# Milnor sphere

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)

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].