It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere.
[1] The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005.
The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are precisely exotic spheres.
It is open whether non-trivial smooth homotopy spheres exist in dimension 4.
Its composition is the connected sum and its neutral element is the sphere, while inversion is given by opposite orientation.