In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic").

The first exotic spheres were constructed by John Milnor (1956) in dimension ${\displaystyle n=7}$ as ${\displaystyle S^{3}}$-bundles over ${\displaystyle S^{4}}$. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and Milnor (1963) showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. These groups are known as Kervaire–Milnor groups.

More generally, in any dimension n ≠ 4, there is a finite Abelian group whose elements are the equivalence classes of smooth structures on Sⁿ, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y] = [x + y], where x and y are arbitrary representatives of their equivalence classes, and x + y denotes the smooth structure on the smooth Sⁿ that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.

The unit n-sphere, ${\displaystyle S^{n}}$, is the set of all (n+1)-tuples ${\displaystyle (x_{1},x_{2},\ldots ,x_{n+1})}$ of real numbers, such that the sum ${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n+1}^{2}=1}$. For instance, ${\displaystyle S^{1}}$ is a circle, while ${\displaystyle S^{2}}$ is the surface of an ordinary ball of radius one in 3 dimensions. Topologists consider a space X to be an n-sphere if there is a homeomorphism between them, i.e. every point in X may be assigned to exactly one point in the unit n-sphere by a continuous bijection with continuous inverse. For example, a point x on an n-sphere of radius r can be matched homeomorphically with a point on the unit n-sphere by multiplying its distance from the origin by ${\displaystyle 1/r}$. Similarly, an n-cube of any radius is homeomorphic to an n-sphere.

In differential topology, two smooth manifolds are considered smoothly equivalent if there exists a diffeomorphism from one to the other, which is a homeomorphism between them, with the additional condition that it be smooth — that is, it should have derivatives of all orders at all its points — and its inverse homeomorphism must also be smooth. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians (including Milnor himself) were surprised in 1956 when Milnor showed that consistent local coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-sphere. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an unsolved problem.

The monoid of smooth structures on n-spheres is the collection of oriented smooth n-manifolds which are homeomorphic to the n-sphere, taken up to orientation-preserving diffeomorphism. The monoid operation is the connected sum. Provided ${\displaystyle n\neq 4}$, this monoid is a group and is isomorphic to the group ${\displaystyle \Theta _{n}}$ of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on Gluck twists. All homotopy n-spheres are homeomorphic to the n-sphere by the generalized Poincaré conjecture, proved by Stephen Smale in dimensions bigger than 4, Michael Freedman in dimension 4, and Grigori Perelman in dimension 3. In dimension 3, Edwin E. Moise proved that every topological manifold has an essentially unique smooth structure (see Moise's theorem), so the monoid of smooth structures on the 3-sphere is trivial.

The group ${\displaystyle \Theta _{n}}$ has a cyclic subgroup

represented by n-spheres that bound parallelizable manifolds. The structures of ${\displaystyle bP_{n+1}}$ and the quotient