A homotopy sphere

My question is part of an exercise in Hatcher's 'Algebraic Topology'.

Consider a CW complex $X$, constructed from a circle and two 2-disks $e_2$ and $e_3$, attached to that circle by maps of degree 2 and 3, respectively. Can someone show that $X$ is homotopy equivalent to a 2-sphere? Its 2-homology is generated by $3e_2 - 2e_3$, hence this homotopy equivalence $S^2 \to X$ must be at least 2-to-1 in a generic point.

It is easy to construct maps $S^2 \overset{f}{\to} X \overset{g}{\to} S^2$ that have degree one, hence compose to homotopy identity, but I am really stuck with $X \overset{g}{\to} S^2 \overset{f}{\to} X$... Is there a nice explanation why this should be homotopy identity?..

Solution 1:

After attaching the first disk with a map of degree 2, you have a projective plane, where the original circle can be thought of as non-contractible loop. We attach the second disk by a map of degree 3, but this is the projective plane, so we can homotope the attaching map to one of degree one. So we just glue a disk into the original circle.

But now we can reverse roles, and homotope the original attaching map. We are now attaching a disk into a closed disk by a map of degree 2, but a closed disk is contractible, so we can homotope the attaching map to any loop, in particular to a map of degree one. This gives us a sphere.

This is all using proposition 0.18 of Hatcher, as suggested in the comments.