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A smooth manifold homeomorphic (and piecewise-linearly isomorphic), but not diffeomorphic, to the sphere  $  S  ^ {n} $. 
The first example of such a manifold was constructed by J. Milnor in 1956 (see [[#References|[1]]]); the same example was the first example of homeomorphic but not diffeomorphic manifolds.

==Construction of a Milnor sphere.==
Any compact smooth oriented closed manifold, homotopically equivalent to  $  S  ^ {n} $, 
$  n \geq  5 $, 
is homeomorphic (and even piecewise-linearly isomorphic) to  $  S  ^ {n} $ (see [[Poincaré conjecture|Poincaré conjecture]], generalized; [[H-cobordism| $  h $-cobordism]]). The index of a closed smooth almost [[Parallelizable manifold|parallelizable manifold]] of dimension  $  4 k $
is divisible by a number  $  \sigma _ {k} $
which exponentially increases with  $  k $. 
For any  $  k $
there is a parallelizable manifold  $  P  ^ {4k} $
of index 8 (namely, the plumbing construction of Milnor) whose boundary  $  M = \partial  P $
is, for  $  k > 1 $, 
a homotopy sphere (see [[#References|[2]]], [[#References|[6]]]). If  $  M $
were diffeomorphic to the sphere  $  S  ^ {4k-1} $, 
then the manifold  $  W  ^ {4k} $
obtained from  $  P  ^ {4k} $
by the addition of a cone over the boundary would be a smooth almost parallelizable closed manifold of index 8. Thus  $  M $
is a Milnor sphere.

There are other examples of Milnor spheres (see [[#References|[5]]]).

==Classification of Milnor spheres.==
In the sequel the term  "Milnor sphere"  will be used also for the standard sphere  $  S  ^ {n} $. 
There are 28 distinct (non-diffeomorphic)  $  7 $-dimensional Milnor spheres.

The set of all smooth structures on the piecewise-linear sphere is equivalent to the set of elements of the group  $  \pi _ {i} (  \mathop{\rm PL} / O) $. 
The latter group is trivial for  $  i < 7 $, 
so in the  $   \mathop{\rm PL} $-case any Milnor sphere of dimension less than 7 is diffeomorphic to the standard sphere.

Let  $  \theta _ {n} $
be the set of classes of  $  h $-cobordant  $  n $-dimensional smooth manifolds which are homotopically equivalent to  $  S  ^ {n} $. 
The operation of [[Connected sum|connected sum]] transforms this set into a group, where the zero is the  $  h $-cobordism class of  $  S  ^ {n} $. 
For  $  n > 5 $
the elements of  $  \theta _ {n} $
are in one-to-one correspondence with the diffeomorphism classes of  $  n $-dimensional Milnor spheres. To calculate the groups  $  \theta _ {n} $, 
$  n > 5 $, 
one specifies (see [[#References|[3]]]) a trivialization of the stable normal bundle (a framing) of the Milnor sphere  $  M  ^ {n} $. 
This is possible since  $  M  ^ {n} $
is stably parallelizable. The framed manifold obtained defines an element of the stable homotopy group  $  \Pi _ {n} = \lim\limits _ {i}  \pi _ {i+n} ( S  ^ {i} ) $. 
This element depends, in general, on the choice of the framing ( $  \theta _ {n} \rightarrow \Pi _ {n} $
is a  "multi-valued mapping" ). Let  $  \theta _ {n} ( \partial  \pi ) $
be the subgroup in  $  \theta _ {n} $
consisting of Milnor spheres that bound parallelizable manifolds. This multi-valued mapping induces a homomorphism  $  \alpha :  \theta _ {n} / \theta _ {n} ( \partial  \pi ) \rightarrow  \mathop{\rm Coker}  J _ {n} $, 
where  $  J _ {n} :  \pi _ {n} (  \mathop{\rm SO} ) \rightarrow \Pi _ {n} $
is the stationary [[Whitehead homomorphism|Whitehead homomorphism]] and  $  \alpha $
is an isomorphism. The calculation of the group  $  \theta _ {n} / ( \theta _ {n} ( \partial  \pi ) ) $
reduces to the problem of calculating  $  \Pi _ {n} $
and  $  \theta _ {n} ( \partial  \pi ) $ (unsolved, 1989), which is done by means of surgery (cf. [[Morse surgery|Morse surgery]]) of the manifold (preserving the boundary). Let  $  [ M  ^ {n} ] \in \theta _ {n} ( \partial  \pi ) $, 
that is,  $  M  ^ {n} = \partial  W  ^ {n+1} $
and  $  W  ^ {n+1} $
is parallelizable. If  $  W $
is a contractible manifold, then after cutting out from  $  W $
a small disc, the manifold  $  M $
is  $  h $-cobordant to  $  S  ^ {n} $, 
that is,  $  [ M  ^ {n} ] = 0 \in \theta _ {n} $. 
If  $  n $
is even, then it is possible to modify  $  W $
by means of surgery so that the new manifold  $  W _ {1} $
with  $  \partial  W _ {1} = M $
is contractible (here one requires parallelizability of  $  W $
and  $  n \geq  4 $). 
Thus  $  \theta _ {2n} ( \partial  \pi ) = 0 $.

The case  $  n + 1 = 4 k $. 
If the index  $  \sigma ( W) $
of  $  W $
is  $  0 $, 
then  $  W $
can be transformed by surgery into a contractible manifold, so that in this case  $  M $
is a standard sphere. If  $  M = \partial  W $
and  $  M _ {1} = \partial  W _ {1} $, 
then  $  M \# ( - M _ {1} ) = \partial  ( W \# ( - W ) ) $
and  $  \sigma ( W \# ( - W _ {1} ) ) = \sigma ( W) - \sigma ( W _ {1} ) $ (here  $  A \# B $
is the connected sum or the boundary connected sum of two manifolds  $  A $
and  $  B $). 
If  $  \sigma ( W) = \sigma ( W _ {1} ) $, 
then  $  [ M ] = [ M _ {1} ] $, 
so that the invariant  $  \sigma ( W) $
defines an element  $  [ M ] \in \theta _ {n} $. 
If  $  [ M ] = 0 \in \theta _ {4k-1} ( \partial  \pi ) $
and  $  M = \partial  W $, 
then  $  \sigma ( W ) $
is divisible by  $  \sigma _ {k} $. 
Conversely, for any  $  k > 1 $
there is a smooth closed manifold  $  B  ^ {4k} $
with  $  \sigma ( B  ^ {4k} ) = \sigma _ {k} $; 
therefore, if  $  M = \partial  W $
and  $  \sigma ( W ) = n \sigma _ {k} $, 
then  $  M = \partial  ( W \# ( - n B  ^ {4k} ) ) $, 
where  $  W \# ( - n B  ^ {4k} ) $
is parallelizable and  $  \sigma ( W \# ( - n B  ^ {4k} ) ) = 0 $. 
The element  $  [ M ] \in \theta _ {4k-1} ( \partial  \pi ) $
is completely determined by the residue of  $  \sigma ( W ) $
modulo  $  \sigma _ {k} $, 
and different residues determine different manifolds. Since  $  \sigma ( W) $
takes any value divisible by  $  8 $, 
$   \mathop{\rm ord}  \theta _ {4k-1} ( \partial  \pi ) = \sigma _ {k} / 8 $. 
E.g.,  $  \theta _ {7} ( \partial  \pi ) = \mathbf Z _ {28} $, 
and  $  \textrm{ Coker  }  J _ {7} = 0 $, 
so  $  \theta _ {7} = \mathbf Z _ {28} $.

The case  $  n = 4 k + 1 $. 
Let  $  M = \partial  W  ^ {4k+2} $. 
If the [[Kervaire invariant|Kervaire invariant]] of  $  W $
is zero, that is,  $  \psi ( W ) = 0 $, 
then  $  W $
can be converted by surgery into a contractible manifold, that is,  $  [ M ] = 0 $. 
Now let  $  \psi ( W ) \neq 0 $. 
Since for  $  4 k + 2 \neq 2  ^ {i} - 2 $
there is no smooth closed almost-parallelizable (which in dimension  $  4 k + 2 $
is equivalent to stably-parallelizable) manifold with Kervaire invariant not equal to zero,  $  M $
is not diffeomorphic to  $  S  ^ {4k+1} $. 
In this case  $  \theta _ {4k+1} ( \partial  \pi ) \neq 0 $, 
that is,  $  \theta _ {4k+1} ( \partial  \pi ) = \mathbf Z _ {2} $. 
For  $  4 k + 2 = 2  ^ {i} - 2 $
and those  $  i $
for which there is a manifold with non-zero Kervaire invariant,  $  M \approx S  ^ {4k+1} $, 
that is,  $  \theta _ {4k+1} ( \partial  \pi ) = 0 $, 
but the question of describing all such  $  i $
has not been solved (1989). However, for  $  i \leq  6 $
the answer is positive. Thus  $  \theta _ {4k+1} ( \partial  \pi ) $
is  $  \mathbf Z _ {2} $
or  $  0 $.

There is another representation of a Milnor sphere. Let  $  W $
be an algebraic variety in  $  \mathbf C  ^ {n+1} $
with equation

$$ 
z _ {1} ^ {a _ {1} } + \dots + z _ {n+1 }^ {a _ {n+1} }  =  0
$$

and let  $  S _  \epsilon  $
be the  $  ( 2 n + 1 ) $-dimensional sphere of (small) radius  $  \epsilon $
with centre at the origin. For suitable values of  $  a _ {k} $, 
$  M = W \cap S _  \epsilon  $
is a Milnor sphere (see [[#References|[4]]]). For example, for  $  n = 4 $
and  $  a _ {1} = 6 k - 1 $, 
$  a _ {2} = 3 $, 
$  a _ {3} = a _ {4} = a _ {5} = 2 $
and  $  k = 1, \dots, 28 $, 
all 28  $  7 $-dimensional Milnor spheres are obtained.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. Milnor,   "On manifolds homeomorphic to the 7-sphere"  ''Ann. of Math.'' , '''64'''  (1956)  pp. 399–405</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Milnor,   "Bernoulli numbers, homotopy groups, and a theorem of Rohlin"  J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press  (1960)  pp. 454–458 {{ZBL|0119.38503}}</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Kervaire,   J.W. Milnor,   "Groups of homotopy spheres"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 504–537 {{ZBL|0115.40505}}</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  J. Milnor,   "Singular points of complex hypersurfaces" , Princeton Univ. Press  (1968)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  J.W. Milnor,   J.D. Stasheff,   "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  W.B. Browder,   "Surgery on simply-connected manifolds" , Springer  (1972)</TD></TR>
</table>

====Comments====
The general problem of constructing different smooth structures on a topological manifold has received much attention since the above article was written (around 1982). In particular, it has been proven that  $  \mathbf R  ^ {4} $
has different smooth structures (but not  $  \mathbf R  ^ {n} $
for  $  n \neq 4 $). 
A general reference is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.S. Freed,   K.K. Uhlenbeck,   "Instantons and four-manifolds" , Springer  (1984)</TD></TR></table>