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Here is the problem I am stuck on: If $X$ is a connected Hausdorff space that is a union of a finite number of $2$-spheres, any two of which intersect in at most one point, then show that $X$ is homotopy equivalent to a wedge sum of $S^1$ and $S^2$.

Intuitively this is pretty easy to see, but the formal details are eluding me. I think the right way to go with this problem is to realize the unions of 2-spheres as a cell complex and then shrink away contractible subcomplexes until we arrive at the desired decomposition. Is there an easy way to make this more formal?

James Miller
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  • Could you phrase the question a little more specifically? is the disjoint union of two spheres considered a 'wedge sum of $S^1$ and $S^2$'? In any case, it's usually a good idea in these kinds of problems to 'stretch out' the intersection points in to intervals so you have a graph where the vertices have been replaced by spheres. – Dan Rust Apr 11 '13 at 16:55
  • You might also like to read the comments and answers to this question http://math.stackexchange.com/questions/70251/question-in-hatcher?rq=1 which references the same question in Hatcher but only asks about the relevance of the Hausdorff hypothesis. – Dan Rust Apr 11 '13 at 17:02

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I think the result you want is stated on pg. 11 of Hatcher's Algebraic Topology, restated here for convenience.

If $(X,A)$ is a CW pair consisting of a CW complex $X$ and a contractible subcomplex $A$, then the quotient map $X\to X/A$ is a homotopy equivalence.

The exact problem that you are referring to is done on pg. 12 (with helpful pictures) in the case of a necklace of $n$ two spheres, which gives a wedge of $S^1$ with $n$ two spheres.

In the general case, I believe we can use induction. Suppose $X$ consists of $n$ two spheres. Distinguish any one of the spheres by 'drawing it away' from the others by lengthening its $m$ points of intersection with other spheres into line segments. On the distinguished sphere, contract all of the $m$ points which intersect the line segments to one point. Now the remaining $n-1$ spheres are homotopy equivalent to a wedge sum of circles and two spheres, and by contracting along subcomplexes, we can assume the $m$ line segments from the distinguished sphere all intersect at the wedge point. Then, just contract one of the $m$ line segments to obtain the desired result (this last contraction adds one two sphere and $m-1$ circles).

Jared
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