Why is the connected sum of homotopy spheres a homotopy sphere?

Question

In the paper "Groups of Homotopy Spheres I" Kevaire and Milnor say, that it is obvious, that the connected sum of two homotopy spheres is again a homotopy sphere. Using, that any smooth homotopy sphere is (topologically) a sphere this really is obvious but back when they published their paper the dimensions 3 and 4 of this statement were unsolved. So is there an elementary argument, why the connected sum of homotopy spheres is a homotopy sphere?

Answer 1

Suppose that a CW-complex $X$ is simply-connected and shares the same (integral) homology groups with $S^m$, $m>1$. By inductively applying Hurewicz, we get that $X$ shares the same homotopy groups as $S^m$ up to degree $m$. Then in particular, $\pi_m(X)\cong\mathbb{Z}$, and so there is a map $f:S^m \rightarrow X$ representing $1$. Choosing an isomorphism $\pi_m(S^m)\cong\mathbb{Z}$ such that the identity on $S^m$ maps to $1$, the induced map $f_*:\pi_m(S^m) \rightarrow \pi_m(X)$ sends $1 \mapsto 1$, and so $f$ induces an isomorphism in homotopy in degree m.

We have the following commutative diagram $\require{AMScd}$ \begin{CD} \pi_m(S^m) @>f_*>> \pi_m(X)\\ @V h V V @VV h V\\ H_m(S^m) @>>f_*> H_m(X) \end{CD} where $h$ denotes the Hurewicz homomorphisms. Both $S^m$ and $X$ are $(m-1)$-connected, so the vertical arrows are isomorphisms. From the above, the top arrow is an isomorphism. Thus, the bottom arrow is an isomorphism, and so $f$ induces an isomorphism in homology in degree $m$. The only non-trivial homology groups of $S^m$ are in degrees $0$ and $m$, and both $X$ and $S^m$ are 0-connected, so $f$ induces isomorphisms in homology in all degrees.

From the "Homology Whitehead Theorem" (see, for example, these notes), a map between simply-connected CW-complexes that induces isomorphisms in all integral homology groups is a homotopy equivalence, so $X \simeq S^m$.

The Kosinski reference pointed out by Quique outlines a proof that given homotopy m-spheres $M_1$ and $M_2$, the homology groups of $M_1\#M_2$ match those of $S^m$, and $M_1\#M_2$ is simply-connected. Any topological manifold is homotopy equivalent to a CW-complex, and so $M_1\#M_2$ is also a homotopy m-sphere.