A homotopy sphere

Question

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?..

As it is, you're only looking for maps that evidence $S^2$ and $X$ as homotopy retracts of each other. It should be the same maps $f:S^2 \rightarrow X$ and $g:X \rightarrow S^2$ that have $fg \simeq 1_X$ and $gf\simeq 1_{S^2}$. — Aaron Mazel-Gee, Sep 21 '12 at 12:29
@AaronMazel-Gee Well, you are right, I'm a bit inaccurate, but if $X$ was really a homotopy sphere, this would not make any difference, since any degree-1 maps would do. — Alexander Shamov, Sep 21 '12 at 12:40
I'm facing the same problem. I believe the map f:S 2 →X is given so that the upper semisphere is mapped to e2, and lower semisphere is mapped to e3. But I cannot give the homotopy inverse of this map. — lee, Dec 05 '12 at 08:42
The crucial point is that when two attaching maps f,g:X->Y are homotopic, you will get two homotopic spaces which are obtained by attaching X to Y via f,g. — lee, Dec 09 '12 at 12:08
@lee: I couldn't figure out how this helps. :) However, an answer using homotopy theory would be that since both spaces are simply connected, any map $f$ that induces isomorphism on homology is a homotopy equivalence. — Alexander Shamov, Jan 16 '13 at 18:02

score 6 · Answer 1 · edited Nov 20 '19 at 03:42

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.

A homotopy sphere

1 Answers1