Jeff Jul 3 '10 at 20:59
One week's worth of effort, as well as the failure to find anything useful on wikipedia, google-book's preview, and in Caltech's own library says otherwise. Also, none of my peers who have taken Math 109a can handle it either. I have asked, trust me. Do you have a hint? I first tried elementary things like inclusion maps and projections (remember the wedge product is a quotient space.) I then thought about examples (I can do well-behaved cases in R^n) and I thought about showing a homotopy with a dumb-bell like object where the wedged point becomes a line. –  Jeff Jul 3 '10 at 21:08
I mean, what's the question? True: If $X$, $Y$, and $Z$ are based spaces and $X$ and $Y$ are homotopy equivalent in the based sense then $X\vee Z$ and $Y\vee Z$ are homotopy equivalent in the same sense. False: If $X$, $Y$, and $Z$ are path connected spaces and $X$ and $Y$ are homotopy equivalent then $X\vee Z$ and $Y\vee Z$ are homotopy equivalent no matter what base points you use to stick things together. –  Tom Goodwillie Jul 3 '10 at 21:30
Tom, I think he's not varying the homeomorphism type of the two factors, only the placement of the basepoint. –  Richard Kent Jul 3 '10 at 21:35