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A smooth manifold homeomorphic (and piecewise-linearly isomorphic), but not diffeomorphic, to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638001.png" />. 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.
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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.==
 
==Construction of a Milnor sphere.==
Any compact smooth oriented closed manifold, homotopically equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638003.png" />, is homeomorphic (and even piecewise-linearly isomorphic) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638004.png" /> (see [[Poincaré conjecture|Poincaré conjecture]], generalized; [[H-cobordism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638005.png" />-cobordism]]). The index of a closed smooth almost [[Parallelizable manifold|parallelizable manifold]] of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638006.png" /> is divisible by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638007.png" /> which exponentially increases with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638008.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m0638009.png" /> there is a parallelizable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380010.png" /> of index 8 (namely, the plumbing construction of Milnor) whose boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380011.png" /> is, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380012.png" />, a homotopy sphere (see [[#References|[2]]], [[#References|[6]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380013.png" /> were diffeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380014.png" />, then the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380015.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380016.png" /> by the addition of a cone over the boundary would be a smooth almost parallelizable closed manifold of index 8. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380017.png" /> is 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]]]).
 
There are other examples of Milnor spheres (see [[#References|[5]]]).
  
 
==Classification of Milnor spheres.==
 
==Classification of Milnor spheres.==
In the sequel the term  "Milnor sphere"  will be used also for the standard sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380018.png" />. There are 28 distinct (non-diffeomorphic) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380019.png" />-dimensional 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380020.png" />. The latter group is trivial for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380021.png" />, so in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380022.png" />-case any Milnor sphere of dimension less than 7 is diffeomorphic to the standard sphere.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380023.png" /> be the set of classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380024.png" />-cobordant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380025.png" />-dimensional smooth manifolds which are homotopically equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380026.png" />. The operation of [[Connected sum|connected sum]] transforms this set into a group, where the zero is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380027.png" />-cobordism class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380028.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380029.png" /> the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380030.png" /> are in one-to-one correspondence with the diffeomorphism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380031.png" />-dimensional Milnor spheres. To calculate the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380033.png" />, one specifies (see [[#References|[3]]]) a trivialization of the stable normal bundle (a framing) of the Milnor sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380034.png" />. This is possible since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380035.png" /> is stably parallelizable. The framed manifold obtained defines an element of the stable homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380036.png" />. This element depends, in general, on the choice of the framing (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380037.png" /> is a  "multi-valued mapping" ). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380038.png" /> be the subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380039.png" /> consisting of Milnor spheres that bound parallelizable manifolds. This multi-valued mapping induces a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380041.png" /> is the stationary [[Whitehead homomorphism|Whitehead homomorphism]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380042.png" /> is an isomorphism. The calculation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380043.png" /> reduces to the problem of calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380045.png" /> (unsolved, 1989), which is done by means of surgery (cf. [[Morse surgery|Morse surgery]]) of the manifold (preserving the boundary). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380046.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380048.png" /> is parallelizable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380049.png" /> is a contractible manifold, then after cutting out from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380050.png" /> a small disc, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380051.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380052.png" />-cobordant to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380053.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380054.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380055.png" /> is even, then it is possible to modify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380056.png" /> by means of surgery so that the new manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380057.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380058.png" /> is contractible (here one requires parallelizability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380060.png" />). Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380061.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380062.png" />. If the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380064.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380066.png" /> can be transformed by surgery into a contractible manifold, so that in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380067.png" /> is a standard sphere. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380069.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380071.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380072.png" /> is the connected sum or the boundary connected sum of two manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380074.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380076.png" />, so that the invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380077.png" /> defines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380078.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380080.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380081.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380082.png" />. Conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380083.png" /> there is a smooth closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380084.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380085.png" />; therefore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380087.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380089.png" /> is parallelizable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380090.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380091.png" /> is completely determined by the residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380092.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380093.png" />, and different residues determine different manifolds. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380094.png" /> takes any value divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380096.png" />. E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380097.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380098.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m06380099.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800100.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800101.png" />. If the [[Kervaire invariant|Kervaire invariant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800102.png" /> is zero, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800103.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800104.png" /> can be converted by surgery into a contractible manifold, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800105.png" />. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800106.png" />. Since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800107.png" /> there is no smooth closed almost-parallelizable (which in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800108.png" /> is equivalent to stably-parallelizable) manifold with Kervaire invariant not equal to zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800109.png" /> is not diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800110.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800111.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800112.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800113.png" /> and those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800114.png" /> for which there is a manifold with non-zero Kervaire invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800115.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800116.png" />, but the question of describing all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800117.png" /> has not been solved (1989). However, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800118.png" /> the answer is positive. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800119.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800120.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800121.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800122.png" /> be an algebraic variety in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800123.png" /> with equation
+
There is another representation of a Milnor sphere. Let $  W $
 +
be an algebraic variety in $  \mathbf C  ^ {n+} 1 $
 +
with equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800124.png" /></td> </tr></table>
+
$$
 +
z _ {1} ^ {a _ {1} } + \dots + z _ {n+} 1 ^ {a _ {n+} 1 }  = 0
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800125.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800126.png" />-dimensional sphere of (small) radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800127.png" /> with centre at the origin. For suitable values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800129.png" /> is a Milnor sphere (see [[#References|[4]]]). For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800134.png" />, all 28 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800135.png" />-dimensional Milnor spheres are obtained.
+
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====
 
====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</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</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>
 
<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</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</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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800136.png" /> has different smooth structures (but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800137.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063800/m063800138.png" />). A general reference is [[#References|[a1]]].
+
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====
 
====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>
 
<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>

Revision as of 08:00, 6 June 2020


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 [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, generalized; $ h $- cobordism). The index of a closed smooth almost 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 [2], [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 [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 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 [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 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) 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 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 [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

[1] J.W. Milnor, "On manifolds homeomorphic to the 7-sphere" Ann. of Math. , 64 (1956) pp. 399–405
[2] 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
[3] M.A. Kervaire, J.W. Milnor, "Groups of homotopy spheres" Ann. of Math. , 77 (1963) pp. 504–537
[4] J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)
[5] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)
[6] W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972)

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 [a1].

References

[a1] D.S. Freed, K.K. Uhlenbeck, "Instantons and four-manifolds" , Springer (1984)
How to Cite This Entry:
Milnor sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Milnor_sphere&oldid=47839
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article