A smooth manifold homeomorphic (and piecewise-linearly isomorphic), but not diffeomorphic, to the sphere . The first example of such a manifold was constructed by J. Milnor in 1956 (see ); 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 , , is homeomorphic (and even piecewise-linearly isomorphic) to (see Poincaré conjecture, generalized; -cobordism). The index of a closed smooth almost parallelizable manifold of dimension is divisible by a number which exponentially increases with . For any there is a parallelizable manifold of index 8 (namely, the plumbing construction of Milnor) whose boundary is, for , a homotopy sphere (see , ). If were diffeomorphic to the sphere , then the manifold obtained from by the addition of a cone over the boundary would be a smooth almost parallelizable closed manifold of index 8. Thus is a Milnor sphere.
There are other examples of Milnor spheres (see ).
Classification of Milnor spheres.
In the sequel the term "Milnor sphere" will be used also for the standard sphere . There are 28 distinct (non-diffeomorphic) -dimensional Milnor spheres.
The set of all smooth structures on the piecewise-linear sphere is equivalent to the set of elements of the group . The latter group is trivial for , so in the -case any Milnor sphere of dimension less than 7 is diffeomorphic to the standard sphere.
Let be the set of classes of -cobordant -dimensional smooth manifolds which are homotopically equivalent to . The operation of connected sum transforms this set into a group, where the zero is the -cobordism class of . For the elements of are in one-to-one correspondence with the diffeomorphism classes of -dimensional Milnor spheres. To calculate the groups , , one specifies (see ) a trivialization of the stable normal bundle (a framing) of the Milnor sphere . This is possible since is stably parallelizable. The framed manifold obtained defines an element of the stable homotopy group . This element depends, in general, on the choice of the framing ( is a "multi-valued mapping" ). Let be the subgroup in consisting of Milnor spheres that bound parallelizable manifolds. This multi-valued mapping induces a homomorphism , where is the stationary Whitehead homomorphism and is an isomorphism. The calculation of the group reduces to the problem of calculating and (unsolved, 1989), which is done by means of surgery (cf. Morse surgery) of the manifold (preserving the boundary). Let , that is, and is parallelizable. If is a contractible manifold, then after cutting out from a small disc, the manifold is -cobordant to , that is, . If is even, then it is possible to modify by means of surgery so that the new manifold with is contractible (here one requires parallelizability of and ). Thus .
The case . If the index of is , then can be transformed by surgery into a contractible manifold, so that in this case is a standard sphere. If and , then and (here is the connected sum or the boundary connected sum of two manifolds and ). If , then , so that the invariant defines an element . If and , then is divisible by . Conversely, for any there is a smooth closed manifold with ; therefore, if and , then , where is parallelizable and . The element is completely determined by the residue of modulo , and different residues determine different manifolds. Since takes any value divisible by , . E.g., , and , so .
The case . Let . If the Kervaire invariant of is zero, that is, , then can be converted by surgery into a contractible manifold, that is, . Now let . Since for there is no smooth closed almost-parallelizable (which in dimension is equivalent to stably-parallelizable) manifold with Kervaire invariant not equal to zero, is not diffeomorphic to . In this case , that is, . For and those for which there is a manifold with non-zero Kervaire invariant, , that is, , but the question of describing all such has not been solved (1989). However, for the answer is positive. Thus is or .
There is another representation of a Milnor sphere. Let be an algebraic variety in with equation
and let be the -dimensional sphere of (small) radius with centre at the origin. For suitable values of , is a Milnor sphere (see ). For example, for and , , and , all 28 -dimensional Milnor spheres are obtained.
|||J.W. Milnor, "On manifolds homeomorphic to the 7-sphere" Ann. of Math. , 64 (1956) pp. 399–405|
|||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|
|||M.A. Kervaire, J.W. Milnor, "Groups of homotopy spheres" Ann. of Math. , 77 (1963) pp. 504–537|
|||J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)|
|||J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)|
|||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 has different smooth structures (but not for ). A general reference is [a1].
|[a1]||D.S. Freed, K.K. Uhlenbeck, "Instantons and four-manifolds" , Springer (1984)|
Milnor sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Milnor_sphere&oldid=12635