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)

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)