In mathematics, especially differential topology and cobordism theory, a Kervaire–Milnor group is an abelian group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation as inversion.
It controls the existence of smooth structures on topological and piecewise linear (PL) manifolds.
[1] Concerning the related question of PL structures on topological manifolds, the obstruction is given by the Kirby–Siebenmann invariant, which is a lot easier to understand.
They are named after the French mathematician Michel Kervaire and the US american mathematician John Milnor, who first described them in 1962.
An important property of spheres is their neutrality with respect to the connected sum of manifolds.
[2] Expanding this monoid structure with a composition and a neutral element to a group structure requires the restriction on manifolds, for which a connected sum can result in a sphere, hence which intuitively doesn't have holes.
This is possible with homotopy spheres, which are closed smooth manifolds with the same homotopy type as a sphere, with restriction to h-cobordism classes being useful for application.
Inversion is then given by changing their orientation,[2] which results in a group structure.
[3] An alternative definition in higher dimensions is given by the description of topological, PL and smooth structures.
be the topological group of diffeomorphisms of euclidean space
An inductive limit yields topological groups
(which is homotopy equivalent to the infinite orthogonal group
is also a topological manifold, which is classified by a continuous map
Analogous for a PL and a smooth manifold, there are classifying maps
Some low-dimensional Kervaire–Milnor groups are given by:[5][6] After the contruction of Milnor spheres in 1956, it was already known that Kervaire–Milnor groups don't have to be trivial with the more exact result
A generating exotic sphere was constructed by Egbert Brieskorn as special case of a Brieskorn manifold in 1966.
It posesses more unique properties and is also called Gromoll–Meyer sphere.
It is still unknown (in 2025) whether exotic spheres exist in four dimensions with the result
This is because Kervaire–Milnor groups only also describe the diffeomorphism classes of spheres for
Often the set of diffeomorphism classes of homotopy spheres is denoted