Show that a space obtained from $S^2$ by attaching n 2-cells along any collection of n circles in $S^2$ is homotopy equivalent to the wedge of n+1 spheres.
I'm a little confused here. I'm imagining a sphere, and putting 2 dimensional discs inside of it. Attaching one disc along one circle of $S^2$ and then collapsing that disc to a point would appear to me to create a wedge of two spheres.
However, if I attached 2 discs along 2 different circles of $S^2$ and collapsed each to a point, to me it seems I would have created a wedge of four spheres, one sphere for each division created by the two discs inside the sphere.
What is wrong with my thinking here? Thanks!!
Best Answer
If you form $Z= S^2 \cup_f e^2$ then the attaching map- $f: S^1 \to S^2$ is null homotopic, as $S^2$ is simply connected. So $Z \simeq S^2 \vee S^2 $. The general result you need is that if $Z= B \cup_f X$ where $f: A \to B$, and the inclusion $i : A \to X$ is a closed cofibration, then the homotopy type of $Z$ depends only on the homotopy class of $f$.