The homotopy-sphere group ${\mathrm{\Theta }}_n$

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For $n\ge 5$, let ${\mathrm{\Theta }}_n$ be the set of oriented h-cobordism classes of oriented smooth homotopy $n$-spheres (Smooth homotopy sphere). Addition is the oriented connected sum of homotopy spheres, the zero class is the class of the standard sphere $S^n$ with its standard orientation, and the inverse of the class of $\mathrm{\Sigma }$ is the class of $-\mathrm{\Sigma }$.

These classes form a set: compactness gives a finite coordinate atlas for each manifold. The finite chart domains are open subsets of ${\mathbb{R}}^n$ and the transition maps are functions between such subsets, so all finite oriented atlas data range over a set. Their quotients represent every compact oriented smooth $n$-manifold, and hence taking the homotopy-sphere subcollection and its quotient by the relation is a set operation. The relation is indeed an equivalence here: an h-cobordism has dimension $n+1\ge 6$ and simply connected faces, so the relative product theorem gives an orientation-preserving diffeomorphism (The smooth simply connected h-cobordism theorem); conversely any such diffeomorphism supplies a product h-cobordism. Thus reflexivity, symmetry and transitivity follow from those of oriented diffeomorphism.

The operation is well defined on h-cobordism classes, associative and commutative with the class of $S^n$ as two-sided identity by Connected sum descends to oriented h-cobordism classes, and $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$ is oriented h-cobordant to $S^n$ by Orientation reversal is the connected-sum inverse; hence these data form an abelian group. The inverse is well defined on classes because an oriented h-cobordism between $\mathrm{\Sigma }$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ can be composed with the given one and reversed in orientation. All choices enter only through the verified connected-sum and inverse lemmas, and no choice principle stronger than ${\mathrm{A}\mathrm{C}}_{\omega }$ is used.